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Universite de Nımes

Laboratoire MIPA

Universite de Nımes, Site des Carmes

Place Gabriel Peri, 30021 Nımes, France

http://mipa.unimes.fr

Convergence of a class of nonlinear

reaction-di↵usion equations and stochastic

homogenization

by

O. Anza Hafsa, J.-P. Mandallena and G. Michaille

September 2016

CONVERGENCE OF A CLASS OF NONLINEAR REACTION-DIFFUSION

EQUATIONS AND STOCHASTIC HOMOGENIZATION

OMAR ANZA HAFSA, JEAN PHILIPPE MANDALLENA, AND GERARD MICHAILLE

Abstract. We establish a convergence theorem for a class of nonlinear reaction-di↵usion equationswhen the di↵usion term is the subdi↵erential of a convex functional of the calculus of variations andthe reaction term is structured in such a way that the dependence on the space-time variable and thestate variable is separate, but cover a wide variety of models related to environment. As a consequencewe prove a homogenization theorem for this class of nonlinear reaction-di↵usion equations under a newstochastic homogenization framework. In particular, the stochastic homogenization of a di↵usive Fisherfood-limited population model with Allee e↵ect is treated.

1. Introduction

A large number of problems in physics, biology, ecology, sociology, economics etc. are modeled bymeans of reaction-di↵usion equations, as for problems arising from chemical physics in order to describechimical substance concentration or temperature distribution, or from population dynamics to describethe density of some species. In the first examples, heat and mass transfer are expressed by the di↵usionterm while the reaction term expresses the rate of heat or mass production. In the second examples,the di↵usion term corresponds to a motion of individuals, spreading out from an area of high concen-tration to an area of low concentration, and the reaction term describes their rate reproduction. Theconnection between the first derivative of the concentration u (t, x), the flux J (t, x) and the reactionterm F (t, u (t, x)) is classically obtained via the conservation of mass (or the number of individuals),and is expressed by

du

dt(t, x) + divxJ (t, x) = F (t, u (t, x)) .

According to Fick’s law (Fourier’s law in the context of thermal conduction), the flux has locally thedirection of the negative spatial gradient and is given in its simpler expression by

J (t, x) = �D (x)ru (t, x) ,

where the coe�cient D (namely the di↵usivity) describes the rate of movement, so that the concentrationu satisfies the reaction-di↵usion equation

du

dt(t, x)� divx (D (x)ru (t, x)) = F (t, u (t, x)) .

For di↵usion models derived from random walk, we refer the reader to [20, 15].Concerning the coe�cient D, the dependence on the spatial variable reflects the fact that the general

direction of movement may take place in heterogeneous media. It also may depend on some small orlarge parameter, as well as the reaction term. From a purely formal point of view, we will denote it byn. For instance, we write n for "n where "n is a small parameter intended to tend toward 0, associatedwith the size of small spatial discontinuities or the size of a mosaic of small habitat in patch dynamic

(Omar Anza Hafsa) UNIVERSITE DE NIMES, Laboratoire MIPA, Site des Carmes, Place Gabriel Peri, 30021

Nımes, France//Laboratoire LMGC, UMR-CNRS 5508, Place Eugene Bataillon, 34095 Montpellier, France.(Jean Philippe Mandallena, Gerard Michaille) UNIVERSITE DE NIMES, Laboratoire MIPA, Site des Carmes,

Place Gabriel Peri, 30021 Nımes, France.E-mail addresses: (Omar Anza Hafsa) [email protected], (Jean Philippe Mandallena, Gerard Michaille)

[email protected], [email protected] Mathematics Subject Classification. 35K57, 35B27, 49J45.Key words and phrases. Convergence of reaction-di↵usion equations, stochastic homogenization, separate variables

reaction functionals.

1

2 CONVERGENCE OF A CLASS OF NONLINEAR REACTION-DIFFUSION EQUATIONS

approach to nonhomogeneous ecosystem. The above equation then becomes

dun

dt(t, x)� divx (Dn (x)run (t, x)) = Fn (t, un (t, x)) ,

with suitable boundary and initial conditions.

In view of these considerations, we are led to consider convergence issues for reaction-di↵usion prob-lems, in a more abstract setting. Given a sequence (Pn)n2N of reaction-di↵usion problems defined inL2

�

0, T, L2 (⌦)�

where ⌦ is a bounded regular domain in RN , and T is any positive real number, thepurpose of this paper is to investigate the convergence of the sequence (un)n2N of solutions of Cauchyproblems of the type

(Pn)

8

>

<

>

:

dun

dt(t, ·) + @�n (un (t, ·)) 3 Fn (t, un (t, ·)) for a.e. t 2 (0, T )

un (0, ·) = u0

n, u0

n 2 dom (@�n).

It should be noted that the boundary condition is implicitly included in the domain of @�n. Indeed, itis well known that for almost every t in (0, T ), the solution un (t) belongs to dom (@�n), which clearlycaptures boundary conditions, even if the initial condition belongs to dom (@�n). For this result, we referthe reader to the section Regularizing e↵ect in [4, Section 17.2], and to the proof of Theorem 2.1. Most ofour study is particularly devoted to the identification of the Cauchy evolution problem (P) which governsthe limit of the sequence (un)n2N in C

�

[0, T ], L2 (⌦)�

. To summarize, we say that (Pn) converges to(P) when (un)n2N uniformly converges in C

�

[0, T ], L2 (⌦)�

to the unique solution of (P). The di↵usionpart of (Pn) is of gradient flow type associated with a sequence (�n)n2N of convex proper and lowersemicontinuous functionals �n : L2 (⌦) ! R [ {+1}. When �n is a standard functional of the calculusof variations with convex integrand Wn (x, ⇠), the di↵usion term takes the form �divx@⇠Wn (·,run). Weimpose a special form on the reaction terms which is structured in such a way that the dependence onthe time-space variable (t, x) and the state variable u is separate (see Definition 3.1). Nevertheless thesereaction functionals give rise to bounded solutions and cover a wide variety of applications as illustratedin Section 3.3 (see examples 3.1, 3.2, 3.3, 3.4). This paper is a first attempt in the convergence andhomogenization study of reaction-di↵usion problems, although aware that it would not cover the morerealistic cases of sequences of time delay reaction-di↵usion equations, time-dependent subdi↵erentialoperator, and those with coupled reaction-di↵usion systems, this will be investigated in forthcomingworks.

With regard to the di↵usion term governed by the subdi↵erentials @�n, the convergence of (Pn)towards (P) is obtained without resorting to the theory of semigroups of contraction and their approx-imation by Trotter or Brezis (see [9] and references therein), which is poorly adjusted to our purposes.Indeed, our main intended application concerns convex potentials �n which are random energies, andthe stochastic homogenization of corresponding reaction-di↵usion problems. The idea is to make use ofthe sequential continuity of the map � 7! @� when the class of convex functionals � is equipped with the�-convergence associated both with strong and weak topology of L2 (⌦), namely, the Mosco-convergence,and the class of subdi↵erential is equipped with the graph convergence (see [4, Theorem 17.4.4]). In thisway, we take advantage of standard results involving �-convergence of the functionals of the calculus ofvariations. Actually, to overcome the di�culty due to the reaction term which is not a subdi↵erentialoperator, we do not use directly the continuity of � 7! @�, but the equivalent assertion which expressesthe bicontinuity of the Fenchel duality transformation, together with the Fenchel extremality condition(see the various references in the proof of Theorem 4.1).

The main result of the paper, Theorem 4.1, is stated and proved in Section 4. Theorem 4.1 is a generalconvergence theorem which can be seen as a first attempt to deal with the convergence of nonlinearreaction-di↵usion problems. Stochastic homogenization of reaction-di↵usion problems is addressed inSection 5 where we set up the basic concepts concerning ergodic dynamical systems. Theorem 5.1, whichis our main homogenization result, establishes a new stochastic homogenization framework in nonlinearreaction-di↵usion problems. As an example, we treat in two situations the stochastic homogenization ofthe reaction-di↵usion problem describing the food-limited population model whose reaction functional isthat of the Fisher model with Allee e↵ect. In the first situation the spatial heterogeneities are distributedfollowing a random patch model, i.e., in the probabilistic setting, the discrete dynamical system is that

CONVERGENCE OF A CLASS OF NONLINEAR REACTION-DIFFUSION EQUATIONS 3

of a random checkerboard-like environment. In the second situation, the discrete dynamical systemdescribes spatial heterogeneities distributed following a Poisson point process.

2. Existence and uniqueness for reaction-diffusion Cauchy problems in Hilbert spaces

In this section, X denotes an Hilbert space equipped with its scalar product denoted by h·, ·i and itsassociated norm k · kX . In all along the paper we use the same notation | · | to denote the norms of theeuclidean spaces Rd, d � 1, and by ⇠ · ⇠0 the standard scalar product of two elements ⇠, ⇠0 in Rd.

Let � : X ! R [ {+1} be a convex proper lower semicontinuous (lsc in short) functional that weassume to be Gateaux di↵erentiable so that its subdi↵erential is single valued. We make this choice inorder to simplify the notation but we could use the subdi↵erential of � in place of its Gateaux derivative,denoted by D�, without additional di�culties. We denote by dom (�) and dom (D�) the domain of �and D� respectively.

On the other hand, we are given a Borel measurable map F : [0,+1) ⇥ X ! X fulfilling the twofollowing conditions:

(C1

) there exists L 2 L2

loc

(0,+1) such that kF (t, u)�F (t, v) kX L (t) ku� vkX for all (u, v) 2 X2

and all t > 0;

(C2

) the map t 7! kF (t, 0) kX belongs to L2

loc

(0,+1).

We sometimes assume that F satisfies the following additional condition for every T > 0:

(C3

) the function L in (C1

) belongs to W 1,1 (0, T ), and for each v 2 C ([0, T ], X) there exists 'v 2L1 (0, T ) such that

kF (t, v (t))� F (s, v (t)) kX ˆ t

s

'v (⌧) d⌧,

for all (s, t) 2 [0, T ]2 with s < t.

The map F is referred as the reaction part, and D� as the di↵usion part of the following Cauchy problem,where T > 0 and u

0

are given in R and dom (D�) respectively:

(P)

8

>

<

>

:

du

dt(t) +D� (u (t)) = F (t, u (t)) for a.e. t 2 (0, T )

u (0) = u0

, u0

2 dom (D�).

We say that u is a solution of (P) if u 2 L2 ([0, T ], X) is absolutely continuous and satisfies (P), wheredudt denotes the distributional derivative of u. In all the paper, the space C ([0, T ], X) is endowed withthe sup-norm.

2.1. Local existence and uniqueness. The results stated in the theorem below are somewhat wellknown. For the sake of completeness we provide a complete proof based on the following Lemma and astandard fixed point procedure (for a proof we refer the reader to [4, Theorems 17.2.5, 17.2.6], or to [9,Theorem 3.7]).

Lemma 2.1. Let T > 0 and X be an Hilbert space. Let � : X ! R [ {+1} be a convex proper lsc

functional, f 2 L2 (0, T,X), and u0

2 dom (@�). Then there exists a unique solution u 2 C ([0, T ], X)of the Cauchy problem

8

>

<

>

:

du

dt(t) +D� (u (t)) = f (t) for a.e. t 2 (0, T )

u (0) = u0

, u0

2 dom (@�)

which satisfies (L1

) and (L2

). If furthermore f 2 W 1,1 (0, T,X), then u (t) 2 dom (D�) for all t 2]0, T ],u admits a right derivative

du+

dt(t) at every t 2 (0, T ) and

du+

dt(t) +D� (u (t)) = f (t).

Theorem 2.1 (local existence and uniqueness). Assume that F satisfies (C1

) and (C2

). Then, thereexists T > 0 small enough which depends only on L, such that (P) admits a unique solution u 2C ([0, T ], X). Moreover, u fulfills the following properties:

(L1

) u (t) 2 dom (D�) for a.e. t 2 (0, T ),


(L2

) u is almost everywhere di↵erentiable in (0, T ) and u0 (t) = dudt (t) for a.e. t 2 (0, T ).

Assume furthermore that F fulfills condition (C3

) for T above, then u satisfies

(L3

) u (t) 2 dom (D�) for all t 2]0, T ],u admits a right derivative

du+

dt(t) at every t 2 (0, T ) and

du+

dt(t) +D� (u (t)) = F (t, u (t)).

Proof. For each u 2 C ([0, T ], X), we denote by ⇤u the solution in C ([0, T ], X) of the Cauchy problem

(Pu)

8

>

<

>

:

d⇤u

dt(t) +D� (⇤u (t)) = F (t, u (t)) for a.e. t 2 (0, T )

⇤u (0) = u0

, u0

2 dom (@�)

whose existence is guaranteed by Lemma 2.1.

Step 1. We show that for T > 0 small enough, ⇤ : C ([0, T ], X) ! C ([0, T ], X) is a contraction. Let(u, v) 2 C ([0, T ], X)⇥ C([0, T ], X). Then, for a.e. t 2 (0, T ) we have

d⇤u

dt(t) +D� (⇤u (t)) = F (t, u (t)) ,

d⇤v

dt(t) +D� (⇤v (t)) = F (t, v (t)) .

By subtracting these two equalities and taking the scalar product with ⇤u (t)� ⇤v (t) in X we obtain⌧

d

dt(⇤u� ⇤v) (t) , (⇤u� ⇤v) (t)

�

+ hD� (⇤u (t))�D� (⇤v (t)) ,⇤u (t)� ⇤v (t)i= hF (t, u (t))� F (t, v (t)) ,⇤u (t)� ⇤v (t)i .

Then, using the fact that D� is a monotone operator, we infer that for a.e. t 2 (0, T )

1

2

d

dtk (⇤u� ⇤v) (t) k2X hF (t, u (t))� F (t, v (t)) ,⇤u (t)� ⇤v (t)i .

Thus, for a.e. t 2 (0, T ),

d

dtk (⇤u (t)� ⇤v (t)) k2X 2kF (t, u (t))� F (t, v (t)) kXk⇤u (t)� ⇤v (t) kX

kF (t, u (t))� F (t, v (t)) k2X + k⇤u (t)� ⇤v (t) k2X L (t)2 ku (t)� v (t) k2X + k⇤u (t)� ⇤v (t) k2X (1)

L2 (t) ku� vk2C([0,T ],X)

+ k⇤u (t)� ⇤v (t) k2X .

By integration over (0, s) for s 2 [0, T ], and taking into account that ⇤u (0) = ⇤v (0) = u0

, we obtain

k⇤u (s)� ⇤v (s) k2X ku� vk2C([0,T ],X)

ˆ s

0

L2 (t) dt+

ˆ s

0

k⇤u (t)� ⇤v (t) k2Xdt.

By using Gronwall’s lemma, we deduce that for1 a.e. s 2 (0, T )

k⇤u (s)� ⇤v (s) k2X ku� vk2C([0,T ],X)

ˆ s

0

L2 (t) dt exp (T ) ,

that is

k⇤u� ⇤vkC([0,T ],X)

ku� vkC([0,T ],X)

ˆ T

0

L2 (t) dt

!

1/2

exp (T/2) ,

Consequently the map ⇤ is a contraction provided that

ˆ T

0

L2 (t) dt

!

1/2

exp (T/2) < 1 which is possible

for T small enough since limT!0

ˆ T

0

L2 (t) dt

!

1/2

exp (T/2) = 0.

1actually for all s 2 [0, T ] since s 7! k⇤u (s)� ⇤v (s) kX is continuous


Step 2. According to the fact that C([0, T ], X) is a complete normed space, the map ⇤ admits a fixedpoint which is a solution of (P) and inherits all the properties of the problem (Pu), in particular (L

1

)and (L

2

).

Step 3. For T obtained above, we establish the uniqueness of the solution of (P) in C ([0, T ], X). Letu and v be two solutions, then, from (1) (with ⇤u = u and ⇤v = v), we infer that for a.e. t 2 (0, T )

d

dtku (t)� v (t)k2X

⇣

L (t)2 + 1⌘

ku (t)� v (t)k2X .

By using Gronwall’s Lemma after integrating over (0, T ), we obtain

ku (t)� v (t)k2X ku (0)� v (0)k2X exp

ˆ T

0

L (t)2 + 1 dt

!

and the claim follows from the fact that u (0) = v (0) = 0.

Step 4. We establish assertion (L3

). According to Lemma 2.1 we have to establish that when themap F fulfills condition (C

3

), the function f : t 7! F (t, u (t)) belongs to W 1,1 (0, T,X). Since X isreflexive, it is equivalent to show that f is absolutely continuous in [0, T ] (see [9, Corollary A2], and [4,Definition 17.2.2]). For any (s, t) 2 (0, T ), s < t, according to (C

1

) and (C3

), we have

kF (t, u (t))� F (s, u (s))kX kF (t, u (t))� F (s, u (t))kX + kF (s, u (t))� F (s, u (s))kXˆ t

s

'u (⌧) d⌧ + L (s) ku (t)� u (s)kX

ˆ t

s

'u (⌧) d⌧ +

L (0) +

ˆ T

0

�

�

�

�

dL

d⌧(⌧)

�

�

�

�

d⌧

!

ku (t)� u (s)kX

But since u is absolutely continuous, we have ku (t)�u (s) kX ˆ t

s

�

�

�

�

du

d⌧(⌧)

�

�

�

�

X

d⌧ so that the inequality

above yields

kF (t, u (t))� F (s, u (s)) kX ˆ t

s

"

'u (⌧) +

L (0) +

ˆ T

0

�

�

�

�

dL

d⌧(⌧)

�

�

�

�

d⌧

!

�

�

�

�

du

d⌧(⌧)

�

�

�

�

X

#

d⌧ (2)

from which we easily deduce that f is absolutely continuous on [0, T ]. ⌅

2.2. Global existence and uniqueness. From now on, we assume that D� (0) = 0. We are going toprove the existence of a global solution of (P). Before that, we establish the following bound on the localsolution.

Lemma 2.2. Let T > 0 small enough so that (P) admits a unique solution u in C ([0, T ], X). Then wehave the following estimate for all t 2 [0, T ]:

ku (t) k2X ✓

ku0

k2X + 2

ˆ t

0

kF (s, 0) k2X ds

◆

exp

✓ˆ t

0

�

2L2 (s) + 1�

ds

◆

.

Proof. Taking u (t) as a test function, for a.e. t 2 (0, T ) we have⌧

du

dt(t) , u (t)

�

+ hD� (u (t)) , u (t)i = hF (t, u (t)) , u (t)i .

Hence, using the fact that hD� (u (t)) , u (t)i � 0 (recall that D� is monotone), we infer that

d

dtku (t) k2X 2 hF (t, u (t)) , u (t)i

(kF (t, 0) kX + L (t) ku (t) kX)2 + ku (t) k2X 2kF (t, 0) k2X +

�

2L2 (t) + 1� ku (t) k2X .

By integrating over (0, s) for 0 s T , we deduce

ku (s) k2X ku0

k2X + 2

ˆ s

0

kF (t, 0) k2Xdt+

ˆ s

0

�

2L2 (t) + 1� ku (t) k2Xdt


(note that t 7! �

2L2 (t) + 1� ku (t) k2X belongs to L1 (0, T ) since ku (t) kX kukC([0,T ],X)

and L 2L2 (0, T )). We obtain the desired estimate by using Gronwall’s lemma. ⌅

Theorem 2.2 (Global existence). Assume that F satisfies condition (C1

), (C2

), and that D� (0) = 0and infX � > �1. Then, there exists a unique solution u 2 C ([0,+1), X) of (P). Moreover, for allT > 0 the restriction of u to [0, T ] satisfies assertions (L

1

) and (L2

). Assume furthermore that F satisfiescondition (C

3

), then for all T > 0 the restriction of u to [0, T ] satisfies assertion (L3

).

Proof. Let T ⇤ > 0 denote a small enough real number given by Theorem 2.1 so that (P) admits aunique solution in C ([0, T ⇤], X). Under the initial condition u

0

2 dom (D�) we are not assured that thecorresponding solution du

dt belongs to L2 (0, T ⇤, X). Neverthelessptdudt 2 L2 (0, T ⇤, X) (see [4, Theorem

17.2.5] or [9, Theorem 3.6]). Hence, for 0 < � < T ⇤, dudt belongs to L2 (�, T ⇤, X). Set

E := {T > � : 9u 2 C ([0, T ], X) solution of (P)} .According to Theorem 2.1, since T ⇤ 2 E, we have E 6= ;. Set T

Max

:= supE (in R+

). In the followingwe denote by u the maximal solution of (P) in C ([0, T

Max

), X) and argue by contradiction by assumingthat T

Max

< +1.

Step 1. We show that limt!TMax

u (t) exists in X.Let T 2 E, then for a.e. t 2 (0, T ) we have

⌧

du

dt(t) ,

du

dt(t)

�

+

⌧

D� (u (t)) ,du

dt(t)

�

=

⌧

F (t, u (t)) ,du

dt(t)

�

.

We know that for a.e. t 2 (�, T ), ddt� (u (t)) =

⌦

D� (u (t)) , dudt (t)

↵

(see [4, Proposition 17.2.5]) so thatby integration on (�, T ) we obtain

ˆ T

�

�

�

�

�

du

dt(t)

�

�

�

�

2

X

dt+ � (u (t))� � (u (�)) ˆ T

0

kF (t, u (t)) k2Xdt

!

12 ˆ T

�

�

�

�

�

du

dt(t)

�

�

�

�

2

X

dt

!

12

(3)

From Lemma 2.2 we haveˆ T

0

kF (t, u (t)) k2Xdt 2

ˆ TMax

0

kF (t, 0) k2Xdt+ 2kuk2C([0,T ],X)

ˆ TMax

0

L2 (t) dt

2

ˆ TMax

0

kF (t, 0) k2Xdt+ C

ˆ TMax

0

L2 (t) dt

where

C :=

ku0

k2X + 2

ˆ TMax

0

kF (s, 0) k2X ds

!

exp

ˆ TMax

0

2L2 (s) + 1 ds

!

.

Therefore, since inf � > �1, (3) yields

ˆ T

�

�

�

�

�

du

dt(t)

�

�

�

�

2

X

dt C (�, �, TMax

)

0

@1 +

ˆ T

�

�

�

�

�

du

dt(t)

�

�

�

�

2

X

dt

!

12

1

A

where the constant C (�, �, TMax

) does not depend on T . Consequently we infer that

ˆ TMax

�

�

�

�

�

du

dt(t)

�

�

�

�

2

X

dt = supT2E

ˆ T

�

�

�

�

�

du

dt(t)

�

�

�

�

2

X

dt < +1. (4)

From (4) we deduce that u : [�, TMax

) ! X is uniformly continuous. Indeed, for s < t in [�, TMax

) we have

ku (t)� u (s) kX ˆ t

s

�

�

�

�

du

dt(⌧)

�

�

�

�

X

d⌧ (t� s)12

ˆ TMax

�

�

�

�

�

du

dt(t)

�

�

�

�

2

X

dt

!

12

so that u is more precisely 1

2

-Holder continuous. Since X is a complete normed space, accordingto the continuous extension principle, u possesses a unique continuous extension u in [�, T

Max

], i.e.,limt!T

Max

u (t) = u (TMax

).


Step 2. (Contradiction) Consider the Cauchy problem

(P 0)

8

>

<

>

:

dv

dt(t) +D� (v (t)) = F (t, v (t)) for a.e. t 2 (0, T )

v (0) = u (TMax

) .

Note that u (TMax

) 2 dom (D�). Indeed, since u (t) 2 dom (D�) for a.e t 2 (0, T ) and u (TMax

) =limt!T

Max

u (t) (choose tn ! TMax

with tn outside the negligible set in which u (t) 62 dom (D�)). Thenaccording to Theorem 2.1, there exists T ⇤⇤ > 0 small enough such that (P 0) admits a solution v 2C ([0, T ⇤⇤], X). Set

eu (t) =

(

u (t) if t 2 [0, TMax

]

v (t� TMax

) if t 2 [TMax

, TMax

+ T ⇤⇤].

Then eu 2 C ([0, TMax

+ T ⇤⇤], X) is a solution of (P), which leads to a contradiction with the maximalityof T

Max

.

Step 3. It remains to establish the uniqueness and to prove that u satisfies conditions (L1

), (L2

)and (L

3

), which is a straightforward consequence of Theorem 2.1. ⌅

3. Existence and uniqueness of bounded solution for a class of reaction-diffusionproblems

From now on ⌦ is a domain of RN of class C1 and LN denotes the Lebesgue measure on RN . Wedenote by @⌦ its boundary and by HN�1

the N�1-dimensional Hausdor↵ measure. To shorten thenotation, we sometimes write X to denote the Hilbert space L2 (⌦) equipped with its standard scalarproduct and its associated norm, denoted by h·, ·iX and k · kX respectively.

3.1. Di↵usion terms associated with convex functionals of the calculus of variations. Fromnow, in all the paper, we focus on the specific case of a standard convex functional � of the calculus ofvariations, i.e., a functional � : L2 (⌦) ! R [ {+1} defined by

� (u) =

8

>

>

<

>

>

:

ˆ⌦

W (x,ru (x)) dx+1

2

ˆ@⌦

a0

u2dHN�1

�ˆ@⌦

hu dHN�1

if u 2 H1 (⌦)

+1 otherwise,

where2 h 2 L2

HN�1(@⌦), a

0

2 L1HN�1

(@⌦) with a0

� 0 HN�1

-a.e. in @⌦ and a0

� � on � ⇢ @⌦ with

HN�1

(�) > 0 for some � > 0, and W : RN ⇥ RN ! R is a measurable function which satisfies thefollowing conditions:

• there exist ↵ > 0 and � > 0 such that for a.e. x 2 RN and every ⇠ 2 RN

↵|⇠|2 W (x, ⇠) ��

1 + |⇠|2� ,• for a.e. x 2 RN , ⇠ 7! W (x, ⇠) is a Gateaux di↵erentiable and convex function (we denote by

D⇠W (x, ·) its Gateaux derivative),• W (x, 0) = D⇠W (x, 0) = 0.

By using the subdi↵erential inequality together with the growth conditions fulfilled by the convexfunction ⇠ 7! W (x, ⇠), it is easy to show that there exist nonnegative constants L (�) and C (�) suchthat, for all (⇠, ⇠0) 2 RN ⇥ RN ,

8

<

:

|W (x, ⇠)�W (x, ⇠0) | L (�) |⇠ � ⇠0| (1 + |⇠|+ |⇠0|) ,

|D⇠W (x, ⇠)| C (�) (1 + |⇠|) .From the second estimate, we infer that if u 2 H1 (⌦), then the function D⇠W (·,ru) belongs to L2 (⌦)N .

Consider the space H (div) := {� 2 L2 (⌦)N : div� 2 L2 (⌦)}. It is well known that when ⌦ is anopen domain of class C1, with outer unit normal ⌘, the normal trace

�⌘ : H (div) \ C�

⌦� ! H� 1

2 (@⌦) \ C (@⌦)

2In the two previous last integral, we still denote by u the trace of u.


defined by �⌘ (�) = (� · ⌘) b@⌦, has a continuous extension from H (div) onto H� 12 (@⌦), still denoted by

�⌘. Moreover, the following Green’s formula holds: for every ' 2 H1 (⌦) whose trace denoted by �0

(')

belongs to H12 (@⌦), we have

ˆ⌦

div�'dx = �ˆ⌦

� ·r' dx+ h�⌘ (�) , �0 (')iH� 1

2(@⌦),H

12(@⌦)

.

In all the paper, as usual, for simplicity of notation, for any � 2 H (div) and any ' 2 H1 (⌦), we(improperly) write,

´@⌦� · ⌘ ' dHN�1

the last term h�⌘ (�) , �0 (')iH� 1

2(@⌦),H

12(@⌦)

, and, as for regular

functions, we denote by � · ⌘ and ' the normal trace and the trace of � and ' respectively. We start byexpliciting the subdi↵erential of the functional � (actually its Gateau derivative), whose domain containsmixed Dirichlet-Neumann boundary conditions.

Lemma 3.1. The subdi↵erential of the functional � is the operator A = @� defined by8

<

:

dom (A) =�

v 2 H1(⌦) : divD⇠W (·,rv) 2 L2 (⌦) , a0

v +D⇠W (·,rv) · ⌘ = h on @⌦

A (v) = �divD⇠W (·,rv) for v 2 dom (A)

where a0

v +D⇠W (·,rv) · ⌘ must be taken in the trace sense.

Proof. The strategy of the proof consists in establishing that A is a maximal monotone operator includedin the subdi↵erential @�, which, in turn, is a maximal monotone operator (for this last point see Theorem17.4.1 in [4]).

Let (u, v) 2 dom (A)2. Note that dom (A) ⇢ H (div). Then, according to the Green formula, to theconvexity of ⇠ 7! W (x, ⇠), and to the boundary condition expressed in dom (A), we infer that

hA (u)�A (v) , u� vi = �ˆ⌦

(divD⇠W (x,ru (x))� divD⇠W (x,rv (x))) (u (x)� v (x)) dx

=

ˆ⌦

(D⇠W (x,ru (x))�D⇠W (x,rv (x))) . (ru (x)�rv (x)) dx

�ˆ@⌦

(D⇠W (x,ru (x))�D⇠W (x,rv (x))) · ⌘ (x) (u (x)� v (x)) dHN�1

=

ˆ⌦


�ˆ@⌦

(h� a0

u� h+ a0

v) (u� v) dHN�1

=

ˆ⌦


+

ˆ@⌦

a0

(u� v)2 dHN�1

� 0,

from which we deduce the monotonicity of A.

Let us establish that A ⇢ @�. Due to the definition of @�, it is enough to prove that for any u 2dom (A) and any v 2 dom (�) = H1 (⌦), the following inequality holds: � (v) � � (u) + hA (u) , v � ui .From convexity of ⇠ 7! W (x, ⇠), for a.e. x 2 RN we have

W (x,rv (x)) � W (x,ru (x)) +D⇠W (x,ru (x)) · (rv (x)�ru (x)) .

By integrating over ⌦ and adding the surface energy1

2

ˆ@⌦

a0

v2dHN�1

�ˆ@⌦

hv dHN�1

, we deduce

� (v) �ˆ⌦

W (x,ru (x)) dx

+

ˆ⌦

D⇠W (x,ru (x)) · (rv (x)�ru (x)) dx+1

2

ˆ@⌦

a0

v2dHN�1

�ˆ@⌦

hv dHN�1

.


Hence, using Green’s formula in the second integral, and D⇠W (x,ru (x)) · ⌘ = h � a0

u on @⌦ (recallthat u 2 dom (A)), we infer that

� (v) �ˆ⌦

W (x,ru (x)) dx+

ˆ⌦

(�divD⇠W (x,ru (x)) (v (x)� u (x))) dx

+

ˆ@⌦

D⇠W (x,ru (x)) · ⌘ (v (x)� u (x)) dHN�1

+1

2

ˆ@⌦

a0

v2dHN�1

�ˆ@⌦

hv dHN�1

=

ˆ⌦

W (x,ru (x)) dx+ hA (u) , v � ui+ˆ@⌦

D⇠W (x,ru (x)) · ⌘ (v (x)� u (x)) dHN�1

+1

2

ˆ@⌦

a0

v2dHN�1

�ˆ@⌦

hv dHN�1

=

ˆ⌦

W (x,ru (x)) dx+

ˆ@⌦

(h� a0

u) (v � u) dHN�1

+1

2

ˆ@⌦

a0

v2dHN�1

�ˆ@⌦

hv dHN�1

+ hA (u) , v � ui

=

ˆ⌦

W (x,ru (x)) dx+

ˆ@⌦

a0

✓

u2 +1

2v2 � uv

◆

dHN�1

�ˆ@⌦

hu dHN�1

+ hA (u) , v � ui� � (u) + hA (u) , v � ui

where we have used inequality u2 + 1

2

v2 � uv � 1

2

u2 in the second integral.

We claim that the operator A is maximal. By Minty’s theorem (see Theorem 17.2.1 in [4] and referencestherein) it remains to prove that R (I +A) = L2 (⌦) where R (I +A) denotes the range of the operator(I +A). Equivalently, for any f in L2 (⌦), we have to establish the existence of a solution u 2 H1 (⌦) ofthe coercive homogeneous mixed Dirichlet-Neumann problem :

8

<

:

u� divD⇠W (·,ru) = f in ⌦

a0

u+D⇠W (·,ru) · ⌘ = h on @⌦

which is a classical result. ⌅Assume that h = 0. Let � be a subset of @⌦ with HN�1

(�) > 0 and extend a0

in [0,+1] in thefollowing way:

a0

(x) =

(

0 if x 2 @⌦ \ �+1 if x 2 �.

Then, the integral´@⌦

a0

u2dHN�1

may be considered as a penalization which forces the function u tobelong to H1

�

(⌦) = {u 2 H1 (⌦) : u = 0 on �}. By convention the functional � becomes

� (u) =

8

>

>

<

>

>

:

ˆ⌦

W (x,ru (x)) dx if u 2 H1

�

(⌦)

+1 otherwise.

The subdi↵erential of � contains now the homogeneous Dirichlet-Neumann boundary conditions as statedin the following lemma which can be proved by an easy adaptation of the proof of Lemma 3.1:

Lemma 3.2. The subdi↵erential of the functional � is the operator A = @� defined by8

<

:

dom (A) =�

v 2 H�

(⌦) : divD⇠W (·,rv) 2 L2 (⌦) , D⇠W (·,rv) · ⌘ = 0 on @⌦ \ �

A (v) = �divD⇠W (·,rv) for v 2 dom (A) .

3.2. The comparison principle. Let us set V := {v 2 H1 (⌦) : divD⇠W (·,rv) 2 L2 (⌦)}, andconsider two functionals F

1

, F2

: [0,+1)⇥ L2 (⌦) ! L2 (⌦) defined by

F1

(t, u) (x) = f1

(t, x, u (x)) , F2

(t, u) (x) = f2

(t, x, u (x))

where f1

, f2

: [0,+1) ⇥ RN ⇥ R ! R are two measurable functions, f2

being Lipschitz continuousuniformly with respect to (t, x), i.e., fulfills the condition |f

2

(t, x, ⇣)�f2

(t, x, ⇣ 0) | L|⇣� ⇣ 0|. Moreover,

we are given two functions u0

and v0

in H1 (⌦) and two functions h1

and h2

in L2

⇣

0, T, L2

HN�1(@⌦)

⌘

.


The following comparison result will be used for proving existence of bounded solutions of reaction-di↵usion problems associated with special reaction functionals (see Subsection 3.3). For similar notionand applications of sub and supersolution related to elliptic boundary valued problems we refer thereader to [6, 7] and for parabolic problems, to [16].

Proposition 3.1. Let T > 0, u 2 C�

[0, T ], L2 (⌦)�

and v 2 C�

[0, T ], L2 (⌦)�

be a subsolution and asupersolution of the reaction-di↵usion problems with respect to the data (u

0

, h1

, F1

) and (v0

, h2

, F2

), i.e.,

P (u0

, h1

, F1

)

8

>

>

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

>

>

:

u (t) 2 V,du

dt(t) 2 L2 (⌦) for a.e. t 2 (0, T ) ,

du

dt(t)� divD⇠W (·,ru (t)) F

1

(t, u (t)) for a.e. t 2 (0, T ) ,

u (0) = u0

2 L2 (⌦) ,

a0

u (t) +D⇠W (·,ru (t)) · ⌘ = h1

(t) on @⌦ for a.e. t 2 (0, T ) ,

P (v0

, h2

, F2

)

8

>

>

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

>

>

:

v (t) 2 V,dv

dt(t) 2 L2 (⌦) for a.e. t 2 (0, T ) ,

dv

dt(t)� divD⇠W (·,rv (t)) � F

2

(t, v (t)) for a.e. t 2 (0, T ) ,

v (0) = v0

2 L2 (⌦) ,

a0

v (t) +D⇠W (·,rv (t)) · ⌘ = h2

(t) on @⌦ for a.e. t 2 (0, T ) .

Then the following comparison principle holds:

u0

v0

in L2 (⌦) ,h1

(t) h2

(t) on @⌦, for a.e. t 2 (0, T ) ,F1

F2

9

=

;

=) u (t) v (t) for all t 2 [0, T ].

Proof. Set w = v � u. We are going to prove that w (t)� = 0 for a.e. t 2 (0, T ). Indeed, for a.e.t 2 (0, T ) we have

dw

dt(t)� [divD⇠W (·,rv (t))� divD⇠W (·,ru (t))] � F

2

(t, v (t))� F1

(t, u (t)) .

Take w (t)� as a test function. By integrating over ⌦, and using Green’s formula we obtain

ˆ⌦

dw

dt(t)w (t)� dx+

ˆ⌦

(D⇠W (x,rv (t))�D⇠W (x,ru (t))) ·rw (t)� dx

�ˆ@⌦

(D⇠W (x,rv (t))�D⇠W (x,ru (t))) · ⌘ w (t)� dHN�1

�ˆ⌦

(f2

(t, x, v (t))� f1

(t, x, u (t)))w (t)� dx.

Noticing that D⇠W (x,ru (t)) · ⌘ = h1

(t) � a0

u (t), and D⇠W (x, v (t)) · ⌘ = h2

(t) � a0

v (t) on @⌦, weinfer that ˆ

⌦

dw

dt(t)w (t)� dx+

ˆ⌦

(D⇠W (x,rv (t))�D⇠W (x, u (t))) ·rw (t)� dx

+

ˆ@⌦

(h1

(t)� h2

(t))w (t)� dHN�1

+

ˆ@⌦

a0

(v � u)w�dHN�1

�ˆ⌦

(f2

(t, x, v (t))� f1

(t, x, u (t)))w (t)� dx,


from which we deduce

�ˆ⌦

dw�

dt(t)w (t)� dx�

ˆ[w(t)0]

(D⇠W (x,rv (t))�D⇠W (x, u (t))) · (rv (t)�ru (t)) dx

�ˆ@⌦

(h2

(t)� h1

(t))w (t)� dHN�1

�ˆ[w(t)0]\@⌦

a0

(v � u)2 dHN�1

�ˆ⌦

(f2

(t, x, v (t))� f1

(t, x, u (t)))w (t)� dx.

Noticing that the three last integrals of the first member are nonnegative, and f1

f2

, we obtain

1

2

d

dt

ˆ⌦

|w (t)� |2dx ˆ⌦

(f2

(t, x, u (t))� f2

(t, x, v (t)))w (t)� dx. (5)

We have used the relations�

dwdt (t)

�

+

= dw+

dt (t),�

dwdt (t)

��= dw�

dt (t), dwdt (t) = dw+

dt (t) � dw�dt (t) and

dw+

dt (t)w (t)� = 0 in the distributional sense. From (5) and the Lipschitz continuity of the function f2

,we deduce that

1

2

d

dt

ˆ⌦

|w (t)� |2dx L

ˆ⌦

|w (t) |w (t)� dx = L

ˆ⌦

|w (t)� |2dx.Integrating this inequality over (0, s) for s 2 [0, T ], we obtainˆ

⌦

|w (s)� |2 dx�ˆ⌦

|w (0)� |2dx L

ˆ s

0

✓ˆ⌦

|w (s)� |2 dx

◆

dt.

According to Gronwall’s lemma we finally obtain that for a.e. s 2 (0, T )ˆ⌦

|w (s)� |2 dx ˆ⌦

|w (0)� |2dx exp (Ls) ,

from which we deduce, since w (0)� = (v0

� u0

)� = 0, that w (s)� = 0 in L2 (⌦) for a.e. s 2 (0, T ),then since s 7! w� (s) is continuous, w� (s) = 0 in L2 (⌦) for all s 2 [0, T ], i.e., u (s) v (s) for alls 2 [0, T ]. ⌅

Let us consider the case

a0

(x) =

(

0 if x 2 @⌦ \ �+1 if x 2 �,

with h = 0, and set eV := {v 2 H1 (⌦) : divD⇠W (·,rv) 2 L2 (⌦) , D⇠W (·,rv) · ⌘ = 0 on @⌦ \�}. Thenan easy adaptation of the previous proof leads to the following comparison principle

Proposition 3.2. Let T > 0, u 2 C�

[0, T ], L2 (⌦)�

and v 2 C�

[0, T ], L2 (⌦)�

be a subsolution and asupersolution of the reaction-di↵usion problems with respect to the data (u

0

, F1

) and (v0

, F2

), i.e.,

P (u0

, h1

, F1

)

8

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

:

u (t) 2 eV ,du

dt(t) 2 L2 (⌦) for a.e. t 2 (0, T ) ,

du

dt(t)� divD⇠W (·,ru (t)) F

1

(t, u (t)) for a.e. t 2 (0, T ) ,

u (0) = u0

2 L2 (⌦) ,

P (v0

, h2

, F2

)

8

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

:

v (t) 2 eV ,dv

dt(t) 2 L2 (⌦) for a.e. t 2 (0, T ) ,

dv

dt(t)� divD⇠W (·,rv (t)) � F

2

(t, v (t)) for a.e. t 2 (0, T ) ,

v (0) = v0

2 L2 (⌦) .

Then the following comparison principle holds:

u0

v0

in L2 (⌦) ,u (t) v (t) on �, for a.e. t 2 (0, T ) ,F1

F2

9

=

;

=) u (t) v (t) for all t 2 [0, T ].


3.3. Reaction functionals with separate variables. The reaction-di↵usion problems modeling awide class of applications for which the mathematical treatment of homogenization (periodic or stochas-tic) is feasible, involve a special class of functionals that we define below.

Definition 3.1. A functional F : [0,+1)⇥L2 (⌦) ! R⌦ is called separate variables reaction functional(SVR-functional in short), if there exists a measurable function f : [0,+1) ⇥ RN ⇥ R ! R such thatfor all t 2 [0,+1) and all v 2 L2 (⌦), F (t, v) (x) = f (t, x, v (x)), and fulfilling the following structureconditions:

f (t, x, ⇣) = r (t, x) · g (⇣) + q (t, x)

with

• g : R ! Rl is a locally Lipschitz continuous function;

• r 2 L1 �

[0,+1)⇥ RN ,Rl�

and er : t 7! r (t, ·) belongs to W 1,1�

0, T, L1 �

RN ,Rl��

for all T > 0;

• q 2 L2

�

0, T, L2

loc

�

RN��

for all T > 0, and eq : t 7! q (t, ·) belongs to W 1,1�

0, T, L2

loc

�

RN��

, i.e.,for all T > 0, and all bounded Borel set B of RN , eq is absolutely continuous from [0, T ] intoL2 (B).

Furthermore f must satisfy the following condition:

(CP) there exist a pair�

f, f�

of functions f, f : [0,+1) ⇥ R ! R with f 0 and f � 0, and a pair�

⇢, ⇢�

in R2 with ⇢ ⇢, such that each of the two following ordinary di↵erential equations

ODE

⇢

y0 (t) = f�

t, y (t)�

for a.e. t 2 [0,+1)y (0) = ⇢

ODE

⇢

y0 (t) = f (t, y (t)) for a.e. t 2 [0,+1)y (0) = ⇢

admits at least one solution denoted by y for ODE and by y for ODE satisfying for a.e. (t, x) 2(0,+1)⇥ R

f�

t, y (t)� f

�

t, x, y (t)�

and f (t, x, y (t)) f (t, y (t)) .

The functional F is referred as a SVR-functional associated with (r, g, q).

We sometimes write r and q to denote er and eq when it causes no ambiguity. Note that, since y and yare nonincreasing and nondecreasing respectively, for any T > 0 we have

y (T ) y (0) = ⇢ ⇢ = y (0) y (T ) .

Remark 3.1. 1) Since the space L1 �

RN ,Rl�

is not reflexive, the space of absolutely continuous

functions from [0, T ] into L1 �

RN ,Rl�

denoted by fW 1,1�

0, T, L1 �

RN ,Rl��

strictly contains thespace W 1,1

�

0, T, L1 �

RN ,Rl��

(see [9, Definition A4, Remark A1]).

2) Apart Section 5, in the definition of SVR-functionals it would be su�cient to consider the absolutecontinuity of eq, taking its values into L2 (⌦). We prefer to introduce the absolute continuity intoL2

loc

�

RN�

because of the specific form of sequences of SVR-functionals F" in the framework ofhomogenization where the scaling x 7! x

" appears.

3) The reason why we introduce condition (CP) may be summarized as follows: even if SVR-functionals do not satisfy the Lipschitz condition (C

1

) invoked in Theorem 2.1, according to thecomparison principle (Proposition 3.1 and Proposition 3.2), we can prove that reaction-di↵usionproblems associated with a SVR-functional admits a solution which satisfies y (T ) u y (T ),see Section 3.4.

Examples 3.1. Let us examine a first class of examples of SVR-functionals for which condition (CP) isreadily checked. Assume that for a.e. (t, x) 2 (0,+1) ⇥ R, f (t, x, 0) � 0 and that there exists ⇢ > 0such that f (t, x, ⇢) 0. Then (CP) is satisfied. Indeed, take f = f = 0 and ⇢ = 0, ⇢ = ⇢. Then y = 0and y = ⇢ are solution of ODE and ODE respectively, and

f�

t, y (t)�

= 0 f (t, x, 0) = f�

t, x, y (t)�

f (t, x, ⇢) = f (t, x, y (t)) 0 = f (t, y (t)) .

For various discussions and references about examples c), d), e) and f) below, we refer the reader to [16].

a) Example derived from food limited population models.


The Fisher logistic growth model. The reaction function is

f (t, x, ⇣) = r (t, x) ⇣

✓

1� ⇣

K

◆

where r (t, x) � 0 andK > 0. The function g defined by g (⇣) = ⇣⇣

1� ⇣K

⌘

is locally Lipschitz continuous.

Moreover, f (t, x, 0) = 0 and f (t, x, ⇢) 0 for ⇢ � K. Therefore the functional F is a SVR-functionalassociated with (r, g, 0).The interpretation of this model is the following:

• u (t, x) is the population density of some specie at time t located at x,

• r (t, x) is the growth rate of the population at time t, located at x,

• K is the carrying capacity, i.e., the capacity of the environment to sustain the population,

• 1

u

du

dtis the per-capita growth rate.

The same conclusion holds for the following extension of the previous logistic function proposed byTurner-Bradley-Kirk

f (t, x, ⇣) = r (t, x) ⇣1+�(1��)

1�✓

⇣

K

◆�!�

where � > 0, � > 0 and � < 1 + 1

� (this last condition ensures that the maximal growth is obtained for

⇣ > 0). For the analysis of this function and various logistic growth models, we refer the reader to [19].

The logistic growth model with immigration (or stocking). The reaction function is

f (t, x, ⇣) = r (t, x) ⇣

✓

1� ⇣

K

◆

+ q (t, x)

The interpretation is that of the logistic growth model where in addition q (t, x) � 0 denotes the immi-gration rate. We have f (t, x, 0) � 0. Assuming that S := sup

(t,x)2[0,+1)⇥RNqr (t, x) < +1, we see that

f (t, x, ⇢) 0 for ⇢ � K1+

p1+

4SK

2

.We will consider the logistic growth model with emigration (or harvesting) in Example 3.4 because it

does not fall into this category.

The Fisher logistic growth model with Allee e↵ect. The reaction function is

f (t, x, ⇣) = r (t, x) ⇣

✓

1� ⇣

K

◆✓

⇣ �A (t, x)

K

◆

where 0 A K. The function f may be written

f (t, x, ⇣) = r (t, x)⇣2

K

✓

1� ⇣

K

◆

� r (t, x)A (t, x)⇣

K

✓

1� ⇣

K

◆

and f (t, x, 0) = 0, f (t, x, ⇢) 0 for ⇢ � K. Therefore the functional F is a SVR-functional associated

with⇣

(ri, gi)i=1,2 , 0⌘

where g1

(⇣) = ⇣2

K

⇣

1� ⇣K

⌘

, g2

(⇣) = ⇣K

⇣

1� ⇣K

⌘

and r1

= r, r2

= �rA.

The interpretation of this model is the same as the one of Fisher model with the additional criticaldensity A below which the per-capita growth rate turns negative. We can also consider the logisticgrowth model with Allee e↵ect and immigration by setting

f (t, x, ⇣) = r (t, x) ⇣

✓

1� ⇣

K

◆✓

⇣ �A (t, x)

K

◆

+ q (t, x)

with the stocking rate q (t, x) � 0. We have f (t, x, 0) � 0 and f (t, x, ⇢) 0 for ⇢ large enough dependingon sup

(t,x)2[0,+1)⇥RNqr (t, x).

b) Example derived from haematopoiesis (Wazewska-Czyziewska & Lasota model). Thereaction function is

f (t, x, ⇣) = �µ (t, x) ⇣ + P (t, x) exp (��⇣)


with µ > 0, P > 0, and � > 0. In this example g1

(⇣) = ⇣, g2

(⇣) = exp (��⇣), r1

= �µ and r2

= P .

Moreover, f (t, x, 0) = 0 and f (t, x, ⇢) 0 for ⇢ � sup(t,x)2[0,+1)⇥RN

P (t,x)µ(t,x) which is assumed to be

finite. Therefore the functional F is a SVR-functional associated with⇣

(ri, gi)i=1,2 , 0⌘

.

The interpretation of this the model is the following:

• u (t, x) is the number of red-blood cell at time t located at x,

• µ (t, x) is the probability of death of red-blood cells, P and � are two coe�cients related to theproduction of red-blood cells per unit time.

For a generalization of this function in the context of delay ordinary di↵erential equations, we refer thereader to [18].

c) Example derived from nuclear reactor dynamics and heat conduction

First model. The reaction function is

f (t, x, ⇣) = r (t, x) ⇣ (a� b⇣) + q (x)

with a > 0, b > 0, and q � 0. In this example i = 1, g1

(⇣) = ⇣ (a� b⇣), r1

= r. Moreover, f (t, x, 0) =q (t, x) � 0 and f (t, x, ⇢) 0 for

⇢ �a+

q

a2 + 4�

qr

�

2b

where⇣q

r

⌘

= sup(t,x)2[0,+1)⇥RN

q (t, x)

r (t, x)which is assumed to be finite. Therefore the functional F is a

SVR-functional associated with ((r1

, g1

) , q).


• u (t, x) is the one velocity neutron flux at time t located at x, i.e., the total path length coveredby all neutrons in one cubic centimeter during one second, of the beam of neutrons traveling in asingle direction. Mathematically, u (t, x) = m (t, x) v (t, x) where m (t, x) is the neutron density�

neutrons/cm3

�

and v (t, x) the neutron velocity (cm/sec).• q (t, x) is an additional source.

d) Example derived from heat transfer: the Stefan’s-Boltzmann fourth-power law in heat

transfer. The reaction function is

f (t, x, ⇣) = r (t, x)�

a4 � ⇣4�

with a > 0 and r > 0. In this example i = 1, g1

(⇣) = a4� ⇣4, r1

= r. We have f (t, x, 0) = r (t, x) a4 > 0and f (t, x, ⇢) 0 for ⇢ � a.


• T is temperature radiated by a black body,

• a is the temperature of surroundings,

• r is related to the radiating area and the emissivity of the radiator.

e) Example derived from chemical reactor and combustion models. The reaction function is

f (t, x, ⇣) = �r (t, x) ⇣p

with p � 1 and r (t, x) � 0, or its generalization f (t, x, ⇣) = � [r1

(t, x) ⇣p1 + r2

(t, x) ⇣p2 ] with p1

� 1,p2

� 1, and r1

(t, x) � 0, r2

(t, x) � 0. In this example, i = 1, 2, gi (⇣) = ⇣pi . We have f (t, x, 0) = 0, andf (t, x, ⇢) 0 for any ⇢ > 0.

The interpretation of this the model is the following for i = 1:

• u (t, x) is the mass concentration of the combustible material at time t, located at x in a non-isothermal reaction,

• r is given according to Arrhenius kinetics by r (t, x) = exp⇣

� � �v(t,x)

⌘

where v (t, x) is the

temperature and � is the Arrhenius number, and p is the order of the reaction.


f) Example derived from enzyme kinetics models in biochemical system. The reactionfunction is

f (t, x, ⇣) = �r (t, x)⇣

1 + a⇣, or f (t, x, ⇣) = �r (t, x)

⇣

1 + a⇣ + b⇣2,

with a > 0, b > 0, and r � 0. In this example, i = 1, g1

(⇣) = ⇣1+a⇣ or ⇣

1+a⇣+b⇣2 that we extend by 0 for

⇣ < 0. We have f (t, x, 0) = 0 and f (t, x, ⇢) 0 for any ⇢ > 0.

The interpretation of this the model is the following for i = 1:

• u (t, x) is the substrate concentration,

• r depends on the total amount of enzyme and various rates of the reaction,

• r depends on various rates of the reaction.

Examples 3.2. We examine now a second class of examples. We assume, as in the previous examples,that for a.e. (t, x) 2 (0,+1) ⇥ R, f (t, x, 0) � 0, but the second condition is no longer satisfied.Nevertheless we assume that there exists a constant M � 0 such that f M . Then (CP) is satisfied.Indeed, take f = 0, ⇢ = 0 as for the previous class of examples, and f = M and ⇢ any positive ⇢. Theny = 0 and y = Mt+ ⇢ are solution of ODE and ODE respectively, with

f�

t, y (t)�

= 0 f (t, x, 0) = f�

t, x, y (t)�

f (t, x,Mt+ ⇢) = f (t, x, y (t)) M = f (t, y (t)) .

Example derived from thermal explosions in the theory of combustion. The reaction func-tion is

f (t, x, ⇣) =

8

<

:

r (t, x) exp

✓

�

✓

1� 1

⇣

◆◆

if ⇣ > 0

0 if ⇣ 0.

In this example i = 1, r1

= r and g1

= f/r. We have f (t, x, 0) = 0 and f M with M =sup

(t,x)2[0,+1)⇥RN r (t, x) exp (�) where sup(t,x)2[0,+1)⇥RN r (t, x) is assumed to be finite.

The interpretation of this model is the following:

• u (t, x) is temperature at time t located at x in thermal explosion,

• � and r are physical coe�cients (see [16] and references therein).

Examples 3.3. We deal with a third class of examples where we still assume that for a.e. (t, x) 2(0,+1) ⇥ R, f (t, x, 0) � 0, but f does not satifies the second condition fulfilled by the two previousclass of examples, but satisfies f (t, x, ⇣) a⇣p for some a > 0 and p � 1. Then (CP) is satisfied. Indeed,take f = 0, ⇢ = 0 as for the previous class of examples, f (t, ⇣) = a⇣p and ⇢ any positive ⇢. Then y = 0is solution of ODE and y defined by

y (t) =

(

⇢ exp (at) when p = 1�

(1� ⇢) at+ ⇢1�p�

11�p when p > 1

is solution of ODE (when p > 1, ODE is the classical Bernouilli o.d.e y0 = ayp) with

f�

t, y (t)�

= 0 f (t, x, 0) = f�

t, x, y (t)�

f (t, x, y (t)) ay (t)p = f (t, y (t)) .

Example derived from nuclear reactor dynamics and heat conduction or from chemical

reactor. The reaction function is

f (t, x, ⇣) = r (t, x) ⇣p

where p � 1. In this example i = 1, r1

= r and g1

= ⇣p. We have f (t, x, 0) = 0 and f r⇣p, wherer = sup

(t,x)2[0,+1)⇥RN r (t, x) is assumed to be finite.


• u (t, x) represents the one velocity neutron flux at time t located at x in case there is a positivetemperature feedback. A second interpretation occurs in the scope of chemical reactor, whereu (t, x), this time, is the concentration of a chemical labile specie (see [16] and references therein).For the case p = 2 see [17].


Examples 3.4. We finally complete our examples by examining a last class for which the first conditionf (t, x, 0) � 0 is not satisfied. We assume that there exists ⇢ > 0 such that for all (t, x) 2 [0,+1)⇥RN ,f (t, x, ⇢) 0 and f (t, x, ⇣) � ⇣ (1� ⇣) � a for some a > 1

4

. Since ⇣ (1� ⇣) � a < 0, it is not assured

that f (t, x, 0) � 0. Nevertheless we claim that condition (CP) is satisfied. Indeed take ⇢ = ⇢, f = 0 asin Examples 3.1. On the other hand, take f (t, ⇣) = ⇣ (1� ⇣)� a and ⇢ any negative number. Then y isthe solution to the ordinary di↵erential equation

ODE

⇢

y0 = y�

1� y�� a

y (0) = ⇢,

and is given by

y (t) =⇢� 1�2⇢��2

2� tan�

�t2

�

1� 1�2⇢

� tan�

�t2

�

,

where � =p4a� 1.

Example derived from food limited population models with emigration (or harvesting, or

extraction). The reaction function is

f (t, x, ⇣) = r (t, x) ⇣

✓

1� ⇣

K

◆

� q (t, x) .

The interpretation is that of the logistic growth model where in addition 0 q (t, x) denotes the em-

igration rate. The change of variable ⇣K = s, and the change of function ef (t, x, s) = 1

r K f (t, x,Ks)

where r (t, x) = inf(t,x)2[0,+1)⇥RN r (t, x) is assume to be positive, leads to ef (t, x, s) � s (1� s) �

sup(t,x)2[0,+1)⇥RN q (t, x). We are in the general situation described above provided that we assume

a := sup(t,x)2[0,+1)⇥RN q (t, x) > r K

4

. For further examples on logistic growth models with migration,we refer the reader to [5].

3.4. Existence and uniqueness of a bounded solution. Combining Theorem 2.2 with the compar-ison principle we can establish the existence of a bounded solution of the Cauchy problem associatedwith SVR-functionals.

Corollary 3.1. Let F be a SVR-functional, with ⇢, ⇢ and y, y given by (CP), and let � be a standardfunctional of the calculus of variations of Section 3.1

� (u) =

8

>

>

<

>

>

:

ˆ⌦

W (x,ru (x)) dx+1

2

ˆ@⌦

a0

u2dHN�1

�ˆ@⌦

hu dHN�1

if u 2 H1 (⌦)

+1 otherwise.

Assume that a0

⇢ h a0

⇢ on @⌦. Then for any T > 0, the Cauchy problem

(P)

8

>

<

>

:

du


u (0) = u0

, ⇢ u0

⇢, u0

2 dom (D�)

admits a unique solution u 2 C�

[0, T ], L2 (⌦)�

satisfying assertions (L2

) and (L3

) of Theorem 2.1. Inparticular for all t 2]0, T ] the solution u (t) satisfies the mixed Dirichlet-Neumann boundary conditiona0

u (t) + D⇠W (·,ru (t)) = h on @⌦. Furthermore u satisfies the following bounds in [0, T ]: y (T ) y (t) u (t) y (t) y (T ).

Proof. The proof of infv2L2

(⌦)

� (v) > �1 is a straightforward.

By definition of SVR-functionals, F : [0,+1) ⇥ L2 (⌦) ! R⌦ is defined for all t 2 [0,+1), allv 2 L2 (⌦), and for a.e. x 2 ⌦ by F (t, v) (x) = f (t, x, v (x)), where for all ⇣ 2 R

f (t, x, ⇣) = r (t, x) · g (⇣) + q (t, x) ,

and where g : R ! Rl is locally Lipschitz continuous. In particular, the restriction of g to the interval[y (T ) , y (T )] is Lipschitz continuous with some lipschitz constant Lg. Consequently ⇣ 7! f (t, x, ⇣) is


Lipschitz continuous with respect to ⇣, uniformly with respect to (t, x), in the interval [y (T ) , y (T )] with

|f (t, x, ⇣)� f (t, x, ⇣ 0) | L|⇣ � ⇣ 0|where L = LgkrkL1

((0,+1)⇥RN ,Rl)

. According to the Mac Shane extension lemma, g may be extended

into a Lipschitz continuous function eg in R, so that the extension ef of f defined by ef (t, x, ⇣) = r (t, x) ·eg (⇣)+q (t, x) is Lipschitz continuous with respect to ⇣ in R, uniformly with respect to (t, x), with the sameLipschitz constant L. Consequently, it is easy to show that the functional eF : [0,+1)⇥L2 (⌦) ! L2 (⌦)

defined by eF (t, v) (x) = ef (t, x, v (x)) fulfills the two conditions (C1

) and (C2

) with L (t) = L. We nextclaim that eF satisfies condition (C

3

).From the Lipschitz continuity of eg we easily deduce that for all ⇣ 2 R |eg (⇣) | C (1 + |⇣|) where C is

a nonnegative constant depending only on Lg and |g (0) |. Thus, for each v 2 C ([0, T ], X), for all s < tin [0, T ], and from the fact that q and r are absolutely continuous, we have 3

k eF (t, v (t))� eF (s, v (t)) kX kq (t)� q (s) kX + LN (⌦)12 kr (t)� r (s) kL1

(⌦,Rl)

keg (v (t)) kX kq (t)� q (s) kX + LN (⌦)

12 C

�

1 + kvkC([0,T ],X)

� kr (t)� r (s) kL1(⌦,Rl

)

ˆ t

s

'T,v (⌧) d⌧ (6)

where the function

⌧ 7! 'T,v (⌧) =

�

�

�

�

dq

dt(⌧)

�

�

�

�

X

+ LN (⌦)12 C

�

1 + kvkC([0,T ],X)

�

�

�

�

�

dr

dt(⌧)

�

�

�

�

L1(⌦,Rl

)

(7)

belongs to L1 (0, T ), which proves the thesis.

Therefore, according to Theorem 2.2, the problem

⇣

eP⌘

8

>

<

>

:

deu

dt(t) +D� (eu (t)) = eF (t, eu (t)) for a.e. t 2 (0, T )

eu (0) = u0

admits a unique solution in C ([0, T ], X) which satisfies (L2

) and (L3

). We are going to prove that eu isactually a solution of (P), which will complete the proof.

From condition (CP), the function y, which does not depend on x, is a subsolution of the reaction-

di↵usion problem with respect to the data⇣

⇢, a0

y (t) , eF⌘

. Indeed since y (t) 2 [y (T ) , y (T )], we have

eF�

t, y (t)�

= F�

t, y (t)�

= f�

t, ·, y (t)� � f�

t, y (t)�

= y0 (t) =dy

dt(t)� divD⇠W

�·,ry (t)�

,

with initial condition y (0) = ⇢, and boundary condition a0

y (t) +D⇠W�

x,ry (t)� · ⌘ = a

0

y (t) on @⌦.

On the other hand eu is a solution of⇣

eP⌘

, hence is a supersolution of the reaction-di↵usion problem

with respect to the data⇣

u0

, h, eF⌘

.

Since ⇢ u0

, and a0

y (t) a0

y (0) = a0

⇢ h, according to the comparison principle, Proposition3.1, we deduce that y (T ) y (t) eu (t) for a.e. t 2 (0, T ). Actually, inequality y (T ) eu (t) holds

for all t 2 [0, T ] (just invoke the continuity of t 7! k �eu (t)� y (T )�� kX). Reasoning similarly with y

which, from (CP), is a supersolution of the reaction-di↵usion problem related to the data⇣

⇢, a0

y (t) , eF⌘

,

we deduce that y (T ) � y (t) � eu (t) for all t 2 [0, T ]. To sum up we have eu (t) 2 [y (T ) , y (T )] for all

t 2 [0, T ]. Therefore eF (t, eu (t)) = F (t, eu (t)) so that eu is solution of (P). The proof of the uniqueness isexactly that of Theorem 2.1. ⌅

An easy adaptation of the proof above, applying this time Proposition 3.2, gives

3for simplicity of notation we write eg (v (t)) to denote the function x 7! eg (v (t, x))


Corollary 3.2. Let F be a SVR-functional, with ⇢, ⇢ and y, y given by (CP), and let � be the functionalof the calculus of variations

� (u) =

8

>

>

<

>

>

:

ˆ⌦


�

(⌦)

+1 otherwise.

Assume that ⇢ 0 ⇢. Then for any T > 0, the Cauchy problem

(P)

8

>

<

>

:

du


u (0) = u0

, ⇢ u0

⇢, u0

2 dom (D�)

admits a unique solution u 2 C�

[0, T ], L2 (⌦)�

satisfying assertions (L2

) and (L3

) of Theorem 2.1.In particular for all t 2]0, T ] the solution u (t) satisfies the homogeneous Dirichlet-Neumann boundaryconditions u = 0 on � and D⇠W (·,ru (t)) = 0 on @⌦. Furthermore u fulfills the following bounds in[0, T ]: y (T ) y (t) u (t) y (t) y (T ).

Remark 3.2. 1) In the proof of Corollary 3.1, we have established that if u is the solution of (P),then, for all t 2 [0, T ], the function F (t, u (t)) belongs to L2 (⌦) since F (t, u (t)) = eF (t, eu (t))for all t 2 [0, T ].

2) In Corollary 3.1, the mixed Dirichlet-Neumann boundary condition fulfilled by the solution u att 2]0, T ], is expressed in condition (L

3

) by the fact that u (t) 2 dom (D�) for all t 2]0, T ], and isgiven by:

a0

u (t) +D⇠W (·,ru (t)) · ⌘ = g on @⌦.

Therefore problem (P) may be written as

(P)

8

>

>

>

>

>

<

>

>

>

>

>

:

du

dt(t)� divD⇠W (·,ru (t)) = F (t, u (t)) for all t 2]0, T ] �equality in L2 (⌦)

�

u (0) = u0

, ⇢ u0

⇢, u0

2 dom (D�),

a0

u (t) +D⇠W (·,ru (t)) · ⌘ = h on @⌦ for all t 2]0, T ].3) As regards Corollary 3.2, the same remark holds, i.e., problem (P) may be written as

(P)

8

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

:

du


�

u (0) = u0

, ⇢ u0

⇢, u0

2 dom (D�),

u (t) = 0 on � for all t 2]0, T ],

D⇠W (·,ru (t)) · ⌘ = 0 on @⌦ \ � for all t 2]0, T ].

4. General convergence theorem for a class of nonlinear reaction-diffusion problems

We are given a sequence (�n)n2N of functionals of the calculus of variations �n : L2 (⌦) ! R[ {+1}defined by

�n (u) =

8

>

>

<

>

>

:

ˆ⌦

Wn (x,ru (x)) dx+1

2

ˆ@⌦

a0,nu

2dHN�1

�ˆ@⌦

hnu dHN�1

if u 2 H1 (⌦)

+1 otherwise

where hn 2 L2

HN�1(@⌦), a

0,n 2 L1HN�1

(@⌦), a0,n � 0 HN�1

-a.e. in @⌦, a0,n � �n on � ⇢ @⌦ with

HN�1

(�) > 0 for some �n > 0, and Wn : RN ⇥ RN ! R is a measurable function which fulfills thefollowing conditions:


(D1

) there exist {↵n}n2N ⇢ R⇤+

and {�n}n2N ⇢ R⇤+

such that for a.e. x 2 RN , all ⇠ 2 RN , and alln 2 N

↵n|⇠|2 Wn (x, ⇠) �n�

1 + |⇠|2� ,(D

2

) for a.e. x 2 RN , ⇠ 7! Wn (x, ⇠) is a di↵erentiable and convex function,

(D3

) Wn (x, 0) = D⇠Wn (x, 0) = 0.

(D4

) (Wn)n2N is uniformly strongly convex, i.e., for some � > 0, for all ⇠ 2 RN , it holds

infn2N

infx2RN

D⇠Wn (x, ⇠) · ⇠ � �|⇠|2.

In the following, we fix T > 0 and we are given a sequence (Fn)n2N of SVR-functionals, each of thembeing associated with (rn, gn, qn), i.e., Fn (t, v) (x) = fn (t, x, v (x)) for all t 2 [0, T ], a.e. x 2 ⌦ and allv 2 L2 (⌦), where

fn (t, x, ⇣) = rn (t, x) · gn (⇣) + qn (t, x) for all (t, x, ⇣) 2 [0,+1)⇥ RN ⇥ R. (8)

We assume that for all n 2 N, gn is a locally Lipschitz function, uniformly with respect to n, i.e., forevery interval I ⇢ R there exists LI � 0 such that for every (⇣, ⇣ 0) 2 R2

supn2N

|gn (⇣)� gn (⇣0)| LI |⇣ � ⇣ 0|. (9)

This condition is fulfilled for example by gn = (gn,i)i=1,...,l where the scalar functions gn,i are convexand satisfy for all ⇣ 2 R, 0 gn,i (⇣) �i (1 + |⇣|pi) for some �i > 0 and pi � 1. This is the case ofExample 3.1 b) where �n > 0 is substitute for � > 0.

We assume that the absolute continuity of the functions ern : t 7! rn (t, ·) and eqn : t 7! qn (t, ·) holdsuniformly with respect to n, i.e.,

supn2N

ˆ T

0

�

�

�

�

drndt

(t, ·)�

�

�

�

L1(⌦,Rl

)

dt < +1,

supn2N

ˆ T

0

�

�

�

�

dqndt

(t, ·)�

�

�

�

L2(⌦)

dt < +1.

(10)

Finally, we assume that

a := infn2N

yn(T ) > �1 and b := sup

n2Nyn (T ) < +1, (11)

and, for all n 2 N,a0,n⇢n hn a

0,n⇢n on @⌦ (12)

where ynand yn are given by condition (CP) fulfilled by each Fn. Recall that these two functions are

solution of suitable o.d.e. with initial condition ⇢nand ⇢n respectively. When considering the case

a0,n (x) =

(

0 if x 2 @⌦ \ �+1 if x 2 �,

and hn = 0, for all n 2 N, then (12) has to be replaced by

⇢n 0 ⇢n for all n 2 N. (13)

In order to establish a convergence result for reaction-di↵usion problems (Pn) with di↵usion part D�n

and reaction part Fn, we take advantage of standard results involving �-convergence of the functionals�n to �, and particularly in homogenization framework (see [4, Subsection 12.4]). More precisely, itis convenient to establish the convergence of the sequence of problems (Pn) under the hypothesis ofthe Mosco-convergence of the sequence (�n)n2N, introduced in [13, 14], i.e., the �-convergence of thefunctionals �n when L2 (⌦) is equipped both with its strong and its weak topology. For the definition andvariational properties of this notion we refer the reader to [4, Section 17.4.2], and for the connection with

Moreau-Yosida approximations we refer to [3, 11]. We will denote by �nM! � the Mosco-convergence

of the sequence (�n)n2N to �. A first important lemma concerns the Mosco-convergence of functionalsdefined in L2 (0, T,X).


Lemma 4.1. Let (X, k · kX) be a reflexive Banach X whose norm together with its dual norm is strictlyconvex, and such that weak convergence of sequences and convergence of their norms imply strong con-vergence. Let ( n)n2N, be a sequence of convex uniformly proper lower semicontinuous functions from

X into R [ {+1} such that nM! and consider ( n)n2N, : L2 (0, T,X) ! R [ {+1} defined by

n (v) :=

ˆ T

0

n (v (t)) dt; (v) :=

ˆ T

0

(v (t)) dt.

Then nM! .

Recall that the sequence ( n)n2N, is said to be uniformly proper if is proper and if there exists abounded sequence (un,0)n2N in X such that supn2N n (un,0) < +1.

Proof. Since ( n)n2N, is uniformly proper, according to [4, Lemma 17.4.5], there exists µ > 0 suchthat n + µ (k · kX + 1) � 0 and + µ (k · kX + 1) � 0 so that the integrals n and are well defined.

Furthermore, for sequences of convex proper and lower semicontinuous functions from a reflexiveBanach X spaces into R [ {+1}, where the norm of X together with its dual norm is strictly convex,and such that weak convergence of sequences and convergence of their norms imply strong convergence,there is equivalence between the Mosco-convergence and convergence of the sequences of the Moreau-Yosida approximations (see [3, Theorem 3.26] or [11]). We are going to apply this result to the spacesX and L2 (0, T,X) which fulfill these conditions and to the functionals n, , n, which are convexproper and lower semicontinuous.

Step 1. Denote by �n,

�, �n, and

� the Moreau-Yosida approximation of index � > 0 of n, , n, and respectively (for the definition and properties of Moreau-Yosida approximation see [4,Proposition 17.2.1]). We establish that for every u 2 L2 (0, T,X):

�n (u) =

ˆ T

0

�n (u (t)) dt and � (u) =

ˆ T

0

� (u (t)) dt.

This result is an elementary case of interchange of infimum and integral and we can apply generalinterchange theorems on the topic (see for instance [2]). In order to make the reading self-contained wegive a direct proof of the second equality, the proof of the first one is similar. We have

� (u) = infv2L2

(0,T,X)

(ˆ T

0

✓

(v (t)) +1

2�kv (t)� u (t) k2X

◆

dt

)

=

ˆ T

0

✓

(J�u (t)) +1

2�kJ�u (t)� u (t) k2X

◆

dt

�ˆ T

0

infw2X

⇢

(w) +1

2�kw � u (t) k2X

�

dt

=

ˆ T

0

� (u (t)) dt.

We have used the fact that the infimum in the Moreau-Yosida approximation � (u) is attained at aunique point denoted by J�u (see [4, Proposition 17.2.1], or [3, Theorem 3.24]).

Conversely ˆ T

0

� (u (t)) dt =

ˆ T

0

infw2X

⇢

(w) +1

2�kw � u (t) k2X

�

dt

=

ˆ T

0

✓

(J� (u (t))) +1

2�kJ� (u (t))� u (t)k2X

◆

dt (14)

where we have used the fact that the infimum in the definition of � (u (t)) is achieved at J� (u (t)) forthe same reason. Assume that we have established that the map t 7! J� (u (t)) belongs to L2 (0, T,X).Then (14) yields

ˆ T

0

� (u (t)) dt � infv2L2

(0,T,X)

(ˆ T

0

✓

(v (t)) +1

2�kv (t)� u (t) k2X

◆

dt

)

:= � (u) ,

which proves the claim.


It remains to prove that t 7! J� (u (t)) belongs to L2 (0, T,X). According to Proposition 17.2.1 of [4],J� : X ! X is non expansive, then continuous. Therefore the measurability of J� � u from [0, T ] intoX follows from the measurability of u from [0, T ] into X. On the other hand, since is proper, thereexists w

0

in X such that (w0

) < +1. We have

1

2�kJ� (u (t))� u (t) k2X � (J� (u (t))) (w

0

) +1

2�kw

0

� u (t) k2X .

from which we easily deduce that

ˆ T

0

kJ� (u (t))k2 dt < +1 since u belongs to L2 (0, T,X).

Step 2. We claim that nM! =) n

M! . We have (see [3, Theorem 3.24])

nM! () 8u 2 X 8� > 0 �

n (u) ! � (u) .

Let u 2 L2 (0, T,X), then from above, for a.e. t 2 (0, T ) and for all � > 0, we have

nM! =) �

n (u (t)) ! � (u (t)) .

Let v be chosen arbitrary in dom ( ). Since nM! , there exists a sequence (vn)n2N in X such that

vn ! v and n (vn) ! (v). Then there exists N 2 N which depends only on (v) and kvkX such thatfor all n � N

�n (u (t)) n (vn) +

1

2�kvn � u (t) k2X (v) + 1 +

1

2�kvn � u (t) k2X

(v) + 1 +1

�kvnk2X +

1

�ku (t) k2X

(v) + 2 +1

�kvk2X +

1

�ku (t) k2X ,

where (v) + 2 + 1

�kvk2X + 1

�ku (·) k2X belongs to L1 (0, T ). Then, according to the Lebesgue dominatedconvergence theorem, we deduce that for all � > 0

ˆ T

0

�n (u (t)) dt !

ˆ T

0

� (u (t)) dt,

that is, from step 1, �n (u) ! � (u), which ends the proof since u is arbitrary chosen in L2 (0, T,X). ⌅

From (L3

)), we know that the right derivative of the solution of gradient flow problems with secondmember in W 1,1 (0, T,X), exists for all t 2 (0, T ). The following lemma provides a sharp estimate ofits norm. This estimate together with the strong convexity ((D

4

)) will be crucial for establishing therelative compactness of the set Et := {un (t) : n 2 N} for each t 2 [0, T ] (step 2 of the proof of Theorem4.1 below).

Lemma 4.2. [9, Theorem 3.7] Let X be an Hilbert space, T > 0, G 2 W 1,1 (0, T,X) and � : X !R [ {+1} be a convex proper lower semicontinuous functional. Let u satisfy

8

>

<

>

:

du

dt(t) + @� (u (t)) 3 G (t) for a.e. t 2 (0, T ) ,

u (0) 2 @� (u).

Then the right derivative of u satisfies for all t 2]0, T [ the following estimate�

�

�

�

du+

dt(t)

�

�

�

�

X

kG (t) kX +1

tdist (u (0) ,K) +

ˆ t

0

�

�

�

�

dG

dt(s)

�

�

�

�

X

s2

t2ds

+

p2

t

✓ˆ t

0

�

�

�

�

dG

dt(s)

�

�

�

�

X

s ds

◆

12✓

dist (u (0) ,K) +

ˆ t

0

kG (s) kX ds

◆

12

where K := {v 2 X : � (v) = 0} and dist (u (0) ,K) = infv2K ku (0)� vk.Here is the main result of Section 4.


Theorem 4.1 (General convergence theorem). Assume that (Wn)n2N satisfies (D1

), (D2

), (D3

), and(D

4

), and that the sequence of SVR-functionals (Fn)n2N, of the form (8), satisfies (9), (10), (11), (12)or (13) when, for all n 2 N, hn = 0 and

a0,n (x) =

(

0 if x 2 @⌦ \ �+1 if x 2 �.

Let un be the unique solution of the Cauchy problem

(Pn)

8

>

<

>

:

dun

dt(t) +D�n (un (t)) = Fn (t, un (t)) for a.e. t 2 (0, T )

un (0) = u0

n, ⇢n u0

n ⇢n, u0

n 2 dom (�n) .

Assume that

(H1

) �nM! � and sup

n2NkhnkL2

HN�1(@⌦)

< +1;

(H2

) supn2N

�n

�

u0

n

�

< +1;

(H3

) u0

n ! u0 strongly in L2 (⌦);

(H4

) gn pointwise converge to g;

(H5

) supn2N

krnkL1([0,+1)⇥RN ,Rl

)

< +1 and there exists r 2 L1 �

[0,+1)⇥ RN ,Rl�

such that rn * r

in L2

�

0, T, L2

�

⌦,Rl��

;

(H6

) for all t 2 [0, T ], supn2N

kqn (t, ·) kL2(⌦)

< +1 and qn * q in L2

�

0, T, L2 (⌦)�

.

Then (un)n2N uniformly converges in C�

[0, T ], L2 (⌦)�

to the unique solution of the problem

(P)

8

>

>

<

>

>

:

du


u (0) = u0, infn2N

yn(T ) u0 sup

n2Nyn (T ) , u

0

2 dom (�)

The reaction functional F : [0,+1) ⇥ L2 (⌦) ! R⌦ is defined, for all t 2 [0, T ], all v 2 L2 (⌦) and fora.e. x 2 ⌦, by

F (t, v) (x) = f (t, x, v (x)) and f (t, x, ⇣) = r (t, x) · g (⇣) + q (t, x) .

Moreover, dundt * du

dt weakly in L2

�

0, T, L2 (⌦)�

and infn2N yn(T ) u supn2N yn (T ).

If furthermore ��

u0

n

� ! ��

u0

�

, rn ! r strongly in L2

�

0, T, L2

�

⌦,Rl��

, and qn ! q strongly in

L2

�

0, T, L2 (⌦)�

, then dundt ! du

dt strongly in L2

�

0, T, L2 (⌦)�

.

Proof. We only establish the proof for � given of the first form, i.e., when the domain of the subdi↵erntialcontains mixed Dirichlet-Neumann boundary conditions. The proof of the second case is slightly shorter,with some easy adaptations. Note that in the statement of Theorem 4.1, we assume that u0

n 2 H1 (⌦) =dom (�n). But dom (�n) ⇢ dom (�n) = dom (D�n) (for this last point we refer to [4, Lemma 17.4.1]),thus u0

n 2 dom (D�n). Therefore, according to Corollary 3.1, (Pn) has a unique solution un whichsatisfies (L

2

) and (L3

) of Theorem 2.1, and the bounds yn(T ) un yn (T ).

Step 1. We establish

supn2N

kunkC(0,T,X)

max (|a|, |b|) (15)

supn2N

�

�

�

�

dun

dt

�

�

�

�

L2(0,T,X)

< +1 (16)

(recall that a := infn2N yn (T ) and b := supn2N yn (T ) which belong to R from (11)).


Inequality (15) follows directly from a yn(T ) un yn (T ) b. Let us establish (16). In what

follows the letter C denotes a constant which can vary from line to line. From (Pn) we deduce that fora.e. t 2 (0, T ),

�

�

�

�

dun

dt(t)

�

�

�

�

2

X

+

⌧

D�n (un (t)) ,dun

dt(t)

�

=

⌧

Fn (t, un (t)) ,dun

dt(t)

�

.

We have used the fact that Fn (t, un (t)) belongs to L2 (⌦) as stated in Remark 3.2. By integrating thisequality over (0, T ), we obtainˆ T

0

�

�

�

�

dun

dt(t)

�

�

�

�

2

X

dt+

ˆ T

0

⌧

D�n (un (t)) ,dun

dt(t)

�

dt =

ˆ T

0

⌧

Fn (t, un (t)) ,dun

dt(t)

�

dt. (17)

But, since u0

n 2 dom (�n), thendundt belongs to L2 (0, T,X) and t 7! �n (un (t)) is absolutely continuous

(see [9, Theorem 3.6]). Therefore for a.e. t 2 (0, T ), ddt� (un (t)) =

⌦

D�un (t) ,dundt (t)

↵

(see [4, Proposi-tion 17.2.5]). Recall that there exists µ > 0 such that �n + µ (k · kX + 1) � 0. Therefore from (17) and(15) we deduceˆ T

0

�

�

�

�

dun

dt(t)

�

�

�

�

2

X

dt = ��n (un (T )) + �n

�

u0

n

�

+

ˆ T

0

⌧

Fn (t, un (t)) ,dun

dt(t)

�

dt (18)

µ (kun (T ) kX + 1)+supn�n

�

u0

n

�

+

ˆ T

0

kFn (t, un (t)) k2Xdt

!

12 ˆ T

0

�

�

�

�

dun

dt(t)

�

�

�

�

2

X

dt

!

12

C + supn�n

�

u0

n

�

+

ˆ T

0

kFn (t, un (t)) k2Xdt

!

12 ˆ T

0

�

�

�

�

dun

dt(t)

�

�

�

�

2

X

dt

!

12

. (19)

By using the structure of the SVR-functional Fn, we have

kFn (t, un (t)) k2X =�

�

�

eFn (t, un (t))�

�

�

2

X 2kFn (t, 0) k2X + 2L2kun (t) k2C([0,T ],X)

(20)

where L = LI supn2N krnkL1((0,+1)⇥RN ,Rl

)

, and LI is the Lipschitz constant of {gn}n2N in the interval

I = [�max (|a|, |b|) ,max (|a|, |b|)] .On the other hand, we have clearly

kFn (t, 0) k2X C�

1 + kqn (t, ·) k2X�

(21)

where, from hypothesis (H4

) and (H5

), C is a nonnegative constant which does not depend on n. Henceˆ T

0

kFn (t, 0) k2Xdt C⇣

1 + kqnk2L2(0,T,X)

⌘

so that, according to hypothesis (H6

)

supn2N

ˆ T

0

kFn (t, 0) k2Xdt < +1. (22)

Combining (15) and (22), (20) yields

supn2N

ˆ T

0

kFn (t, un (t)) k2Xdt < +1,

so that, from (19) and hypothesis (H2

), we infer that

ˆ T

0

�

�

�

�

dun

dt(t)

�

�

�

�

2

X

dt C

0

@1 +

ˆ T

0

�

�

�

�

dun

dt(t)

�

�

�

�

2

X

dt

!

12

1

A

where C is a nonnegative constant which does not depend on n, from which we deduce (16).

Step 2. We prove that there exist u 2 C ([0, T ], X), and a subsequence of (un)n2N not relabeled,satisfying un ! u in C ([0, T ], X).


We apply the Ascoli-Arzela compactness theorem. From (15), (un)n2N is bounded in C ([0, T ], X).Moreover, for s < t, (s, t) 2 [0, T ], we have

kun (t)� un (s) kX ˆ t

s

�

�

�

�

dun

dt(⌧)

�

�

�

�

X

d⌧ (t� s)12

�

�

�

�

dun

dt

�

�

�

�

L2(0,T,X)

(t� s)12 supn2N

�

�

�

�

dun

dt

�

�

�

�

L2(0,T,X)

which, according to (16), proves the equicontinuity of the sequence (un)n 2 N. It remains to establishfor each t 2 [0, T ], the relative compactness in X of the set Et := {un (t) : n 2 N}. For t = 0 thereis nothing to prove because of hypothesis (H

3

) on the initial condition. For t 2]0, T ] this is the mostinvolved point of the proof, that requires the sharp result of Lemma 4.2.

According to Corollary 3.1, un satisfies (H3

), then possesses a right derivative at each t 2]0, T ] (att = T , this is due to the fact that un belongs to C ([0,+1), X) so that the right derivative of un att = T is nothing but the right derivative of the restriction of un to [0, T ]). Moreover,

du+

n

dt(t) +D�n (un (t)) = Fn (t, un (t)) for all t 2]0, T ].

Taking un (t) as a test function, we infer that for all t 2]0, T ]⌧

du+

n

dt(t) , un (t)

�

+ hD�n (un (t)) , un (t)i = hFn (t, un (t)) , un (t)i ,

hence, according to the Green formula and to the fact that un (t) 2 domD�n for all t 2]0, T ],ˆ⌦

D⇠Wn (x,run (t)) ·run (t) dx

=

ˆ@⌦

D⇠Wn (x,r un (t)) · ⌘un (t) dHN�1

�ˆ⌦

du+

n

dt(t)un (t) dx+

ˆ⌦

Fn (t, un (t))un (t) dx

=

ˆ@⌦

(hn � a0,nun (t))un (t) dHN�1

�ˆ⌦

du+

n

dt(t)un (t) dx+

ˆ⌦

Fn (t, un (t))un (t) dx

ˆ@⌦

hnun (t) dHN�1

�ˆ⌦

du+

n

dt(t)un (t) dx+

ˆ⌦

Fn (t, un (t))un (t) dx.

Take 0 < ⌫ < 2�Ctrace

where � is the positive constant of the uniform strong convexity condition (D4

), andC

trace

is the constant of continuity of the trace operator. From (D4

), and from (15), we deduce that forall t 2]0, T ],

�

ˆ⌦

|run (t) |2dx khnkL2HN�1

(@⌦)

kunkL2HN�1

(@⌦)

+ LN (⌦)12 max (|a|, |b|)

✓

�

�

�

�

du+

n

dt(t)

�

�

�

�

X

+ kFn (t, un (t)) kX◆

Ctrace

2⌫khnk2L2

HN�1(@⌦)

+C

trace

⌫

2kunk2H1

(⌦)

+ LN (⌦)12 max (|a|, |b|)

✓

�

�

�

�

du+

n

dt(t)

�

�

�

�

X


Ctrace

2⌫khnk2L2

HN�1(@⌦)

+C

trace

⌫

2

✓ˆ⌦

|run (t) |2dx+ LN (⌦)max (|a|, |b|)◆

+ LN (⌦)12 max (|a|, |b|)

✓

�

�

�

�

du+

n

dt(t)

�

�

�

�

X


.

Hence✓

� � Ctrace

⌫

2

◆ˆ⌦

|run (t) |2dx Ctrace

2⌫supn2N

khnkL2HN�1

(@⌦)

2 + LN (⌦)max (|a|, |b|)

+ LN (⌦)12 max (|a|, |b|)

✓

�

�

�

�

du+

n

dt(t)

�

�

�

�

X


. (23)

For establishing the claim, it su�ces to prove that

supn2N

✓

�

�

�

�

du+

n

dt(t)

�

�

�

�

X


< +1. (24)


Indeed, from (23), (15), and the compactness embedding H1 (⌦) ! L2 (⌦) we will conclude to thecompactness of the set Et for each t 2]0, T ]. For proving (24), it remains to establish

supn2N

kFn (t, un (t)) kX < +1; (25)

supn2N

�

�

�

�

du+

n

dt(t)

�

�

�

�

X

< +1. (26)

Proof of (25). This estimate follows straightforwardly from (15), (20), (21), and hypothesis (H6

).

Proof of (26). Set Gn (t) := Fn (t, un (t)), then, from (2), (6), and (7) we have

kGn (t)�Gn (s) kX ˆ t

s

✓

'T,un (⌧) +

�

�

�

�

dun

d⌧(⌧)

�

�

�

�

X

◆

d⌧ (27)

where

'T,un (⌧) =

�

�

�

�

dqndt

(⌧)

�

�

�

�

X

+ LN (⌦)12 C

�

1 + kunkC([0,T ],X)

�

�

�

�

�

drndt

(⌧)

�

�

�

�

L1(⌦,Rl

)

.

From (10) and (15) we infer that

supn2N

ˆ T

0

'T,un (t) dt < +1.

Consequently (27) yields that the total variation of Gn satisfies

Var (Gn, [0, T ]) =

ˆ T

0

�

�

�

�

Gn

dt(t)

�

�

�

�

X

dt C

1 +

ˆ T

0

�

�

�

�

dun

dt(t)

�

�

�

�

X

dt

!

(28)

where C is a nonnegative constant which does not depend on n. Set Kn := {v 2 X : �n (v) = 0}.Since �n (0) = 0, we have dist

�

u0

n,Kn

� �

�u0

n

�

�

Xso that from hypothesis (H

3

) supn2N dist�

u0

n,Kn

� supn2N

�

�u0

n

�

�

X< +1. Applying Lemma 4.2, and according to (25) and (28), we infer that there exists

a nonnegative constant C (t, T ), which depends only on t and T , such that

�

�

�

�

du+

n

dt(t)

�

�

�

�

X

C (t, T )

2

41 +

ˆ t

0

�

�

�

�

dun

dt(t)

�

�

�

�

X

dt+

ˆ T

0

�

�

�

�

dun

dt(t)

�

�

�

�

X

dt

!

12

3

5

and (26) follows by applying Cauchy-Bunyakovsky-Schwarz inequality and using (16).

Step 3. We assert that dundt * du

dt weakly in L2 (0, T,X) for a non relabeled subsequence, and thata u b. The first claim is a straightforward consequence of (16) and Step 2. The second one followseasily from inequality a un b and un ! u in C ([0, T ], X).

Step 4. We prove that u is the unique solution of (P). From Step 2, there exists u 2 C ([0, T ], X)and a (non relabeled) subsequence such that un ! u in C ([0, T ], X). To simplify the notation, we stillwrite Gn (t) = Fn (t, un (t)) and we use the subsequence obtained in Step 3, that we do not relabel.According to the Fenchel extremality condition (see [4, Proposition 9.5.1]) (Pn) is equivalent to

�n (un (t)) + �⇤n

✓

Gn (t)� dun

dt(t)

◆

+

⌧

dun

dt(t)�Gn (t) , un (t)

�

= 0

for a.e. t 2 (0, T ) (together with the initial condition that we do not write), which is also equivalent toˆ T

0

�n (un (t)) + �⇤n

✓

Gn (t)� dun

dt(t)

◆

+

⌧

dun

dt(t)�Gn (t) , un (t)

��

dt = 0.

Note that equivalence above is due to the Legendre-Fenchel inequality which asserts that inequality�n (un (t)) + �⇤

n

�

Gn (t)� dundt (t)

�

+⌦

dundt (t)�Gn (t) , un (t)

↵ � 0 for a.e. t 2 (0, T ), is always true (see[4, Remark 9.5.1]). Therefore, (Pn) is equivalent toˆ T

0

�n (un (t)) + �⇤✓

Gn (t)� dun

dt(t)

◆

+d

dt

1

2kun (t) k2 � hGn (t) , un (t)i

�

dt = 0,


or, equivalently, toˆ T

0

�n (un (t)) + �⇤n

✓

Gn (t)� dun

dt(t)

◆�

dt+1

2

⇣

kun (T ) k2 ��

�u0

n

�

�

2

⌘

�ˆ T

0

hGn (t) , un (t)i dt = 0.

(29)From hypothesis (H

3

) we have�

�u0

n

�

�

X! ku0kX . (30)

Combining un (T ) = u0

n +

ˆ T

0

dun

dt(t) dt with

dun

dt*

du

dtin L2 (0, T,X), we infer that

kun (T ) k2 ! ku (T ) k2. (31)

We postpone the proof of the following convergence after the end of the proof (see Lemma 4.3)

Gn * G weakly in L2 (0, T,X) (32)

where G (t) = F (t, u (t)) and F (t, u (t)) (x) = r (t, x) · g (u (t, x)) + q (t, x). Passing to the limit in (29),from (30), (31), (32), Step 3, and Lemma 4.1, we obtain4

ˆ T

0

�(u (t)) + �⇤✓

G (t)� du

dt(t)

◆�

dt+1

2

�ku (T ) k2 � ku0k2��ˆ T

0

hG (t) , u (t)i dt 0

or equivalently ˆ T

0

� (u (t)) + �⇤✓

G (t)� du

dt(t)

◆

+

⌧

du

dt(t)�G (t) , u (t)

��

dt 0. (33)

But from the Legendre-Fenchel inequality we have � (u (t))+�⇤ �G (t)� dudt (t)

�

+⌦

dudt (t)�G (t) , u (t)

↵ �0, so that (33) yields that for a.e. t 2 (0, T ), � (u (t)) + �⇤ �G (t)� du

dt (t)�

+⌦

dudt (t)�G (t) , u (t)

↵

= 0which is, according to [4, Proposition 9.5.1], equivalent to

du

dt(t) +D� (u (t)) = G (t) for a.e. t 2 (0, T ) .

We have already proved that a u0

b in Step 3. It remains to establish that u0

2 dom (�). Indeed,from (H

1

) and (H2

), we infer that

� (u0

) lim infn!+1�n

�

u0

n

� supn2N

�n

�

u0

n

�

< +1.

For the proof of uniqueness of

(P)

8

>

<

>

:

du


u (0) = u0, a u0 b, u0 2 H1 (⌦) ,

it is enough to reproduce the proof of uniqueness of Theorem 2.1. Since every subsequence of thesubsequence of (un)n2N obtained above converges to the same limit u in C ([0, T ], X), the sequence

(un)n2N converges to u in C ([0, T ], X). Idem for the sequence�

dundt

�

n2N which converges to dudt weakly

in L2 (0, T,X).

Last step. We show that if ��

u0

n

� ! ��

u0

�

, rn ! r strongly in L2

�

0, T, L2

�

⌦,Rl��

and qn ! q

strongly in L2 (0, T,X), then dundt ! du

dt strongly in L2 (0, T,X). From Step 3 we have dundt * du

dt weakly

in L2 (0, T,X), hence it su�ces to establish that�

�

dundt

�

�

2

L2(0,T,X)

! �

�

dudt

�

�

2

L2(0,T,X)

to prove the claim. By

repeating the proof of Lemma 4.3 under the hypotheses of strong convergence of rn and qn to r and qrespectively, it is easily seen that Gn strongly converges to G in L2 (0, T,X). Therefore, passing to thelimit on (18), i.e., on

ˆ T

0

�

�

�

�

dun

dt(t)

�

�

�

�

2

X

dt = ��n (un (T )) + �n

�

u0

n

�

+

ˆ T

0

⌧

Fn (t, un (t)) ,dun

dt(t)

�

dt,

4From hypotheses (H1

), (H2

) and (H3

), the sequence (�n)n2N, �, is clearly uniformly proper.


and since �nM! �, we deduce that

lim supn!+1

ˆ T

0

�

�

�

�

dun

dt(t)

�

�

�

�

2

X

dt = � lim infn!+1�n (un (T )) + �

�

u0

�

+

ˆ T

0

⌧

F (t, u (t)) ,du

dt(t)

�

dt

�� (u (T )) + ��

u0

�

+

ˆ T

0

⌧

F (t, u (t)) ,du

dt(t)

�

dt

=

ˆ T

0

�

�

�

�

du

dt(t)

�

�

�

�

2

X

dt.

The conclusion follows from the lower semicontinuity of the norm in L2 (0, T,X). ⌅We establish the convergence (32) invoked in Step 4.

Lemma 4.3. The functional Gn = Fn (·, un) weakly converges in L2 (0, T,X) to G defined by G (t) =F (t, u (t)) where F (t, u (t)) (x) = r (t, x) · g (u (t, x)) + q (t, x).

Proof. Recall that Gn (t) = Hn (t) + qn (t) where

Hn (t) (x) = rn (t, x) · gn (un (t, x)) .

Hence, since qn * q in L2 (0, T,X), it remains to prove that Hn * H in L2 (0, T,X) where H (t) (x) =r (t, x) · g (u (t, x)). According to (9) in the interval [�max (|a|, |b|) ,max (|a|, |b|)], we have 5

kgn (un (t))� g (u (t)) kL2(⌦,Rl

)

LIkun (t)� u (t) kX + kgn (u (t))� g (u (t)) kL2(⌦,Rl

)

.

Henceˆ T

0

kgn (un (t))�g (u (t)) k2L2(⌦,Rl

)

dt 2L2

I

ˆ T

0

kun (t)�u (t) k2X dt+

ˆ T

0

kgn (u (t))�g (u (t)) k2L2(⌦,Rl

)

dt.

(34)On the other hand, from (9), and hypothesis (H

4

), we clearly deduce that |gn (⇣) | C (1 + |⇣|) whereC is a nonnegative constant depending only on LI and g (0). Consequently, applying the Lebesguedominated convergence theorem and (H

4

), we infer that

limn!+1

ˆ T

0

kgn (u (t))� g (u (t)) k2L2(⌦,Rl

)

dt = 0.

Passing to the limit in (34) we deduce that gn (un (.)) ! g (u (.)) strongly in L2

�

0, T, L2

�

⌦,Rl��

. Theconclusion of Lemma 4.3 follows from the fact that rn * r weakly in L2

�

0, T, L2

�

⌦,Rl��

. ⌅Remark 4.1. In some cases, we can specify the domain of the limit functional � as follows:

• if ↵n = ↵ for all n 2 N, then dom (�) ⇢ H1 (⌦),

• if lim infn!1 �n < +1, a0,n * a

0

for the �⇣

L1HN�1

(@⌦) , L1

HN�1(@⌦)

⌘

topology, and hn * h

weakly in L2

HN�1(@⌦), then H1 (⌦) ⇢ dom (�).

Let us establish the first assertion. Let v 2 dom (�), then from (H1

), there exists vn ! v strongly inL2 (⌦) such that limn!+1 �n (vn) = � (v) < +1. From the uniform lower growth condition fulfilled by�n, and hypotheses (H

1

) and (H2

), we infer that for any ⌫ > 0,

↵

ˆ⌦

|rvn (t) |2 dx supn2N

�n (vn) + khnkL2HN�1

(@⌦)

kvnkL2HN�1

(@⌦)

supn2N

�n (vn) +C

trace

2⌫khnk2L2

HN�1(@⌦)

+C

trace

⌫

2kvnk2H1

(⌦)

.

Hence choosing ⌫ such that ↵� 1

2

Ctrace

⌫ > 0, we obtain for some constant C > 0,ˆ⌦

|rvn (t)|2 dx C⇣

1 + kvnk2L2(⌦)

⌘

,

and, finally, from (H3

),supn2N

kvnkH1(⌦)

< +1.

5To simplify the notation we write gn (un (t)) for the function x 7! gn (un (t, x)).


Therefore, there exists a subsequence, that we do not relabel, and w 2 H1 (⌦) satisfying vn * w weaklyin H1 (⌦) and strongly in L2 (⌦). Hence v = w 2 H1 (⌦).

For the second assertion, just notice that for v 2 H1 (⌦), according to (H1

) and the growth condition,one has

� (v) lim infn!+1�n (v) lim inf

n!+1 �n

✓

1 +

ˆ⌦

|rv|2 dx

◆

+1

2

ˆ@⌦

a0

v2 dHN�1

�ˆ@⌦

hv dHN�1

< +1.

For each n 2 N consider �n : L2 (⌦) ! R [ {+1} defined by

�n (u) =

8

>

>

<

>

>

:

e�n (u) +1

2

ˆ@⌦

a0,nu

2 dHN�1

�ˆ@⌦

hnu dHN�1

if u 2 H1 (⌦)

+1 otherwise,

where e�n : H1 (⌦) ! R+ is defined by e�n (u) =

ˆ⌦

Wn (x,ru (x)) dx. The following result gives

su�cient conditions for the Mosco-convergence of (�n)n2N when we assume that e�n �-converges to e�with respect to the L2 (⌦) topology.

Proposition 4.1. Assume that

(H01

) • there exist ↵ > 0 and � > 0 such that the sequence (Wn)n2N satisfies (D1

) with ↵n = ↵ and�n = � for all n 2 N;

• e�n �-converges to e� when H1 (⌦) is equipped with the strong convergence of L2 (⌦);• a

0,n ! a0

strongly in L1HN�1

(@⌦);

• hn ! h strongly in L2

HN�1(@⌦).

Then �nM! � where � : L2 (⌦) ! R [ {+1} is given by

� (u) =

8

>

>

<

>

>

:

e� (u) +1

2

ˆ@⌦

a0

u2 dHN�1

�ˆ@⌦

hu dHN�1

if u 2 H1 (⌦)

+1 otherwise.

Proof. The proof fall into two steps.

Step 1. Let vn * v weakly in L2 (⌦), we establish that � (v) lim infn!+1�n (vn).

We assume that lim infn!+1 �n (vn) < +1 and we reason with various subsequences that we do notrelabel, and C denotes various positive constants. From (H0

1

), the uniform lower bound fulfilled by W ,and the continuity of the trace operator, we have, for ⌫ > 0,

↵

ˆ⌦

|rvn|2 dx C + khnkL2HN�1

(@⌦)

kvnkL2HN�1

(@⌦)

C +C

trace

2⌫khnk2L2

HN�1(@⌦)

+C

trace

⌫

2kvnk2H1

(⌦)

.

Hence✓

↵� Ctrace

⌫

2

◆ˆ⌦

|rvn|2 dx C⇣

1 + kvnk2H1(⌦)

⌘

.

Therefore, choosing ⌫ < 2↵Ctrace

, we deduce that

supn2N

kvnkH1(⌦)

< +1.

Consequently, there exist a subsequence and w 2 H1 (⌦) such that vn * w weakly in H1 (⌦) and stronglyin L2 (⌦). Thus w = v so that v 2 H1 (⌦) and vn ! v strongly in L2 (⌦). According to (H0

1

), we inferthat

e� (v) lim infn!+1

e�n (vn) . (35)


On the other handˆ@⌦

a0,nvn dHN�1

=

ˆ@⌦

(a0,n � a

0

) v2ndHN�1

+

ˆ@⌦

a0

v2ndHN�1

� �ka0,n � a

0

kL1HN�1

(@⌦)

supn2N

ˆ@⌦

v2n dHN�1

+

ˆ@⌦

a0

v2n dHN�1

.

According to the weak continuity of the trace operator from H1 (⌦) into L2

HN�1(@⌦) and to the lower

semicontinuity of the map w 7! ´@⌦

a0

w2 dHN�1

, we infer thatˆ@⌦

a0

v2 dHN�1

lim infn!+1

ˆ@⌦

a0,nv

2

n dHN�1

. (36)

Finally, since hn ! h strongly in L2

HN�1(@⌦), and vn * v weakly in L2

HN�1(@⌦), we have

limn!+1

ˆ@⌦

hnvn dHN�1

=

ˆ@⌦

hv dHN�1

. (37)

The proof of the claim is obtained by collecting (35), (36), and (37).

Step 2. Assume that � (v) < +1. We prove that there exists a sequence (vn)n2N strongly convergingto v in L2 (⌦) such that lim supn!+1 �n (vn) � (v).

Since � (v) < +1, vn 2 H1 (⌦), and, according to hypothesis (H01

), there exists a sequence (wn)n2Nin H1 (⌦) strongly converging to v in L2 (⌦), such that

limn!+1

e�n (wn) = � (v) .

By using the well known De Giorgi slicing method, (this is precisely at this point that we use the uniformgrowth condition), we can modify wn into a function vn in H1 (⌦) satisfying vn = v on @⌦ and satisfying

lim supn!+1

e�n (vn) e� (v)

(see proof of [4, Corollary 11.2.1]). Then clearly lim supn!+1 �n (vn) � (v), which proof the claim. ⌅

Proposition 4.1 leads straight to the following corollary of Theorem 4.1 which is applied in Theorem 5.1below.

Corollary 4.1. Under hypotheses of Theorem 4.1 where (H1

) is replaced by (H01

), the same conclusionshold.

Remark 4.2. We can, in some sense, justify our convention consisting in seeing the functional

e� (u) =

8

>

>

<

>

>

:

ˆ⌦


�

(⌦)

+1 otherwise

as a particular case of

� (u) =

8

>

>

<

>

>

:

ˆ⌦

W (x,ru (x)) dx+1

2

ˆ@⌦

a0

u2dHN�1

�ˆ@⌦

hu dHN�1

if u 2 H1 (⌦)

+1 otherwise

with h = 0 and a0

(x) =

(

0 if x 2 @⌦ \ �+1 if x 2 �.

For this we are going to suitably apply Theorem 4.1. Set hn = 0 and a0,n (x) =

(

0 if x 2 @⌦ \ �n if x 2 � .

We have

�n (u) =

8

>

>

<

>

>

:

ˆ⌦

W (x,ru (x)) dx+n

2

ˆ�

u2dHN�1

if u 2 H1 (⌦)

+1 otherwise.


On the other hand, set Fn = F and un0

= u0

, ⇢ u0

⇢, u0

2 dom (D�). Condition a0,n⇢ hn

a0,n⇢ on @⌦ becomes ⇢ 0 ⇢. We claim that �n Mosco converges to e�.

Let vn ! v strongly in L2 (⌦) and assume that lim infn!+1 �n (vn) < +1. In what follows, wereason with various subsequences that we do not relabel. From

supn2N

n2

2

ˆ�

v2ndHN�1

< +1

we infer thatvn ! 0 strongly in L2

HN�1(�) . (38)

On the other hand, from

supn2N

ˆ⌦

W (x,rvn (x)) dx < +1

and the lower bound condition fulfilled by W , we deduce that the sequence (vn)n2N is bounded in H1 (⌦)(recall that vn ! v in L2 (⌦)). Therefore vn * v weakly in H1 (⌦), and, according to the continuity ofthe trace operator from H1 (⌦) into L2

HN�1(@⌦), vn * v weakly in L2

HN�1(�). From (38) we infer that

v = 0 in �, hence v 2 H1

�

(⌦) and e� (v) =

ˆ⌦

W (x,rv (x)) dx. Since for all n 2 N,ˆ⌦

W (x,rv (x)) dx �n (vn)

we deduce that e� (v) lim infn!+1 �n (vn).

Take now v 2 H1

�

(⌦) (otherwise we have nothing to prove), and set vn = v. Since � (v) = �n (v), we

have limn!+1 �n (vn) = e� (v), which proves the claim.

Since all other conditions of Theorem 3 4.1 are fulfilled, we deduce that problem (Pn) with mixedDirichlet-Neumann boundary conditions

(Pn)

8

>

>

>

>

>

<

>

>

>

>

>

:

du


�

u (0) = u0

, ⇢ u0

⇢, u0

2 dom (D�),

n u (t) +D⇠W (·,ru (t)) · ⌘ = 0 on @⌦ for all t 2]0, T ].converges in the sense of Theorem 4.1, to problem (P) with homogeneous Dirichlet-Neumann boundaryconditions

(P)

8

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

:

du


�

u (0) = u0

, ⇢ u0

⇢, u0

2 dom (D�),

u (t) = 0 on � for all t 2]0, T ],

D⇠W (·,ru (t)) · ⌘ = 0 on @⌦ \ � for all t 2]0, T ].5. Application to stochastic homogenization

The behavior of heterogeneous media in physics or mechanics is by now well analyzed from the mathe-matical point of view, through the framework of homogenization. In this context, di↵usion problems withperiodic heterogeneities are now well understood, and di↵usion in random media is fairly well analyzedin [12] and [4, Sections 17.4.4, 17.4.5], where the di↵usion operator is the subdi↵erential of a randomenergy.

By contrast, homogenization of reaction-di↵usion problems does not seem to be adressed. However, inmodeling of biological invasion or for example in food limited models, the interplay between environmentheterogeneities in the individual evolution of propagation species, plays an essential role. Indeed, growthrates, or various thresholds appearing in the models, are mostly influenced by the environment, and varyin each small habitats (forests, marshes, hedges, etc...) which, statistically, are in general homogeneously


distributed. These heterogeneities appear very small compared with the dimension of the domain.Therefore both di↵usion and reaction parts in the problems modeling the propagation, present randomcoe�cients and a small parameter " which account for the dimension of heterogeneities. In order toidentify the e↵ective coe�cients (e↵ective growth rate, various e↵ective thresholds etc...), the purposeof this section is to identify the equivalent homogenized problem when " goes to zero. The procedureconsists in applying Theorem 4.1 in Section 4.

5.1. Probabilistic setting. The notation used here is local to this preamble and should not be confusedwith the notation of the other sections. For any topological space X , we denote by B (X ) its Borel �field, and we return to the basic concepts of [4, Section 12.4.3] (see also references therein) concerningergodic dynamic systems. Let (⌃,A,P) be a probability space. Let (Tz)z2ZN be a group of P-preservingtransformations on ⌃, i.e., for all z 2 ZN , the map Tz : ⌃! ⌃ is A measurable and satisfies Tz#P = P,where we use the standard notation Tz#P to denote the image measure (or push forward) of P by Tz.We denote by F the �-algebra of invariant sets of A by the group (Tz)z2ZN and, for every h in the spaceL1

P (⌃) of P-integrable functions, by EFh the conditional expectation of h with respect to F , i.e., theunique F-measurable function in L1

P (⌃) satisfying for every E 2 FÊ

EFh (!) dP (!) =

Ê

h (!) dP (!) .

If F is made up of sets with probability 0 or 1 then we say that the discrete dynamical system�

⌃,A,P, (Tz)z2ZN

�

is ergodic. Under this condition, we have EFh = Eh where Eh =´⌃

h (!) dP (!) isthe mathematical expectation of h.A su�cient condition to ensure ergodicity is the so called mixing condition which expresses an asymptoticindependence: for all sets E and F of A

lim|z|!+1

P (TzE \ F ) = P (E)P (F ) . (39)

Ergodicity is indeed obtained from (39) by taking E = F in F . In what follows we will also need thefollowing technical standard results.

Invariance and F-measurability. A function h : ⌃ ! R is F-measurable if and only if it is invariantunder the group (Tz)z2Z, i.e., h � Tz = h for all z 2 ZN . For implication

(h is F-measurable =) h is invariant) ,

the claim is indeed the straightforward consequence of

T�1

z h�1 ({h (!)}) = h�1 ({h (!)}) () h (Tz (!)) = h (!) .

The other implication is immediate.

The conditional Lebesgue dominated convergence theorem. Let (hn)n2N be a sequence in LP (⌃) such

that hn ! h, P-a.s. in ⌃, and assume that there exists eh 2 LP (⌃) such that |hn| eh for all n 2 N.Let G be a sub �-algebra of A, then EGhn ! EGh, P-a.s. in ⌃. The proof follows a similar methodas in the proof of the standard Lebesgue dominated convergence theorem, using the conditional FatouLemma instead of the standard Fatou Lemma.

In the two sections below,�

⌃,A,P, (Tz)z2ZN

�

is a given discrete dynamical system.

5.1.1. The random di↵usion part. Given ↵ > 0 and � > 0, we denote by Conv↵,� the class of functionsg : RN ⇥RN ! R, (x, ⇠) 7! g (x, ⇠), satisfying conditions (D

1

), (D2

), (D3

), and (D4

). We equip Conv↵,�with the �-algebra denoted by T

Conv↵,� , trace of the product �-algebra of RRN⇥RN

, i.e., the smallest�-algebra on Conv↵,� such that all the evaluation maps

e(x,⇠) : g 7! g (x, ⇠) , (x, ⇠) 2 RN ⇥ RN

are measurable when R is endowed with its Borel �-algebra.

We consider a random convex integrandW : ⌃⇥RN⇥RN ! R, i.e., a�A⌦ B �

RN�⌦ B �

RN�

,B (R)�

-measurable function such that for every ! 2 ⌃, the function W (!, ·, ·), belongs to the class Conv↵,� .

Since for all (x, ⇠) 2 RN ⇥ RN , ! 7! W (!, x, ⇣) is (A,B (R))-measurable, the map fW : ⌃ ! Conv↵,� ,

! 7! W (!, ·, ·), is �A, TConv↵,�

�

-measurable, and we denote by eP its law, that is eP = fW#P.


We assume that W satisfies the following covariance property with respect to the dynamical system�

⌃,A,P, (Tz)z2ZN

�

: for all z 2 ZN

W (Tz!, x, ⇠) = W (!, x+ z, ⇠) for a.e. x 2 RN , for all ⇠ 2 RN , and for P-a.e. ! 2 ⌃.For all g in Conv↵,� and all z 2 ZN , let us set eTzg (x, ·) = g (x+ z, ·). This defines a group

⇣

eTz

⌘

z2ZN

acting on the class Conv↵,� , and clearly, for all z 2 ZN , eTz : Conv↵,� ! Conv↵,� is TConv↵,� measurable.

Then it is easy to show that the covariance property implies that the law eP of fW is invariant under the

group⇣

eTz

⌘

z2ZN, that is eTz#eP = eP for all z 2 ZN . The random function W is referred as to be periodic

in law.

We write " to denote a sequence ("n)n2N of positive numbers "n going to zero when n ! +1,

and we briefly write " ! 0 instead of limn!+1 "n = 0. Then, the following random functional e�" :⌃⇥ L2 (⌦) �! R+ [ {+1} defined by

e�" (!, u) =

8

>

>

<

>

>

:

ˆ⌦

W⇣

!,x

",ru

⌘

dx if u 2 H1 (⌦)

+1 otherwise.

models a random energy concerning various steady-states situations, where the small parameter " ac-counts for the size of small and randomly distributed heterogeneities in the context of a statisticallyhomogeneous media. The measurability of ! 7! e�" (!, u) for all u 2 H1 (⌦) may be obtained by stan-dard arguments (see for instance [4, Section 12.4.3 and Proposition 12.4.1]).

Under above hypotheses on fW with respect to the discrete dynamical system�

⌃,A,P, (Tz)z2ZN

�

, it isnow standard, using the subadditive ergodic theorem ([12] or [4, Theorem 12.4.3]), that for P-a.s. !

in ⌃ the sequence of functional⇣

e�" (!, ·)⌘

">0

�-converges to the integral functional e�hom (!, ·), e�hom :

⌃⇥ L2 (⌦) �! R+ [ {+1} where

e�hom (!, u) =

8

>

>

<

>

>

:

ˆ⌦

Whom (!,ru) dx if u 2 H1 (⌦)

+1 otherwise,

when L2 (⌦) is equipped with its strong convergence. Let Y denote the unit cell (0, 1)N , then, for everya 2 RN , the density Whom is given, for P-a.s. ! 2 ⌃, by

Whom (!, a) = limn!+1EF inf

⇢

1

nN

ˆnY

W (!, y, a+ru (y)) dy : u 2 H1

0

(nY )

�

= infn2N⇤

EF inf

⇢

1

nN

ˆnY

W (!, y, a+ru (y)) dy : u 2 H1

0

(nY )

�

.

If�

⌃,A,P, (Tz)z2ZN

�

is ergodic, then Whom is deterministic and given for P-a.s. ! 2 ⌃ by

Whom (a) = limn!+1E inf

⇢

1

nN

ˆnY

W (!, y, a+ru (y)) dy : u 2 H1

0

(nY )

�

= infn2N⇤

E inf

⇢

1

nN

ˆnY

W (!, y, a+ru (y)) dy : u 2 H1

0

(nY )

�

.

For a proof we refer the reader to [4, Proposition 12.4.3, Theorem 12.4.7].

Given h 2 L2

HN�1 (@⌦), and a0

2 L1HN�1

(@⌦), a0

� 0, a0

� � on � ⇢ @⌦ with HN�1

(�) > 0, for

some � > 0, we consider the random functionals �" and �hom defined by

�" (!, u) =

8

>

>

<

>

>

:

e�" (!, u) +1

2

ˆ@⌦

a0

u dHN�1

�ˆ@⌦

hu dHN�1

if u 2 H1 (⌦)

+1 otherwise,


and

�hom (!, u) =

8

>

>

<

>

>

:

e�hom (!, u) +1

2

ˆ@⌦

a0

u dHN�1

�ˆ@⌦

hu dHN�1

if u 2 H1 (⌦)

+1 otherwise.

According to Lemma 3.1, for P-a.s. ! 2 ⌃, the subdi↵erential of �" (!, ·) is the operator A" (!) :

L2 (⌦) ! 2L2(⌦) defined for every ! 2 ⌃ by

domA" (!) =n

v 2 H1 (⌦) : divD⇠W⇣

!,.

",rv

⌘

2 L2 (⌦) , a0

v + divD⇠W⇣

!,.

",rv

⌘

· ⌘ = 0 on @⌦o

and, for all v 2 domA" (!),

A" (!) v = �divD⇠W⇣

!,.

",rv

⌘

.

Similarly the subdi↵erential of �hom (!, ·) is the operator Ahom (!) : L2 (⌦) ! 2L2(⌦) defined for every

! 2 ⌃ by

domAhom (!)=�

v 2 H1 (⌦) : divD⇠Whom (!,rv) 2 L2 (⌦) , a

0

v + divD⇠Whom (!,rv) · ⌘ = 0 on @⌦

and, for all v 2 domA (!),

Ahom (!) v = �divD⇠Whom (!,rv) .

When W is ergodic, then Ahom is deterministic and

Ahomv = �divD⇠Whom (rv) .

We emphasize the fact that Ahom (!) is the P-a.s. graph limit of the operator A" (!), and that undera suitable condition on the Fenchel conjugate of ⇠ 7! W (!, x, ⇠), this limit is a P-a.s. pointwise limit.For a proof see [4, Proposition 17.4.6].

5.1.2. The random reaction part. We are given a SVR-random functional , i.e., a functional

F : ⌃⇥ [0,+1)⇥ L2 (⌦) ! ⌦RN

defined by F (!, t, v) (x) = f (!, t, x, v (x)) where

f : ⌃⇥ [0,+1)⇥ RN ⇥ R ! Ris a

�A⌦ B (R)⌦ B �

RN�⌦ B (R) ,B (R)

�

-measurable function such that for P-a.s. ! 2 ⌃, (t, x, ⇣) 7!f (!, t, x, ⇣) is a SVR-function associated with (r (!) , g, q (!, ·)). Furthermore, we make the followingadditional hypotheses on r and q: we assume that for all bounded Borel sets of RN , the real valuedfunctions

! 7! kr (!, t, ·) k2L2(B,Rl

)

for all t 2 [0, T ], (40)

! 7! kq (!, t, ·) k2L2(B)

for all t 2 [0, T ], (41)

! 7!ˆ T

0

�

�

�

�

dq

d⌧(!, ⌧, ·)

�

�

�

�

2

L2(B)

d⌧ (42)

belong to LP (⌃), and we assume that r and q, satisfy the covariance property with respect to thedynamical system

�

⌃,A,P, (Tz)z2ZN

�

, i.e., that for all z 2 ZN , all t 2 [0,+1), a.e. x 2 RN and P-a.s.! 2 ⌃,

r (!, t, x+ z) = r (Tz!, t, x) ,

q (!, t, x+ z) = q (Tz!, t, x) . (43)

We set f" (!, t, x, ⇣) := f�

!, t, x" , ⇣

�

, and define the functional F" by F" (!, t, v) (x) = f�

!, t, x" , v (x)

�

.

Note that in the expression of the condition (CP), the functions f , f , y, y, and the numbers ⇢, ⇢ maydepend on !, and that F" is a SVR-functional whose condition (CP) is exactly that of F , i.e., with thefunctions f , f , y, y, ⇢ and ⇢. Since y and y do not depend on ", condition (11) is automatically satisfied.We assume that condition (13) is satisfied, i.e., ⇢ 0 ⇢. Let us show that condition (10) holds forP-a.s. ! 2 ⌃. This condition is a straightforward consequence of the following more accurate result.


Lemma 5.1. For P-a.s. ! in ⌃, we have

sup"

ˆ T

0

�

�

�

�

dr

dt

⇣

!, t,·"

⌘

�

�

�

�

L1(⌦,Rl

)

dt < +1, (44)

lim sup"!0

ˆ T

0

�

�

�

�

dq

dt

⇣

!, t,·"

⌘

�

�

�

�

L2(⌦)

dt "

TLN (⌦)EFˆ T

0

�

�

�

�

dq

d⌧(!, ⌧, ·)

�

�

�

�

2

L2(Y )

d⌧

#

12

. (45)

Proof. Assertion (44) is straighforward becauseˆ T

0

�

�

�

�

dr

dt

⇣

!, t,·"

⌘

�

�

�

�

L1(⌦,Rl

)

dt =

ˆ T

0

�

�

�

�

dr

dt(!, t, ·)

�

�

�

�

L1( 1"⌦,Rl)

dt ˆ T

0

�

�

�

�

dr

dt(!, t, ·)

�

�

�

�

L1(RN ,Rl

)

dt < +1

since er 2 W 1,1�

0, T, L1 �

RN ,Rl��

.

The proof of (45) is more involved. Consider the set function A from the class Bb

�

RN�

of boundedBorel subsets of RN into the space LP (⌃) of P-integrable real valued functions, defined by

A (B) (·) =ˆ T

0

�

�

�

�

dq

d⌧(·, ⌧, ·)

�

�

�

�

2

L2(B)

d⌧.

From (42), the process A is well defined. Then, for every (A,B) 2 Bb

�

RN�⇥ Bb

�

RN�

with A \B = ;,from additivity of the integral we have

A (A [B) = A (A) +A (B) .

Moreover, from (43) we deduce that

A (z +B) = A (B) � Tz.

Furthermore, A fulfills the following domination property: for all Borel set A included in [0, 1[N ,

A (A) h :=

ˆ T

0

�

�

�

�

dq

d⌧(·, ⌧, ·)

�

�

�

�

2

L2(Y )

d⌧

with h 2 LP (⌃). Therefore, A is an additive process indexed by Bb

�

RN�

, covariant with respect to(Tz)z2ZN (see [1] or [4, Definition 12.4.1] and references therein). According to the additive ergodictheorem [4, Theorem 12.4.1], there exists N 2 A with P (N) = 0 such that for all6 ! 2 ⌃ \N ,

lim"!0

A�

1

"⌦�

LN

�

1

"⌦� = lim

"!0

ˆ T

0

"N

LN (⌦)

�

�

�

�

dq

d⌧(!, ⌧, ·)

�

�

�

�

2

L2( 1"⌦)

d⌧

= EFˆ T

0

�

�

�

�

dq

d⌧(!, ⌧, ·)

�

�

�

�

2

L2(Y )

d⌧.

Hence, after a change of scale,

lim"!0

ˆ T

0

�

�

�

�

dq

d⌧

⇣

!, ⌧,·"

⌘

�

�

�

�

2

L2(⌦)

d⌧ = LN (⌦)EFˆ T

0

�

�

�

�

dq

d⌧(!, ⌧, ·)

�

�

�

�

2

L2(Y )

d⌧. (46)

Finally, from Cauchy-Bunyakovsky-Schwarz inequality we have

ˆ T

0

�

�

�

�

dq

dt

⇣

!, t,·"

⌘

�

�

�

�

L2(⌦)

dt T12

"ˆ T

0

�

�

�

�

dq

d⌧

⇣

!, ⌧,·"

⌘

�

�

�

�

2

L2(⌦)

d⌧

#

12

which combined with (46) gives (45). ⌅

6Strictly speaking the almost sure convergence holds when ⌦ is a convex set. Using approximation of ⌦ by finite unionof convex subset, it is easy to show that the convergence holds for regular ⌦ of class C

1 (see [10, Remark 3.3]).


5.2. General homogenization theorem for a class of nonlinear reaction-di↵usion equations.Given a sequence

�

u0

"

�

"of�A,B �

L2 (⌦)��

-measurable functions u0

" : ⌃! H1 (⌦), by combining Theorem4.1 of the previous section together with the variational convergence of the sequence of random energies�" specified above, we intend to analyze the asymptotic behavior in C

�

0, T, L2 (⌦)�

of the solution u" (!)of the random reaction-di↵usion problem when "! 0:

(P" (!))

8

>

>

>

>

>

>

<

>

>

>

>

>

>

:

du" (!)

dt(t)�divD⇠W

⇣

!,.

",ru" (!) (t)

⌘

=F" (!, t, u" (!) (t)) in L2 (⌦) , for a.e. t 2 (0, T )

u" (!, 0) = u0

" (!) , ⇢ (!) u0

" (!, ·) ⇢ (!) ,

a0

u" (!) (t) + divD⇠W⇣

!,.

",ru" (!) (t)

⌘

· ⌘ = h on @⌦ for all t 2]0, T ].

Theorem 5.1. For each ! 2 ⌃, let us denote by u" (!) the unique solution in C�

[0, T ], L2 (⌦)�

of the

(random) reaction-di↵usion problem (P" (!)). Assume that for P-a.s. ! 2 ⌃, u0

" (!) strongly convergesto u0 (!) in L2 (⌦), and that sup" �"

�

!, u0

" (!)�

< +1. Then, for P-a.s. ! 2 ⌃, u" (!) uniformly

converges in C�

[0, T ], L2 (⌦)�

to the unique solution of the reaction-di↵usion problem

�Phom (!)�

8

>

>

>

>

>

<

>

>

>

>

>

:

du (!)

dt(t)�divD⇠W

hom(!,ru (!) (t))=Fhom (!, t, u (!) (t)) in L2 (⌦) , for a.e. t 2 (0, T )

u (!) (0) = u0 (!) , y (!, T ) u0 (!, ·) y (!, T ) ,

a0

u (!) (t) + divD⇠Whom (!,ru (!) (t)) · ⌘ = h on @⌦ for all t 2]0, T ]

where Fhom is given by Fhom (!, t, v) (x) = fhom (!, t, x, v (x)) with

fhom (!, t, x, ⇣) = r (!, t) · g (⇣) + q (!, t) , r (!, t) = EF ˆ

(0,1)Nr (!, t, y) dy

!

and q (!, t) = EF ˆ

(0,1)Nq (!, t, y) dy

!

.

Moreover, for P-a.s. ! 2 ⌃, du"(!)

dt * du(!)

dt weakly in L2

�

0, T, L2 (⌦)�

and y (!, T ) u (!) y (!, T ).

When the dynamical system�

⌃,A,P, (Tz)z2ZN

�

is ergodic, the initial condition is deterministic, i.e.,

u0

" (!) = u0

" for P-a.s. ! 2 ⌃, together with ⇢, f , ⇢, and f , then�Phom (!) = Phom

�

is deterministicand is given by

�Phom�

8

>

>

>

>

>

<

>

>

>

>

>

:

du

dt(t)� divD⇠W

hom (ru (t)) = Fhom (t, u (t)) in L2 (⌦) , for a.e. t 2 (0, T )

u (0) = u0, y (T ) u0 (·) y (T ) ,

a0

u (t) + divD⇠Whom (ru (t)) · ⌘ = h on @⌦ for all t 2]0, T ]

where Fhom is given by Fhom (t, v) (x) = fhom (t, x, v (x)) with

fhom (t, x, ⇣) = r (t)·g (⇣)+q (t) , r (t) = E

ˆ(0,1)N

r (·, t, y) dy

!

and q (t) = E

ˆ(0,1)N

q (·, t, y) dy

!

.

Moreover, for P-a.s. ! 2 ⌃, du"(!)

dt * dudt weakly in L2

�

0, T, L2 (⌦)�

and y (T ) u y (T ).

Proof. The proof is a straightforward consequence of Theorem 4.1 and consists in checking (H1

), (H5

)and (H

6

).

Proof of (H1

) : �" (!, ·) M! �hom (!, ·). According to [4, Theorem 12.4.7] in the scalar version, we

deduce that for P-a.s. ! in ⌃, the sequence of functional⇣

e�" (!, ·)⌘

">0

�-converges to the random

integral functional e�hom (!, ·) when L2 (⌦) is equipped with its strong convergence. The thesis is then astraightforward consequence of Proposition 4.1.


Proof of (H5

) : For P-a.s. ! 2 ⌃, sup" kr�

!, ·, ·"

� kL1([0,+1)⇥RN ,Rl

)

< +1 and r�

!, ·, ·"

�

* r (!, ·)in L2

�

0, T, L2

�

⌦,Rl��

.The first claim is obvious. For proving assertion r

�

!, ·, ·"

�

* r (!, ·) in L2

�

0, T, L2

�

⌦,Rl��

we need thefollowing lemma

Lemma 5.2. There exists N 2 A with P (N) = 0, such that for every t 2 [0, T ] and every ! 2 ⌃ \N ,

r⇣

!, t,·"

⌘

* r (!, ·) := EF ˆ

(0,1)Nr (!, t, y) dy

!

weakly in L2

�

⌦,Rl�

.

Proof. Fix t 2 [0, T ] \ Q. From (40) we can apply [10, Theorem 4.2]), straightforward consequence ofthe additive ergodic theorem (see [4, Theorem 12.4.1]): there exists Nt 2 A with P (Nt) = 0 such thatfor every ! 2 ⌃ \Nt

r⇣

!, t,·"

⌘

* r (!, t)

weakly in L2

�

⌦,Rl�

. Set N := [t2[0,T ]\Q Nt. We are going to show that for all ! 2 ⌃ \ N , the weakconvergence r

�

!, t, ·"

�

* r (!, t), holds for all t 2 [0, T ]. Let ! 2 ⌃ \ N , ' 2 L2

�

⌦,Rl�

, t 2 [0, T ] and(tn)n2N be a sequence in [0, T ] \Q converging to t with tn t. We have

D

r⇣

!, t,·"

⌘

,'E

L2(⌦,Rl

)

=D

r⇣

!, tn,·"

⌘

,'E

L2(⌦,Rl

)

+D

r⇣

!, t,·"

⌘

� r⇣

!, tn,·"

⌘

,'E

L2(⌦,Rl

)

, (47)

with, from the weak convergence above, lim"!0

⌦

r�

!, tn,·"

�

,'↵

L2(⌦,Rl

)

= hr (!, tn) ,'iL2(⌦,Rl

)

. Let us

set R" (!, t, tn) :=⌦

r�

!, t, ·"

�� r�

!, tn,·"

�

,'↵

L2(⌦,Rl

)

. By the absolute continuity of ⌧ 7! r (!, ⌧, ·) from[0, T ] into L1 �

⌦,Rl�

, we infer that

|R" (!, t, tn) | LN (⌦)12

�

�

�

r⇣

!, t,·"

⌘

� r⇣

!, tn,·"

⌘

�

�

�

L1(⌦,Rl

)

k'kL2(⌦,Rl

)

LN (⌦)12 k'kL2

(⌦,Rl)

ˆ t

tn

�

�

�

�

dr

d⌧

⇣

!, ⌧,·"

⌘

�

�

�

�

L1(⌦,Rl

)

d⌧

LN (⌦)12 k'kL2

(⌦,Rl)

ˆ t

tn

�

�

�

�

dr

d⌧(!, ⌧, ·)

�

�

�

�

L1(RN ,Rl

)

d⌧. (48)

Let "! 0, then n ! +1 in (47). From (48) we deduce that

lim"

D

r⇣

!, t,·"

⌘

,'E

L2(⌦,Rl

)

= limn!+1 hr (!, tn) ,'iL2

(⌦,Rl)

= hr (!, t) ,'iL2(⌦,Rl

)

which ends the proof provided that we justify the convergence

limn!+1 hr (!, tn) ,'iL2

(⌦,Rl)

= hr (!, t) ,'iL2(⌦,Rl

)

,

which is a straightforward consequence of the continuity of t 7! Ýr (!, t, y) dy and the conditional

Lebesgue dominated convergence theorem. ⌅Proof of (H

5

) continued. Fix ! 2 ⌃ \N . Let ' 2 L2

�

0, T, L2

�

⌦,Rl��

. According to Lemma 5.2,for all t 2 [0, T ] we have

D

r⇣

!, t,·"

⌘

,' (t)E

L2(⌦,Rl

)

! hr (!, t) ,' (t)iL2(⌦,Rl

)

,

�

�

�

�

D

r⇣

!, t,·"

⌘

,' (t)E

L2(⌦,Rl

)

�

�

�

�

LN (⌦)12 kr (!, ·, ·) kL1

(0,+1)⇥RN k' (t) kL2(⌦,Rl

)

. (49)

The conclusion follows from the Lebesgue dominated convergence theorem.

Proof of (H6

). First, we must prove that for P-a.s. ! 2 ⌃, sup" kq�

!, t, ·"

� kL2(⌦)

< +1. Forthis, reproducing the proof of Lemma 5.1, and using the additive ergodic theorem for the process B 7!kq (!, t, ·) k2L2

(B)

, which is well defined according to (41), we easily obtain that for P-a.s. ! 2 ⌃,

lim"!0

�

�

�

q⇣

!, t,·"

⌘

�

�

�

2

L2(⌦)

= LN (⌦)EFkq (!, t, ·) k2L2(Y )

, (50)


from which we deduce that sup" kq�

!, t, ·"

� kL2(⌦)

< +1 for P-a.s. ! 2 ⌃.The rest of the proof concerning the weak convergence is exactly that of condition iv) except the very

last point, i.e., the domination condition (49), which must be replaced by�

�

�

D

q⇣

!, t,·"

⌘

,' (t)E

X

�

�

�

�

�

�

q⇣

!, t,·"

⌘

�

�

�

Xk' (t)kX

1

2

�

�

�

q⇣

!, t,·"

⌘

�

�

�

2

X+

1

2k' (t)k2X

and using the fact that for P-a.s. ! 2 ⌃, kq �!, t, ·"

� k2X is bounded by a function which belongs to

L1 (0, T ). Indeed, from (50), for " small enough we have kq �!, t, ·"

� k2X EF kq (!, t, ·)k2L2(Y )

+ 1 for

P-a.s. ! 2 ⌃, and it remains to show that t 7! EFkq (!, t, ·) k2L2(Y )

belongs to L1 (0, T ) which is astraightforward consequence of Lemma 5.3 below. ⌅

Lemma 5.3. let a : ⌃⇥ [0, T ] ! R+ be a A⌦B ([0, T ])-measurable function such that for P-a.s. ! 2 ⌃,t 7! a (!, t) belongs to L1 (0, T ), and ! 7! ´ T

0

a (!, t) dt belongs to LP (⌃). Then

EFˆ T

0

a (!, t) dt =

ˆ T

0

EFa (!, t) dt

Proof. Let E 2 F . Then, fromÊ

EFa (!, t) dP (!) =

Ê

a (!, t) dP (!)

we deduce (a priori equality in [0,+1])ˆ T

0

Ê

EFa (!, t) dP (!) dt =

ˆ T

0

Ê

a (!, t) dP (!) dt,

thus, from Fubini-Tonelli’s theoremÊ

ˆ T

0

EFa (!, t) dtdP (!) =

Ê

ˆ T

0

a (!, t) dtdP (!) ,

equality in R+. Furthermore, the function ! 7! ´ T0

EFa (!, t) dt is F-measurable because invariantunder the group (Tz)z2ZN . Then from definition of the conditional expectation with respect to F we

have EF ´ T0

a (!, t) dt =´ T0

EFa (!, t) dt. ⌅

5.3. Examples of stochastic homogenization of a di↵usive Fisher food limited populationmodel with Allee e↵ect. As an eloquent application, we treat the stochastic homogenization of thereaction-di↵usion problem describing the food limited population model whose reaction function is thatof the Fisher model with Allee e↵ect (see Examples 3.1). We assume that the growth rate r, along withthe critical threshold A below which the per-capita growth rate turns negative, are influenced by theheterogeneities of the spatial environment and change in each small habitats, but we assume that thecarrying capacity K is constant. In a first example, we assume that the heterogeneities are distributedfollowing a regular random patch model, i.e., in the probabilistic setting, the dynamical system is thatof a random checkerboard-like environment. In a second example, the heterogeneities are distributedfollowing a Poisson point process. In the two examples, in order to simplify the model, we assume thatr and A do not depend on the time variable t. Otherwise, it would be su�cient to make the appropriateassumptions concerning the absolute continuity on r and A with respect to the time variable, withoutchanging the constructions below.

5.3.1. Random checkerboard-like environment. Given two triples (r�, A�,W�) and (r+, A+,W+) in[0,+1]⇥ [0,K]⇥Conv↵,� where W�, W+ do not depend on x, and ✓ 2 [0, 1], we consider the product

⌃ = {(r�, A�,W�) , (r+, A+,W+)}ZN

equipped with the �-algebra A, product of the trivial �-algebraof subsets of {(r�, A�,W�) , (r+, A+,W+)}. Each element of ⌃ is then of the form (!z)z2ZN , with!z =

�

!1

z ,!2

z ,!3

z

�

, where !1

z 2 {r�, r+}, !2

z 2 {A�, A+}, and !3

z 2 {W�,W+}.We equip (⌃,A) with the product probability measure P = ⌦z2ZNµz where µz = ✓�

(r�,A�,W�)

+(1� ✓) �

(r+,A+,W+)

for all z 2 ZN . By construction P is invariant under the shift group (Tz)z2ZN


defined by Tz (!t)t2ZN = (!t+z)t2ZN , i.e., Tz#P = P for all z 2 ZN . We set

r (!, x) := !1

z and A (!, x) = !2

z , W (!, x, ·) = !3

z whenever x 2 Y + z,

and f (!, t, x, ⇣) = r (!, x) ⇣⇣

1� ⇣K

⌘⇣

⇣�A(!,x)K

⌘

which define a random SVR-function provided that we

write it

f (!, t, x, ⇣) = r (!, x)⇣2

K

✓

1� ⇣

K

◆

� r (!, x)A (!, x)⇣

K

✓

1� ⇣

K

◆

.

According to this definition it is straightforward to show that f is a random SVR-function, that for all! 2 ⌃, all x 2 RN and all ⇠ 2 RN , f (!, t, x+ z, ·) = f (Tz!, t, x, ⇠) and that conditions (40) and (41)hold. Regarding the random densityW , one can easily show that it verifyW (!, x+ z, ⇠) = W (Tz!, x, ⇠).

Furthermore, it is easily seen that�

⌃,A,P, (Tz)z2ZN

�

is ergodic since its satisfies the mixing condition(39) (notice that (39) is satisfied with the cylinders which generate A).

The random SVR-function f" defined by f" (!, t, x, ⇣) = f�

!, t, x" , ⇣

�

may be seen as the Fisherreaction function defined in a checkerboard-like spatial environment, i.e., the growth rate r and thethreshold A take two values at random on the lattice spanned by the unit cell Y = (0, ")2 modeling amosaic of two kinds of small habitats. The di↵usion is associated with a random density W" defined byW" (!, x, ⇠) = W

�

!, x" , ⇠

�

taking also two values at random on this lattice. The triples (r�, A�,W�)and (r+, A+,W+) represent a sample of two kinds of habitat whose probability of occurring is a and brespectively. Obviously we can easily generalize this model with r, A and W taking countable values.

The reaction-di↵usion problem modeling the evolution of the density u of some specie during a timeT > 0, in a C1-regular domain ⌦ included in a random checkerboard-like environment, is, when no specieis located on the boundary, and when the density at time t = 0 is regular and known equal to u0

" , givenby

8

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

:

du"

dt(!, ·)� div

⇣

D⇠W⇣

!,x

",ru" (!, ·)

⌘⌘

=

r⇣

!,x

"

⌘

u(!, ·)✓

1� u (!, ·)K

◆

u (!, ·)�A�

!, x"

�

K

!

a.e. in (0, T )⇥ ⌦u" (!, 0, x) = u0

", 0 u0

" K,

u" (!, t, x) = 0 for (t, x) 2 [0, T ]⇥ @⌦.

Assuming that the initial density u0

" strongly converges to some u0

in L2 (⌦), according to Theorem 5.1,we can say that when " is very small compared to the size of the domain ⌦, a deterministic model, wellaware with the evolution specie, is given by

8

>

>

>

>

>

<

>

>

>

>

>

:

du

dt� div

�

D⇠Whom (ru)

�

= ru⇣

1� u

K

⌘

u� rAr

K

!

for a.e. (t, x) 2 (0, T )⇥ ⌦u (0, x) = u

0

, 0 u0

" K,

u (t, x) = 0 for (t, x) 2 [0, T ]⇥ @⌦,

where

Whom (a) = infn2N⇤

E inf

⇢

1

n2

ˆnY

W (!, y, a+ru (y)) dy : u 2 H1

0

(Y )

�

,

r = E

✓ˆY

r (!, y) dy

◆

=✓r� + (1� ✓) r+, rA = E

✓ˆY

r (!, y)A (!, y) dy

◆

=✓r�A� + (1� ✓) r+A+.

Everything happens as if the density evolution of the specie took place in a homogeneous environmentfollowing a Fisher di↵usive model with Allee e↵ect and constant coe�cients. Concerning the solution u,it is interesting to note that the growth rate is deterministic and constant in the environment, and that

the critical density eA = rAr which still satisfies 0 eA K, is now a function of the growth rate. This

illustrates the interplay between the environment and the evolution. It is also interesting to note that eAis a monotone function of the probability ✓. The di↵usion operator is now governed by an homogeneousand deterministic operator obtained as an almost sure graph limit.


5.3.2. Environment whose heterogeneities are independently randomly distributed with a frequency �. Asa first step, we are going to define a discrete dynamical system

�

⌃,A,P, (Tz)z2ZN

�

, modeling the envi-ronment whose heterogeneities are spheres and whose centers are randomly distributed with a frequency� per unit area. We assume that the number of centers is locally finite and that the numbers of centersin two disjointed regions are two independent random variables. The growth rate and the thresholdin the Fisher reaction function with Allee e↵ect, together with the density associated with the randomdi↵usion, must take di↵erent values outside or inside the heterogeneities.

Denote by M the set of countable and locally finite sums of Dirac measures in R2, equipped withthe �-algebra generated by all the evaluation maps EB : m 7! m (B) from M into N [ {+1} when Bbelongs to B �

R2

�

. Then, given � > 0, there exists a subset ⌃ of locally finite sequences (!i)i2N in R2, aprobability space (⌃,A,P) and a point process, called Poisson point process, N : ! 7! N (!, ·) from ⌃into M satisfying

(i) N (!, ·) =X

i2N�!i , where we identify the sequence (!i)i2I with the set {!i : i 2 N};

(ii) for every finite and pairwise disjoint family (Bi)i2I of B �

R2

�

, (N (·, Bi))i2I are independentrandom variables;

(iii) for every bounded Borel set B and every k 2 N

P ([N (·, B) = k]) = �kL2

(B)kexp (��L

2

(B))

k!.

Note that for every bounded Borel set B in R2, we have N (!, B) = # (⌃ \B), and that an easycalculation yields E (N (·, B)) = �L

2

(B). For existence of Poisson point processes and an explicitconstruction of the probability space (⌃,A,P), we refer the reader to [8]. We define the group (Tz)z2Z2

of P-preserving transformation on (⌃,A,P), by Tz! = !� z. From (ii), and using the mixing condition(39), we can easily show that

�

⌃,A,P, (Tz)z2ZN

�

is ergodic. We claim, as we will see below, that thedynamical system

�

⌃,A,P, (Tz)z2ZN

�

is a good description of the heterogeneous environment describedabove.

As a second step we are going to define the random di↵usion and the random reaction part. GivenR > 0, (r�, A�,W�) and (r+, A+,W+) in [0,+1] ⇥ [0,K] ⇥ Conv↵,� , where W�, W+ do not dependon x, we define the random density W associated with the random di↵usion part, by

W (!, x, ⇠) = W+ (⇠) +�

W� (⇠)�W+ (⇠)�

min (1,N (!, BR (x))) .

More explicitly we have

W (!, x, ⇠) =

(

W� (⇠) if x 2 [i2N

BR (!i) ,

W+ (⇠) otherwise.

Similarly we define the random growth rate and the random threshold by

r (!, x) = r+ +�

r� � r+�

min (1,N (!, BR (x))) ,

A (!, x) = A+ +�

A� �A+

�

min (1,N (!, BR (x))) .

The random SVR-function is given by

f (!, t, x, ⇣) = r (!, x)⇣2

K

✓

1� ⇣

K

◆

� r (!, x)A (!, x)⇣

K

✓

1� ⇣

K

◆

which is a Fisher reaction function with Allee e↵ect whose growth rate and threshold are (r�, A�) whenx 2 [i2NBR (!i, r), and (r+, A+) otherwise. It is easy to check that conditions (40) and (41) hold.We set W" (!, x, ⇠) = W

�

!, x" , ⇠

�

, and f" (!, t, x, ⇣) = f�

!, t, x" , ⇣

�

. Then, the conclusion of previousexample holds and we let to the reader to explicitly calculate the e↵ective growth rate and the thresholdof the homogenized reaction functional.

References

[1] M. A. Ackoglu, U. Krengel. Ergodic theorem for superadditive processes. J. Reine Angew. Math. 323 (1981), 53–67.[2] O. Anza Hafsa and J.-P. Mandallena. Interchange of infimum and integral. Calc. Var. Partial Di↵erential Equations,

18(4):433–449, 2003.[3] H. Attouch. Variational convergence for functions and operators. Applicable Mathematics Series. Pitman (Advanced

Publishing Program), Boston, MA, 1984.


[4] Hedy Attouch, Giuseppe Buttazzo, and Gerard Michaille. Variational analysis in Sobolev and BV spaces. MOS-SIAMSeries on Optimization. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; MathematicalOptimization Society, Philadelphia, PA, second edition, 2014. Applications to PDEs and optimization.

[5] Robert B. Banks. Growth and di↵usion phenomena, volume 14 of Texts in Applied Mathematics. Springer-Verlag,Berlin, 1994. Mathematical frameworks and applications.

[6] H. Berestycki and P.-L. Lions. Some applications of the method of super and subsolutions. In Bifurcation and nonlineareigenvalue problems (Proc., Session, Univ. Paris XIII, Villetaneuse, 1978), volume 782 of Lecture Notes in Math.,pages 16–41. Springer, Berlin, 1980.

[7] H. Berestycki and P.-L. Lions. Une methode locale pour l’existence de solutions positives de problemes semi-lineaireselliptiques dans RN . J. Analyse Math., 38:144–187, 1980.

[8] N. Bouleau. Processus stochastiques et applications. Collection Methodes. Hermann, 2000.[9] H. Brezis. Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-

Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. North-HollandMathematics Studies, No. 5. Notas de Matematica (50).

[10] E. Chabi and G. Michaille. Ergodic theory and application to nonconvex homogenization. Set-Valued Anal., 2(1-2):117–134, 1994. Set convergence in nonlinear analysis and optimization.

[11] Gianni Dal Maso. An introduction to �-convergence. Progress in Nonlinear Di↵erential Equations and their Applica-tions, 8. Birkhauser Boston Inc., Boston, MA, 1993.

[12] G. Dal Maso, L. Modica. Nonlinear stochastic homogenization and ergodic theory. J. Reine Angew. Math. 363 (1986),27–43.

[13] U. Mosco. Convergence of convex sets and of solutions of variational inequalities. Advances in Math., 3:510–585, 1969.[14] U. Mosco. On the continuity of the Young-Fenchel transform. J. Math. Anal. Appl., 35:518–535, 1971.[15] Akira Okubo. Di↵usion and ecological problems: mathematical models, volume 10 of Biomathematics. Springer-Verlag,

Berlin-New York, 1980. An extended version of the Japanese edition, ıt Ecology and di↵usion, Translated by G. N.Parker.

[16] C. V. Pao. Nonlinear parabolic and elliptic equations. Plenum Press, New York, 1992.[17] S. I. Pohozaev. The Dirichlet problem for the equation �u = u

2. Soviet Math. Dokl., 1:1143–1146, 1960.[18] S. Ruan. Delay di↵erential equations in single species dynamics. In Delay di↵erential equations and applications,

volume 205 of NATO Sci. Ser. II Math. Phys. Chem., pages 477–517. Springer, Dordrecht, 2006.[19] A. Tsoularis. Analysis of logistic growth models. volume 2 of Research Letters in the Information and Mathematical

Sciences, pages 23–46. Massey University, 2001.[20] P. Turchin. Quantitative Analysis of Movement: Measuring and Modeling Population Redistribution in Animals and

Plants. Beresta Books, 2015.

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