Existence of weak solutions to a system of nonlinear partial differential equations...

Nonlinear Analysis: Real World Applications 8 (2007) 267–287www.elsevier.com/locate/na

Existence of weak solutions to a system of nonlinear partialdifferential equations modelling ice streams�

J.I. Díaza,∗, A.I. Muñozb, E. Schiavib

aDepartment of Applied Mathematics, University Complutense of Madrid, 28040 Madrid, SpainbDepartment of Applied Mathematics and Physics and Sciences of Nature, University Rey Juan Carlos, 28933-Móstoles, Madrid

Received 7 July 2005; accepted 26 July 2005

Abstract

This paper deals with the mathematical analysis of a nonlinear system of three differential equations of mixed type. It describes thegeneration of fast ice streams in ice sheets flowing along soft and deformable beds. The system involves a nonlinear parabolic PDEwith a multivalued term in order to deal properly with a free boundary which is naturally associated to the problem of determiningthe basal water flux in a drainage system. The other two equations in the system are an ODE with a nonlocal (integral) term for theice thickness, which accounts for mass conservation and a first order PDE describing the ice velocity of the system. We first consideran iterative decoupling procedure to the system equations to obtain the existence and uniqueness of solutions for the uncoupledproblems. Then we prove the convergence of the iterative decoupling scheme to a bounded weak solution for the original system.� 2005 Elsevier Ltd. All rights reserved.

MSC: 22E46; 53C35; 57S20

Keywords: Ice sheet models; Nonlinear partial differential equations system of mixed type; Free boundaries

1. Introduction

It is well known that the relationships between the ice sheets, the atmosphere and the ocean dynamics have a majoreffect on the climate of the Earth (see for example [19,18]). This fact motivates the scientific community to pursue abetter knowledge of the large scale behavior of ice sheets by means of modelling their complex nonlinear dynamics,which in time constitutes an important application of mathematics to the field of geophysical fluids mechanics and moregenerally to continuum mechanics. In particular, the applied mathematics community is specially interested in lookingfor instability mechanisms that could explain the detected oscillations in the ice flow regime in the West Antarctic IceSheet (WAIS). Two crucial phenomena associated to oscillations in the ice flow regime are the ice surging, defined asthe fast and sudden advance of ice masses, and the ice streaming, associated to the spontaneous generation of fast icestreams when compared with the slow surrounding ice.

Ice streaming is a phenomenon concerning the development of lateral instabilities in the ice flow regime mainlydue to the basal sliding over a underlying deformable layer of sediments. The complexity of the physical processes

� The authors are supported by the project MTM2004-07590-C03-01. The second and third authors are partially supported by the project GDV-2004-03 of the University Rey Juan Carlos of Madrid.

∗ Corresponding author. Tel.: +34 91 39 45 242; fax: +34 91 39 44 607.E-mail addresses: [email protected] (J.I. Díaz), [email protected] (A.I. Muñoz), [email protected] (E. Schiavi).

1468-1218/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.nonrwa.2005.07.003

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268 J.I. Díaz et al. / Nonlinear Analysis: Real World Applications 8 (2007) 267–287

x=L

ICE VELOCITY (meters per year)

Lateral boundary

Control section(margin)

Lateral boundary

x=0, t=0

Ice divide

t=T

500 m/y

1000 Km

900 Km

800 Km

700 Km

600 Km

500 Km

400 Km

300 Km

200 Km

100 Km

0 Km 0 Km

200 Km400 Km

600 Km800 Km

Fig. 1. Results obtained for the ice velocity in [6], where a numerical resolution of the model studied in this paper, is presented. The main flow takesplace in the direction parallel to the t-axis. This figure illustrates the development of a region where the ice velocity takes considerable greater valuesthan in the rest of the domain, i.e., the ice streaming generation.

involved in this phenomenon makes that its mathematical modelling results in models consisting of nonlinear systemsof differential equations involving multivalued terms, which account for the free boundary nature of the problem.

The system of equations we deal with in this paper make up a model, referred to as the multivalued model (proposedin [16]), is related to a parameterized model derived by Fowler and Johnson (for the physics of the problem we referto Fowler [9]) to generate ice streams similar to those detected in the Siple Coast (WAIS). Note that our aim is not tojustify the physics of the ice streaming model, not to provide qualitative information about the behavior of solution.Our aim is to show the existence of bounded weak solutions to the system of equations describing the model. In fact,the main result of the paper which is stated in Section 3 concerns the existence of a bounded weak solution (b.w.s) tothe multivalued model. This result justifies, theoretically, the numerical treatment of the model carried out in [6], wherea finite element algorithm combined with a duality technique were employed in order to cope with the free boundarynature of the model. The uniqueness of b.w.s result appears in [17], where existence was assumed.

The rest of the paper is devoted to the proof of the existence theorem, developed throughout Sections 4 and 5 as itsplits into two main parts. The first part is detailed in Section 4, where we consider an iterative decoupling procedureto the system equations. This strategy leads to the analysis of uncoupled systems, denoted by (Sj ), j ∈ N (parameterwhich denotes the iterative decoupling step). Each system (Sj ) comprises three decoupled problems, one for each ofthe variables which are studied separately. We end this section with the result stating the existence and uniqueness ofb.w.s. to the systems (Sj ). The second part of the proof is given in Section 5 and consists of the convergence of theiterative decoupling procedure, i.e., we prove that a sequence of solutions relative to the decoupled systems convergesto a b.w.s. of the multivalued model (Fig. 1).

2. Some notation and preliminaries

In this section, we shall briefly introduce the strong formulation of the Fowler and Johnson’s model, as the multivaluedmodel constitutes a generalization of it in a sense specified later on. Fowler and Johnson’s model is a stationary two-dimensional model in which the variables do not depend on the vertical coordinate and is derived from conservationequations complemented with suitable constitutive laws. Hence, let T , L > 0 be two positive constants.We shall considerthe rectangular domain �T = (0, T ) × �, (t, x) ∈ �T , x ∈ � = (0, L), resembling the Siple Coast ice flow area. Note

J.I. Díaz et al. / Nonlinear Analysis: Real World Applications 8 (2007) 267–287 269

that the first coordinate t is considered with respect to the direction parallel to the main flow and the second coordinatex is considered with respect to the perpendicular direction to the previous one, i.e., with respect to the cross streamdirection. Let us define the subset �+

T ⊂ �T ,

�+T := {(t, x) ∈ �T , Q(t, x) > 0} ⊂ �T ,

which is a priori unknown. Simple algebraical manipulations of the original Fowler and Johnson’s model lead to anequivalent formulation, referred to as the strong formulation of Fowler and Johnson’s model, in terms of a systemof three equations for the variables: water flux, Q(t, x), ice thickness, h(t) and accumulated velocity, �(t, x). Let �x

and �t denote the partial differentiation operators. The strong formulation consists of the following system of coupledequations for the unknowns Q, h and �, complemented with suitable initial and boundary conditions.

Given Q0(x) > 0 and the positive constants h0 and �0, find three functions Q, h and � satisfying:⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

�tQ − 1

n�x[(Q + Q̄)−1/n�xQ] = f (�, h, �t h, Q) in �+

T , (1.1)

�t h = −Mrh−(1+r)(∫ L

0 (Q + Q̄)s/nr dx)−r in �+T , (1.2)

�t� = (h|�t h|)1/r (Q + Q̄)s/nr in �+T , (1.3)

�xQ(t, 0) = �xQ(t, L) = 0, t ∈ (0, T ), (1.4)

Q(0, x) = Q0(x), h(0, x) = h0, �(0, x) = �0, x ∈ �, (1.5)

with f (�, h, �t h, Q) given by

f (�, h, �t h, Q) = (h|�t h|)1/r (h|�t h| − �−1/2)(Q + Q̄)s/nr + � − �h−1.

The constants that appear in the model, Eqs. (1.1)–(1.5), are the following: 0 < Q̄>1 which stands for a residual basalwater flux (see [11]) and M, the prescribed value for the ice flux at the ice divide. The parameters of the model are� ≈ 0.2, which represents the geothermal water flux and � ≈ 0.4 which measures the importance attributed to theconductive cooling. The exponents r and s are the ones relative to the Boulton and Hindmarsh’s ice rheology (see [8]),where r, s ∈ (0, 1) and n is the exponent considered in Glen’s flow law (see [14]), usually taken to be n = 3. Notethat, according to Fowler’s assumptions, the variable h does not depend on the lateral cross stream coordinate x, i.e.,h = h(t), so hereafter we will use the notation h′ to denote �t h. Eq. (1.1) stands for the basal water flux conservationequation and is a parabolic equation, where the longitudinal downstream coordinate t would be the time-like coordinate,with nonlinear diffusion and a nonlinearity in the forcing term f (�, h, h′, Q) (see [10] for physical interpretations). Weprescribe Q at the ice divide and the lateral boundary conditions given by (1.4), which is of homogeneous Neumanntype and whose physical meaning is no water flux through condition. Eq. (1.2) models the ice mass conservation and isan ODE, in which stands out the presence of a nonlocal integral term. This Eq. (1.2) is complemented with a prescribedvalue at the ice divide (initial condition (1.5)). Finally, (1.3) is a first order partial differential equation complementedwith the condition (1.5) and represents a constitutive law of the Boulton and Hindmarsh type, which relates the icevelocity (�t�) to the shear (given by h|h′|) and to the effective pressure in the drainage system (modelled by the term(Q + Q̄)−1/n).

Note that the domain of application of (1.1)–(1.5), �+T , is a priori unknown and therefore its determination is part of the

problem. So, in order to deal properly with this free boundary problem, Muñoz et al. [16] proposed a new formulation inthe framework of an obstacle problem with a multivalued operator. This weak formulation of the problem, referred to asthe multivalued model, is presented in Section 3 and generalizes the previous Fowler and Johnson’s strong formulation.

3. The multivalued model

The multivalued formulation corresponding to the Fowler and Johnson strong formulation (Eqs. (1.1)–(1.5)) allowsfor a mathematically correct description of the free boundary and also for considering only physically admissiblesolutions, i.e., those with nonnegative water flux. The multivalued formulation of the model is the following:

Qt − 1

n[(Q + Q̄)−1/nQx]x + �(Q) � f (�, h, h′, Q) in �T , (1.1b)


h′ = −Mrh−(1+r)

(∫ L

0(Q + Q̄)s/nr dx

)−r

in �T , (1.2b)

�t = (h|h′|)1/r (Q + Q̄)s/nr in �T , (1.3b)

complemented with the boundary condition (1.4) and the initial conditions (1.5). The multivalued operator � is amaximal monotone graph of R2 which is defined as follows:

�(r) = ∅ if r < 0, �(0) = (−∞, 0] and �(r) = 0 if r > 0, (1)

where the symbol ∅ denotes the empty set. In order to study the free boundary value problem consisting of Eqs.(1.1b)–(1.3b), complemented with (1.4) and (1.5), we shall introduce the new variable

w = 1

n − 1(Q + Q̄)(n−1)/n and the function b(w) = [(n − 1)w]n/(n−1). (2)

Note that b(w) is Lipschitz continuous and that the change of variable given by (2) results in a shift of the obstaclein the sense that now it is w = � with � = Q̄(n−1)/n/(n − 1) > 0. Next we define the closed convex set K which isnaturally associated to the definition of b.w.s to the multivalued model:

K = {v ∈ H 1(�), such that v(x)��, almost everywhere (a.e.) x ∈ �}. (3)

Definition 3.1. It will be said that the initial data (i.e., prescribed data at the ice divide) are admissible when w0 ∈H 1(�), w0 ∈ K and, �0 and h0 are positive constants. And given h0, the data (constants) T , L, M and � are consideredto be admissible if

mh = [hr+20 − (r + 2)T MrL−r (2�)]1/(r+2) > 0. (4)

Remark 3.1. The assumption mh > 0 will allow us to assure that the ice thickness, h, takes only strictly positivevalues. Note also that a solution to the multivalued problem might be considered in fact a local b.w.s as here we proveits existence only for t ∈ (0, T ).

The multivalued formulation (1.1b)–(1.3b) written in terms of w is the following:Given w0(y) > 0, �0 > 0 and admissible data (in the sense of Remark 3.1), find three functions w, h and � such that

the following is satisfied:

�t b(w) − wxx + �(w − �) � f (�, h, h′, w) in �T , (5)

h′ = −Mrh−(1+r)

(∫ L

0((n − 1)w)s/r(n−1) dx

)−r

in �T , (6)

�x� = (h|h′|)1/r ((n − 1)w)s/r(n−1) in �T , (7)

�xw(0, t) = �xw(L, t) = 0, t ∈ (0, T ), (8)

w(x, 0) = w0(x), h(x, 0) = h0, �(0, x) = �0, x ∈ �, (9)

with

f (�, h, h′, w) = (h|h′|)1/r (h|h′| − �−1/2)[(n − 1)w]s/r(n−1) + � − �h−1. (10)

From now on we shall assume the values for the exponents already considered in [12,15,16], i.e., n=3 and s = r =1/2.More general results, considering r ∈ (0, 1) and s/nr ∈ (0, 1), can be obtained with minor changes. As in [1,7],


we shall consider the complementarity formulation associated to (5)–(9), which consists of the following system:

(S)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

w�� in �T ,

�t b(w) − wxx − f (�, h, h′, w)�0 in �T ,

[�t b(w) − wxx − f (�, h, h′, w)](w − �) = 0 in �T ,

h′ = −M1/2h−3/2[∫�(2w)1/2 dx]−1/2 in �T ,

�t� = (hh′)2(2w)1/2 in �T ,

�xw(0, t) = �xw(L, t) = 0, t ∈ (0, T ),

w(x, 0) = w0(x), h(x, 0) = h0, �(x, 0) = �0, x ∈ �.

3.1. Bounded weak solution

In this section, we present the definition of bounded weak solution (b.w.s) to the system (S) and the theorem whichstates the existence of at least one b.w.s to (S).

Definition 3.2. Let V denote the functional space given by V = Vw × Vh × V�, where

Vw := {� : � ∈ L2(0, T ; K) ∩ L∞(�T ), �t b(�) ∈ L2(�T )},Vh := {� : � ∈ C([0, T ]), �′ ∈ L∞(0, T )}

and

V� := { : ∈ W 1,∞(0, T ; L∞(�)) ∩ L2(0, T ; H 1(�))}.It will be said that (w, h, �) ∈ Vw × Vh × V� is a bounded weak solution to the system (S), if the following conditionsholds:∫ T

0

∫�

�t b(w) +∫ T

0

∫�[b(w) − b(w0)]�t = 0, (11)

∀ ∈ L2(0, T ; H 1(�)) ∩ W 1,1(0, T ; L∞(�)) such that (·, T ) = 0,∫ T

0

∫�

�t b(w)(� − w) +∫ T

0

∫�

wx(� − w)x

�∫ T

0

∫�

f (�, h, h′, w)(� − w) ∀� ∈ L2(0, T ; K), (12)

h(t) =[h

5/20 − 5M1/2

2

∫ t

0

(∫�(2w(x, r))1/2 dx

)−1/2

dr

]2/5

, t ∈ [0, T ], (13)

�(x, t) = �0 +∫ t

0(2w(x, s))1/2(h(s)h′(s))2 ds a.e. t ∈ (0, T ) a.e. x ∈ �. (14)

The main result of this paper states:

Theorem 3.1. Let the function w0(x), the constants �0 > 0 and h0 > 0, and the positive constants L, M, T , � beadmissible data in the sense given in Definition 3.1, then the system (S) has, at least, a bounded weak solution(w, h, �) ∈ V .

Next, we shall prove the above theorem through several steps which will be developed in Sections 4 and 5.


4. Proof of Theorem 3.1: iterative decoupling

The first part of the proof consists in the application of an iterative decoupling procedure for the system (S). Let theparameter j = 1, . . . , J → ∞ denote the steps of the iterative decoupling scheme and let us define for each j ∈ N thesystem (Sj ) as follows:

Definition 4.1. In the hypothesis of Theorem 3.1, for j ∈ N we shall consider the following decoupled system (Sj )

for the variables wj ∈ Vw, hj ∈ Vh and �j ∈ V�:

(Sj )

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

wj �� in �T ,

�t b(wj ) − (wj )xx − Fj−1(b(wj ))1/3 − Dj−1 �0 in �T ,

(�t b(wj ) − (wj )xx − Fj−1(b(wj ))1/3 − Dj−1)(wj − �) = 0 in �T ,

h′j = −M1/2h

−3/2j

(∫� Aj dx

)−1/2 in �T ,

�t�j = EjAj in �T ,

�xwj (0, t) = �xwj (L, t) = 0, t ∈ (0, T ),

wj (x, 0) = w0(x), hj (x, 0) = h0, �j (x, 0) = �0, x ∈ �,

where the coefficient functions Aj , Bj , Cj , Dj , Ej and Fj are defined in terms of wj(x, t), �j (x, t) and hj (t), asfollows:

Aj(x, t) = (2wj(x, t))1/2, Bj (t) = (hj (t)h′j (t))

3, Cj (x, t) = (hj (t)h′j (t))

2

(�j (x, t))1/2 ,

Dj(t) = � − �h−1j (t), Ej (t) = (hj (t)h

′j (t))

2, Fj (x, t) = Bj (t) − Cj (x, t).

Note that the system (Sj ) is composed of a problem for the variable wj , given by (Sj )1 − (Sj )3 and (Sj )6 − (Sj )7,which we shall denote by P(wj ) (detailed in Section 4.1). The coefficient functions in P(wj ) do not depend on thevariables hj and �j , but on hj−1 and �j−1 which are obtained in the previous step of the iterative scheme. So oncewe prove the existence of a solution wj to P(wj ), we can pass to solve the problem given by (Sj )4 and (Sj )7 forthe ice thickness hj (see Section 4.2), denoted by P(hj ) and finally we tackle with the problem for the accumulatedvelocity �j , consisting of (Sj )5 and (Sj )7, which will be denoted by P(�j ) (treated in Section 4.3). Then the triple{(wj (x, t), hj (t), �j (x, t))}, with wj(x, t), hj (t) and �j (x, t) being solutions to the problems P(wj ), P(hj ) andP(�j ), respectively, results to be the unique b.w.s to the system (Sj ) in the following sense:

Definition 4.2. It will be said that (wj , hj , �j ) ∈ V is a bounded weak solution to (Sj ) if the following conditionshold: ∫ T

0

∫�

�t b(wj ) +∫ T

0

∫�[b(wj ) − b(w0)]�t = 0,


0

∫�

�t b(wj )(� − wj) +∫ T

0

∫�(wj )x(� − wj)x �

∫ T

0

∫�

fj (� − wj) ∀� ∈ L2(0, T ; K),

where

fj := [Bj−1 − Cj−1]Aj + Dj−1,

hj (t) =[h

5/20 − [5/2]M1/2

∫ t

0

(∫�

Aj(x, r) dx

)−1/2

dr

]2/5

, t ∈ [0, T ],

�j (x, t) = �0 +∫ t

0Aj(x, s)Ej (s) ds a.e. t ∈ (0, T ) a.e. x ∈ �.


Properties 3.1. The coefficient functions Aj , . . . , Fj , j ∈ N, satisfy the following regularity and monotonicity prop-erties:

(1) Aj ∈ L∞(�T ) ∩ L2(0, T ; H 1(�)) ∩ C([0, T ]; L2(�)) and Aj �(2�)1/2 > 0, a.e. t ∈ (0, T ), a.e. x ∈ �,(2) Bj ∈ L∞(0, T ) and Bj (t) > 0, a.e. t ∈ (0, T ),(3) Cj ∈ L∞(�T ), Cj (x, t) > 0, a.e. (x, t) ∈ �T ,(4) Dj ∈ C([0, T ]), dDj/dt = D′

j ∈ L∞(0, T ) and D′j > 0, a.e. t ∈ (0, T ),

(5) Ej ∈ L∞(0, T ) and Ej(t) > 0, a.e. t ∈ (0, T ),(6) Fj ∈ L∞(�T ).

Remark 4.1. Note that the properties are derived from the fact that (wj , hj , �j ) ∈ V . Moreover, we shall see that thenorms in the space L∞(0, T ) of the functions Bj , Cj , Dj , D

′j and Ej are uniformly bounded with respect to j ∈ N.

In order to initiate the iterative decoupling scheme, we shall consider, as usually made in these kind of strategies(see for instance [7]), the functions h0(t) and �0(x, t) obtained by means of extending continuously the data h0 and�0 to the whole domain, i.e., h0(t) = h0 and �0(x, t) = �0, ∀ t ∈ (0, T ). As a consequence we obtain the coefficientfunctions B0, C0 (note that both functions result to be the null function as h′

0(t) ≡ 0) and D0 to be included insystem (S1). Then, once we have defined the system (S1) as starting point of the iterative process, we proceed tostudy the existence of solution to (Sj ) for a jth arbitrary step, assuming that the coefficient functions that come fromthe previous step have some regularity and monotonicity properties. Such assumptions turn out to be real facts asthe coefficient functions entering in system (Sj ) are defined in terms of the solution to (Sj−1) which belong, as weshall prove, to the functional space V . To be precise, the main goal of this section will be to prove the followingtheorem:

Theorem 4.1. For each j ∈ N, let w0(x), �0 > 0 and h0 > 0, and the positive constants L, M, T , �, be admissi-ble data in the sense of Definition 3.1. Then there exists a unique bounded weak solution (wj , hj , �j ) ∈ V to thesystem Sj .

The proof of Theorem 4.1 will amount to proving the existence of solution to each one of the problems P(wj ),P(hj ) and P(�j ). This is done throughout the next three sections. Therefore, we apply an inductive reasoning to builda sequence {(wj (x, t), hj (t), �j (x, t))} consisting of the unique b.w.s. to systems (Sj ), j ∈ N.

4.1. Decoupled problem related to the water flux

In this section, we prove the existence and uniqueness of a solution to (Sj )1 − (Sj )3 and (Sj )6 − (Sj )7, i.e.,P(wj ), departing from the assumption that we have already proved the existence and uniqueness of solution to thesystem Sj−1.

Definition 4.3. For j ∈ N, let the function w0(x) and the positive constants L, M, T , � be admissible data, let thecoefficient functions Dj−1 and Fj−1 satisfy Properties 3.1. We define the unilateral obstacle problem P(wj ) relatedto the equations (Sj )1 − (Sj )4 and to the conditions (Sj )6 − (Sj )7, as follows:

P(wj )

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

wj �� in �T ,

�t b(wj ) − (wj )xx − Fj−1(b(wj ))1/3 − Dj−1 �0 in �T ,

[�t b(wj ) − (wj )xx − Fj−1(b(wj ))1/3 − Dj−1](w − �) = 0 in �T ,

�xwj (0, t) = �xwj (L, t) = 0, t ∈ (0, T ),

wj (x, 0) = w0(x), x ∈ �.


It will be said that wj ∈ Vw is a bounded weak solution to the problem P(wj ) (see the definition given in [1]) if thefollowing conditions hold:∫ T

0

∫�

�t b(wj ) +∫ T

0

∫�[b(wj ) − b(w0)]�t = 0, (15)


0

∫�

�t b(wj )(� − wj) +∫ T

0

∫�(wj )x(� − wj)x

�∫ T

0

∫�[Fj−1(b(wj ))

1/3 + Dj−1](� − wj) ∀� ∈ L2(0, T ; K). (16)

Lemma 4.1. There exists a unique bounded weak solution to the problem P(wj ).

Proof (Existence part). First of all, we note that if wj is a solution to the problem P(wj ) in the sense of Definition4.2 then applying well known results (see for instance [3]) we get the following estimate:

‖b(wj )‖Lp(�T ) �eC∗T{‖b(w0)‖Lp(�) + L1/p

∫ T

0e−C∗s |Dj−1(s)| ds

}, (17)

for 1�p�∞. In particular, the estimate (17) implies the existence of a positive constant W such that ‖wj‖L∞(�T ) < W .Later, it will be proved that W can be chosen uniformly in j ∈ N. Next, we start considering a sequence of reg-ularized problems, denoted by Pn,j (w), n ∈ N, which approximate the problem P(wj ) by means of replacingFj−1(x, t)(b(w))1/3 by

Yn(b(w)) := Yn(x, t, b(w(x, t))) = Fj−1(x, t)pn(b(w)), (18)

where {pn} is a sequence of Yosida approximations (and therefore, Lipschitz continuous) for the function p(s) = s1/3.

Definition 4.4. Under the hypothesis of Definition 4.3 and Yn as in (18), we consider the unilateral obstacle problemPn,j given by

Pn,j

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

w�� in �T ,

�t b(wnj ) − (wnj )xx − Yn(b(wnj )) − Dj−1 �0 in �T ,

[�t b(wnj ) − (wnj )xx − Yn(b(wnj )) − Dj−1](wnj − �) = 0 in �T ,

�xwnj (0, t) = �xwnj (L, t) = 0, t ∈ (0, T ),

wnj (x, 0) = w0(x), x ∈ �.

It will be said that wnj ∈ Vw is a bounded weak solution to Pn,j , if the following conditions hold:∫ T

0

∫�

�t b(wnj ) +∫ T

0

∫�[b(wnj ) − b(w0)]�t = 0, (19)


0

∫�

�t b(wnj )(� − wnj ) +∫ T

0

∫�(wnj )x(� − wnj )x (20)

�∫ T

0

∫�(Yn(b(wnj )) + Dj−1)(� − wnj ) ∀� ∈ L2(0, T ; K). (21)

Let j ∈ N be fixed, let wnj +ion of the problem Pn,j , then wnj (x, t)��, a.e. (x, t) ∈ �T , hence, we only care theproperties of the functions pn and p in the interval [(2�)3/2, ∞). Note that pn(b) (and p(b)) is Lipschitz continuous


for b�(2�)3/2. Then, we can apply well known results (see [3]) to obtain that ∀n ∈ N,

‖b(wnj )‖Lp(�T ) �eC∗T{‖b(w0)‖Lp(�) + L1/p

∫ T

0e−C∗s |Dj−1(s)| ds

},

for 1�p�∞. Therefore, due to the monotonicity of b, we get that there exists a constant W such that

‖wnj‖L∞(�T ) �W .

In order to prove the result of existence of b.w.s. to the problem Pn,j we shall resort to a penalization technique, arguingin a similar way to that performed in [1]. In fact, for n, j ∈ N, we shall consider the following regularized problemsPrnj , which approximate the problem Pn,j .

Definition 4.5. For r ∈ N, let us consider the problem Prnj given by

Prnj

⎧⎪⎨⎪⎩

�t b(wrnj ) − (wrnj )xx + rj(wrnj − Pwrnj ) = Yn(b(wrnj )) + Dj−1 in �T ,

�xwrnj (0, t) = �xwrnj (L, t) = 0, t ∈ (0, T ),

wrnj (x, 0) = w0(x), x ∈ �,

where j is a monotone and convex, duality operator j : H 1(�) → (H 1(�))∗ defined as follows:

〈j(w), 〉 =∫�(w + wxx) ∀w, ∈ H 1(�)

and P is projection operator over the convex set K, P : H 1(�) → K and

〈j(w − Pw), Pw − v〉�0 for v ∈ K.

It will be said that wrnj ∈ Vw is a bounded weak solution to the problem Prnj , if the following conditions hold:

∫ T

0

∫�

�t b(wrnj ) +∫ T

0

∫�[b(wrnj ) − b(w0)]�t = 0, (22)

∀ ∈ L2(0, T ; H 1(�)) ∩ W 1,1(0, T ; L∞(�)) such that (·, T ) = 0.

∫ T

0

∫�

�t b(wrnj ) +∫ T

0

∫�(wrnj )xx + r

∫ T

0〈j(wrnj − Pwrnj ), 〉

=∫ T

0

∫�(Yn(b(wrnj )) + Dj−1) ∀ ∈ L2(0, T ; H 1(�)). (23)

Note that wrnj (·, t) ∈ L∞(�), a.e. t ∈ (0, T ) and hence S(wrnj (·, t)) = [6√2/5](wrnj (·, t))5/2 ∈ L∞(�). Using

the integration by parts formula (see [1]), we obtain that∫ t

0

∫�

�t b(wrnj )wrnj =∫ t

0

∫�

�t [b(wrnj )wrnj ] −∫ t

0

∫�

b(wrnj )�t (wrnj )

= b(wrnj (t))wrnj (t) − b(w0)w0 + 2

3[S(wrnj (t)) − S(w0)]

= S(wrnj (t)) − S(w0).

Then, ∫ t

0

∫�

�t b(wrnj )wrnj =∫�

S(wrnj (t)) −∫�

S(w0) a.e. t ∈ (0, T ).


Concerning the existence of solution for the problem Prnj , it is a result of a well known technique (see for instance [2])that resides in using a time semidiscretization. In fact, we shall replace the term �t b(wrnj ) in Prnj by the backwarddifference quotient �−h

t b(wrnj ), defined by

�−ht b(wrnj (�)) = (b(wrnj (�)) − b(wrnj (� − h)))/h, � ∈ (0, T ),

assuming that wrnj (�)=w0rnj =w0, a.e. � ∈ (−h, 0). This leads to a family of elliptic problems for which the existence

of solution is a well known classical result (note the fundamental fact that Yn(t, x, b) are Lipschitz continuous withrespect to the third variable), moreover those solutions converge to the solution of the problem Prnj (see [2,4]). So, wehave already proved the existence of a unique solution to the problem Prnj .

Lemma 4.2. There exists at least one solution to the problem Pn,j .

Proof. Once we know about the existence of solution to the problem Prnj , we shall obtain some a priori estimates ofthe energy of the solutions wrnj in order to obtain a subsequence of {wrnj } (n, j ∈ N fixed and r → ∞) that willconverge to a function wnj , which in turn will be a solution to Pn,j . To be precise, we shall get two estimates: from thefirst one, we shall deduce the existence of a subsequence of {wrnj } which will converge to a function wnj in the weaktopology of L2((0, T ); H 1(�)), and from the second we will deduce the regularity of the parabolic term, to precise,we shall obtain that �t b(wrnj ) ∈ L2(�T ). For the sake of clarity, we opt for the notation w = wrnj and employ theexpression w(t), t ∈ [0, T ] to denote the function w(t) : � → R such that w(t)(x) = w(x, t).

First estimate: We consider (Prnj )1 and we multiply it by = w − w0, where w0 is the prescribed data at t = 0 andwe integrate over (0, t) × �,∫ t

0

∫�[�t b(w) + wx()x] + r

∫ t

0〈j(w − Pw), 〉 =

∫ t

0

∫�(Yn(b(w)) + Dj−1).

Then we have the following estimates for each of the integral terms:∫ t

0

∫�

�t b(w)�(1 − �)

∫�

S(w(t)) − (1 + C�)

∫�

S(w0),

∫ t

0

∫�

wxx =∫ t

0

∫�

|wx |2 −∫ t

0

∫�

wx(w0)x �(1 − �)

∫ t

0

∫�

|wx |2 − C�

∫ t

0

∫�

|(w0)x |2,

r

∫ t

0〈j(w − Pw), 〉�r

∫ t

0‖w − Pw‖2

H 1(�)

and ∫ t

0

∫�[Yn(b(w)) + Dj−1]�

∫ t

0

∫�

C1S(w) + C2

∫�[1 + S(w0) + S2(w0)],

for some constants C1 and C2, and ∀n, j, r ∈ N. From the above estimates we deduce that

(1 − �)

∫�

S(w(t)) − (1 + C�)

∫ t

0

∫�

S(w0) + (1 − �)

∫ t

0

∫�

|wx |2

− C�

∫ t

0

∫�

|(w0)x |2 + r

∫ t

0‖w − Pw‖2

H 1(�)

�∫ t

0

∫�

C1S(w) + C2

∫�[1 + S(w0) + S2(w0)],

where C1, C2 and C� are positive constants. Ordering the above terms, we get that

(1 − �)

(∫�

S(w(t)) +∫ t

0

∫�

|wx |2)

+ r

∫ t

0‖w − Pw‖2

H 1(�)�C1

∫ t

0

∫�

S(w) + C,


a.e. t ∈ (0, T ) where 0 < �>1 and C a positive constant. The non-negativeness of the second and third terms thatappear in the left-hand side of the above inequality leads to the following inequality:

(1 − �)

∫�

S(w(t))�C1

∫ t

0

∫�

S(w) + C.

Applying Gronwall’s inequality and taking � → 0, we get that there exists a positive constant CE1, such that

sup ess{0<t<T }

∫�

S(w(t)) +∫ t

0

∫�

|wx |2 + r

∫ T

0‖w − Pw‖2

H 1(�)�CE1. (24)

Note that the constant CE1 can be chosen in a way that it does not depend on the parameters r and n, allowing us todeduce the existence of a subsequence of {wrnj } for n and j fixed, that we shall labelled by {wrnj }, converging to afunction wnj in the weak topology of the space L2(0, T ; H 1(�)). Later on it will be shown that we can find a constantCE1 valid ∀j ∈ N.

Second estimate: In this case, we multiply (Prnj )1 by �tw and we integrate over (0, t) × � to obtain that∫ t

0

∫�[�t b(w)�tw + wx(�tw)x] + r

∫ t

0〈j(w − Pw), �tw〉 =

∫ t

0

∫�(Yn(b(w)) + Dj−1)�tw.

Several simple calculations lead to the following estimates:∫ t

0

∫�

�t b(w)�tw�Cb

∫ t

0

∫�

|�t b(w)|2 with Cb a positive constant, (25)

∫ t

0

∫�

wx(�tw)x = 1

2

∫ t

0

∫�

�t [(wx)2] = 1

2

∫�

|wx |2(t) − 1

2

∫�

|(w0)x |2, (26)

r

∫ t

0〈j(w − Pw), �tw〉�(r/2)‖w(t) − Pw(t)‖2

H 1(�). (27)

Regarding the obtention of (27), we approximate the partial derivative �tw by the backward difference quotient,�−ht w = (w(t) − w(t − h))/h, where we assume that w(x, t) = w0(x) for t ∈ (−h, 0) and we estimate the left-hand

side integral term of (27) considering �−ht w instead of �tw. Let us take h satisfying 0 < h>1, then

r

∫ t

0〈j(w − Pw), �−h

t w〉 ds = r

h

∫ t

0〈j(w − Pw), w(s) − Pw(s)〉 ds

+ r

h

∫ t

0〈j(w − Pw), Pw(s) − Pw(s − h)〉 ds

+ r

h

∫ t

0〈j(w − Pw), (Pw − w)(s − h)〉 ds.

Taking into consideration the convexity of j, that Pw(s − h) ∈ K and that 〈j(w − Pw), Pw(s) − Pw(s − h)〉�0, wederive that

r

∫ t

0〈j(w − Pw), �−h

t w〉�(r/2h)

∫ t

0‖w(s) − Pw(s)‖2

H 1(�)− (r/2h)

∫ t

0‖w(s − h) − Pw(s − h)‖2

H 1(�)

= (r/2h)

∫ t

t−h

‖w(s) − Pw(s)‖2H 1(�)

.

Then taking the limit h → 0, we find (27). Finally, we present the relating estimate to the source term, which is∣∣∣∣∫ t

0

∫�[Yn(b(w)) + Dj−1]�tw

∣∣∣∣�CF

(K� + K1

∫�[2 + S(w(t)) + S(w0)] + �

∫ t

0

∫�

|�t (b(w))|2)

+ CD , (28)


where CF , CD (referred to the bounds of Fj−1 and Dj−1 in the spaces L∞(�T ) and L∞(0, T ), respectively), K1 andK� are positive constants. As a consequence of (25)–(28), we obtain

(1 − �CF )

∫ t

0

∫�

|�t b(w)|2 + (r/2)‖w(t) − Pw(t)‖2H 1(�)

+ (1/2)

∫�

|wx |2(t)

�(1/2)

∫�

|(w0)x |2 + CF

(K� + K1

∫�[2 + S(w(t)) + S(w0)]

)+ Cd . (29)

The regularity of the function S(w) allows us to deduce from the inequality (29) that �t (b(w)) ∈ L2(�T ), ∀n, r ∈ N.And finally, we have∫ t

0

∫�

|�t b(w)|2 dx ds + r

2‖w(t) − Pw(t)‖2

H 1(�)+ 1

2

∫�

|wx |2(t) dx�CE2, (30)

a.e. t ∈ (0, T ), ∀n, j, r ∈ N, with CE2 a positive constant CE2 which does not depend on the subindexes. Recall thatwe have been using the notation wrnj = w.

Thanks to the uniform bound in (30) we deduce the existence of a subsequence of {�t (b(wrnj ))}, that we shall labelby {�t (b(wrnj ))}, weakly convergent to a function �t (b(wnj )). Note that wnj is in fact the weak limit of {wrnj } inL2(0, T ; H 1(�)). Moreover∫ T

0

∫�

�t (b(wnj )) = −∫ T

0

∫�[b(wnj ) − b(w0)]�t ∀ ∈ M,

whereM={� : � ∈ L2(0, T ; H 1(�))∩W 1,1(0, T ; L∞(�)), �(·, T )=0}. Note that wrnj is a solution of the penalizationproblem, so we have that∫ T

0

∫�

�t (b(wrnj )) = −∫ T

0

∫�[b(wrnj ) − b(w0)]�t ∀r ∈ N ∀ ∈ M.

Due to the fact that the sequence {�t (b(wrnj ))} is uniformly bounded with respect to r in L2(�T ) by (30), there existsa subsequence, which will be labelled in the same way, that converges weakly to a function �. As � is the weak limitof {�t (b(wrnj ))} and M ⊂ L2(�T ), then

limr→∞

∫ T

0

∫�

�t (b(wrnj )) =∫ T

0

∫�

� ∀ ∈ M.

On the other hand, as wnj is the weak limit of {wrnj } in L2(0, T ; H 1(�)), we have that wrnj → wnj , a.e. (x, t) ∈ �T

and therefore {b(wrnj )} converges almost everywhere to b(wnj ). Then we have that for each ∈ M:

− limr→∞

∫ T

0

∫�[b(wrnj ) − b(w0)]�t = −

∫ T

0

∫�[b(wnj ) − b(w0)]�t.

Then, by the uniqueness property of the limit,∫ T

0

∫�

� = −∫ T

0

∫�[b(wnj ) − b(w0)]�t ∀ ∈ M,

that is to say � = �t b(wnj ) and, therefore, wnj satisfies (19). Again, we resort to well known results (see for instanceBenilan [3]) to obtain that

‖b(wrnj )‖Lp(�T ) �ec∗nT

{‖b(w0)‖Lp(�) + L1/p

∫ T

0e−c∗

ns |Dj−1(s)| ds

}

and

‖wrnj‖L∞(�T ) �W ∀r ∈ N. (31)


Previous estimates allow us to apply well known convergence results so that to obtain a subsequence of {wrnj } thatwill be labelled in the same way and a function wnj , such that

wrnj ⇀ wnj in L2(0, T ; H 1(�)) and wrnj → wnj in Lp(�T ) ∀p ∈ [1, ∞),

�t b(wrnj ) ⇀ �t b(wnj ) in L2(�T ), S(wrnj ) → S(wnj ) in Lp(�T ), p ∈ [1, ∞),

Yn(x, t, b(wrnj )) → Yn(x, t, b(wnj )) in Lp(�T ), p ∈ [1, ∞).

Note that wrnj → wnj , a.e. (x, t) ∈ �T and

‖wrnj (t) − Pwrnj (t)‖2H 1(�)

�(2/r)CE2 → 0 as r → ∞. (32)

Therefore, wnj = Pwnj , a.e. (x, t) ∈ �T i.e., wnj ∈ K.

End of Proof of Lemma 4.2. Now, let wnj be the function just obtained. Let us take limit with respect to r in theestimates (24) and (30), where the constants that appeared there, i.e., CE1 and CE2, do not depend on the subindexes{r, n}. Then we find that

sup ess{0<t<T }

∫�

S(wnj (t)) +∫ t

0

∫�

|(wnj )x |2 �CE1, (33)

∫ t

0

∫�

|�t b(wnj )|2 + 1

2

∫�

|(wnj )x |2(t)�CE2 a.e. t ∈ (0, T ) ∀n, j ∈ N, (34)

where CE1 and CE2 are two suitable positive constants. Now, we can take limits in a weak sense in conditions (22)and (23) to get that wnj satisfies conditions (19), (21) and therefore is a bounded weak solution to the problemPn,j (w). �

Next, we present a result of convergence regarding the sequence of solution {wnj }n, j fixed and n ∈ N, whose proofresides in an application of a weak version of Ascoli–Arzelá’s theorem (see [21]).

Lemma 4.3. Let wnj be a bounded weak solution to the problem Pn,j (w), then there exists a positive constant Csuch that

‖Yn(·, t, b(wnj ))‖L∞(�) �C a.e. t ∈ (0, T ) ∀n ∈ N. (35)

Moreover, the family {b(wnj )} is equi-continuous in the following sense:

‖b(wnj (t)) − b(wmj (t))‖2L1(�)

�M

(1

n+ 1

m

)∀t ∈ [0, T ],

where M is a positive constant that only depends on the constants W and C (see (31) and (35)).

Proof. Let {wnj } be the sequence of solutions associated to the problems Pn,j (w). As a consequence of (31), we havethat

‖b(wnj (·, t))‖L∞(�) �b(W) ∀t ∈ [0, T ] ∀n, j ∈ N. (36)

Taking into account that Yn(b(w)) = Fj−1pn(b(w)), where pn is the Yosida approximation of p, that Yn(b(w))�Fj−1p(b(w)), ∀n ∈ N and (36), we deduce the existence of a positive constant C, such that (35) is verified.


Being −wxx +�(w−�) a m-accretive operator in L1(�) and b(wnj (0))=b(w0), ∀n ∈ N, and we have that (see [21]),

‖b(wnj (t)) − b(wmj (t))‖2L1(�)

�2∫ t

0(b(wnj (s)) − b(wmj (s)), Fj−1(s)[pn(b(wnj (s))) − pm(b(wmj (s)))])+ ds

�2∫ t

0(Jn(b(wnj )) − Jm(b(wmj )), Fj−1[pn(b(wnj )) − pm(b(wmj ))])+ ds

+ 2∫ t

0

(1

npn(b(wnj )) − 1

mpm(b(wmj )), Fj−1[pn(b(wnj )) − pm(b(wmj ))]

)+

ds

�M

(1

n+ 1

m

), (37)

where (·, ·)+ denotes the upper semi-interior product with respect to the usual norm of the space L1(�) (see [21]), Iis the identity operator and Jn is the resolvent. In the inequality (37) it has been used that pn(b) = n(I − Jn)(b) andtherefore, I = (1/n)pn + Jn. Using (35), (36) and the fact that Jn is a contraction (see [5]), we have that there exists apositive constant M such that (36) holds. �

Next, we observe that Lemma 4.3 allows for an application of a weak version of the Ascoli–Arzelá’s theorem (see[21]) to the sequence {b(wnj )}, from which we derive the existence of a subsequence of {b(wnj )}, that we shall denotein the same way, and a function b̃, such that limn→∞b(wnj ) = b̃, strongly in the topology of C([0, T ]; L1(�)). Beingthe norms {‖wnj‖L∞(�)} uniformly bounded in n ∈ N and b strictly monotone, there exists a function wj such thatlimn→∞ wnj = b−1(b̃) = wj in C([0, T ]; L1(�)). Note that the constant M, in (37), at first seems to depend on theparameter j ∈ N, however later on we shall prove that it can be chosen so that it does not. Next, we pass to consider thelast part of the proof of Lemma 4.1, in which we shall check that the function wj is a solution of the problem P(wj ).

End of proof of the existence part of Lemma 4.1. From estimates (32)–(34), we derive that wnj ∈ K and the uni-form boundwith respect to n ∈ N of the norms ‖�t b(wnj )‖L2(�T ) and ‖wnj‖L2(0,T ;H 1(�)). These properties allow usto deduce the existence of a subsequence of {wnj }, that converges to a function wj in the topology of C([0, T ]; L1(�))

and weakly in L2(0, T ; H 1(�)). Moreover,

wnj → wj , b(wnj ) → b(wj ) strongly in Lp(�T ), 1�p < ∞,

�t b(wnj ) ⇀ �t b(wj ) weakly in L2(�T ), wj (t) ∈ K a.e. t ∈ [0, T ],Yn(b(wnj )) = Fj−1pn(b(wnj )) → Fj−1p(b(wj )) strongly in Lp(�T ), 1�p < ∞.

So, if we pass to the limit in a weak sense with respect to n in (33) and (34) then (note that the positive constants CE1and CE2 appearing there did not depend on the parameters n ∈ N) it results that wj satisfies the estimates

sup ess{0<t<T }

∫�

S(wj (t)) +∫ T

0

∫�

|(wj )x |2 �CE1, (38)

∫ t

0

∫�

|�t b(wj )|2 + (1/2)

∫�

|(wj )x |2(t)�CE2 a.e. t ∈ (0, T ), j ∈ N, (39)

and as a consequence we deduce that the functions wj ∈ Vw, j ∈ N and are uniformly bounded in the norms of thespaces considered in Vw. Note also that taking limits in (19) and (21) we obtain that wj is a solution of P(wj ) in thesense given in Definition 4.3. �

Proof of the uniqueness part of Lemma 4.1. Regarding the uniqueness of solution to problem P(wj ), we start pre-senting a comparison result.

Definition 4.6. Under the hypothesis considered in Definitions 4.2 and 4.3, we will consider the obstacle problemsPi , defined as follows:


Pi

⎧⎪⎨⎪⎩

�t b(ui) − (ui)xx + �(ui − �) − Fp(b(ui)) � Di in �T ,

�xui(0, t) = �xui(L, t) = 0, t ∈ (0, T ),

ui(x, 0) = u0,i (x), x ∈ �.

It will be said that ui is a weak solution to Pi if ui ∈ L2(0, T ; K) ∩ L∞(�T ), �t b(ui) ∈ L2(�T ), ui satisfies (19) andthere exists a function i ∈ L1(�T ), i (x, t) ∈ �(ui(x, t) − �), a.e. (x, t) ∈ �T , such that

∫ T

0

∫�

�t b(ui) +∫ T

0

∫�[b(ui) − b(u0,i )]�t = 0,

∀ ∈ L2(0, T ; H 1(�)) ∩ W 1,1(0, T ; L∞(�)) such that (·, T ) = 0,∫ ∫�T

(�t b(ui)� + i�) dx dt =∫ ∫

�T

(ui)x�x dx dt +∫ ∫

�T

(Fp(b(ui)) + Di)� dx dt ,

∀� ∈ L2(0, T ; H 1(�)) ∩ L∞(�). (40)

Note that (40) is an equivalent condition to (16) (see [7]). Let [f ]+ denote the positive part of an arbitrary functionf, i.e., [f ]+ := max{0, f }.

Lemma 4.4. Under the hypothesis considered in Definition 4.6, let u1 and u2 be solutions to the obstacle problemsPi , i = 1, 2, respectively, then there exists a positive constant C∗ such that∫

�[b(u1)(t) − b(u2)(t)]+ dx�eC∗t

∫�[b(u0,1) − b(u0,2)]+ dx

+ eC∗t∫ t

0e−C∗s‖D1 − D2‖L1(�) ds a.e. t ∈ (0, T ). (41)

Proof. Let ui be a solution to Pi , i = 1, 2, then ui(x, t)�� and

|p(b(u1)) − p(b(u2))|�Cp|b(u1) − b(u2)|,

as p(b(z)) = (2z)1/2, for a positive constant Cp. Note that p(b(z)) = (2z)1/2 is locally Lipschitz continuous in (0, ∞),and, therefore, Lipschitz continuous in every compact interval contained in (0, ∞). Let sig+

0 be the function definedas: sig+

0 (s) = 0 if s�0 and sig+0 (s) = 1 if s > 0. Let us define the functions Tn for n ∈ N,

Tn(s) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

0, s�0,

n2s2/2, 0 < s�1/n,

2ns − (n2s2/2) − 1, 1/n < s�2/n,

1, s > 2/n,

which verify the following estimates and convergence properties:

0�T ′n(s)�n, lim

n→∞ sT ′n(s) = 0, |Tn(s)|�1,

limn→∞ Tn(s) = sig+

0 (s) and limn→∞ sT n(s) = [s]+.

Let z = b(u1) − b(u2). Note that u1, u2 ∈ L2(0, T ; H 1(�)) ∩ L∞(�) and Tn is a continuous function, thereforeTn(u1 − u2) ∈ L2(0, T ; H 1(�)) ∩ L∞(�). Let us consider i (x, t) ∈ �(ui(x, t) − �), a.e. (x, t) ∈ �T , i = 1, 2,such that

�t b(ui) − (ui)xx + i − Fp(b(ui)) = Di a.e. (x, t) ∈ �, i = 1, 2, (42)


and subtract (in the weak sense of Definition 4.6) identity (42) applied for i = 2 to the one for i = 1,

�t (b(u1) − b(u2)) + 1 − 2 − (u1 − u2)xx = F [p(b(u1)) − p(b(u2))] + D1 − D2, (43)

a.e. (x, t) ∈ �T . Next, we multiply Eq. (43) by the function Tn(u1 − u2) and integrate in (0, t) × � to obtain thefollowing identity:

−∫ t

0

∫�

Tn(u1 − u2)zt dx ds = −∫ t

0

∫�( 1 − 2)Tn(u1 − u2) dx ds

−∫ t

0

∫�[(u1)x − (u2)x]xTn(u1 − u2) dx ds

−∫ t

0

∫�

F [p(b(u1)) − p(b(u2))]Tn(u1 − u2) dx ds

−∫ t

0

∫�(D1 − D2)Tn(u1 − u2) dx ds. (44)

Now we study the integral terms appearing in (44). Due to the monotonicity of the approximating functions Tn andto the prescription of homogeneous Neumann boundary conditions, the application of the integration by parts formulaleads us to the following estimate:

−∫ t

0

∫�[(u1)x − (u2)x]xTn(u1 − u2) dx ds =

∫ t

0

∫�

T ′n(u1 − u2)|(u1)x − (u2)x |2 dx ds�0,

a.e. t ∈ (0, T ). Note that �t b(u1) and �t b(u2) ∈ L2(�T ), so zt ∈ L2(�T ). Moreover, Tn(u1 −u2) ∈ L2(0, T ; H 1(�))

and T ′n(u1 − u2) ∈ L2(�T ). On the other hand, by the monotonicity of the function b, we have that

limn→∞ Tn(u1 − u2) = sig+

0 (u1 − u2) = sig+0 (b(u1) − b(u2)) a.e. (x, t) ∈ �T .

Hence,

limn→∞

∫ t

0

∫�

Tn(u1 − u2)zt dx ds =∫ t

0

∫�

sig+0 (z)zt dx ds a.e. t ∈ (0, T ),

and therefore∫ t

0

∫�

Tn(z)zt dx ds =∫�

Tn(z(t))z(t) dx −∫�

Tn(z(0))z(0) dx −∫ t

0

∫�

zT ′n(z)zt dx ds,

a.e. t ∈ (0, T ), then, taking limit

limn→∞

∫ t

0

∫�

Tn(z)zt dx ds =∫�

sig+0 (z(t))z(t) −

∫�

sig+0 (z(0))z(0) dx ds, a.e. t ∈ (0, T ),

hence,

limn→∞

∫ t

0

∫�

Tn(u1 − u2)�t (b(u1) − b(u2)) dx ds

=∫�[b(u1) − b(u2)]+(t) dx −

∫�[b(u0,1) − b(u0,2)]+ dx a.e. t ∈ (0, T ).

Taking into account that i (x, t) ∈ �(ui(x, t) − �), a.e. (x, t) ∈ �T , i = 1, 2, we have∫ t

0

∫�( 1 − 2)Tn(u1 −

u2) dx ds�0, a.e. t ∈ (0, T ). Hence, for a.e. t ∈ (0, T ),∫�[b(u1) − b(u2)]+(t) dx�

∫�[b(w01) − b(w02)]+ dx

+∫ t

0

∫�(F [p(b(u1)) − p(b(u2))] + D1 − D2)sig+

0 (u1 − u2) dx ds.


Let C∗ be a positive constant such that F · |p(b(u1)) − p(b(u2))|�C∗|b(u1) − b(u2)|. Let us define v1(x, t) =e−C∗t b(u1(x, t)) and v2(x, t) = e−C∗t b(u2(x, t)). Note that sig+

0 (u1 − u2) = sig+0 (v1 − v2) and �t vi = −C∗vi +

e−C∗t�t b(ui) ∈ L2(�), a.e. t ∈ (0, T ), in a weak sense. Considering the above expression for i =1, 2, and subtracting,we obtain that

�t (v1 − v2) = −C∗(v1 − v2) + e−C∗t�t (b(u1) − b(u2)). (45)

Next, we multiply (45) by the function Tn(v1 − v2) and argue in a analogous way as before to get∫�[v1(t) − v2(t)]+ dx�

∫�[v0,1 − v0,2]+ dx − C∗

∫ t

0

∫�[v1 − v2]+ dx ds

+∫ t

0

∫�[F(p(b(u1)) − p(b(u2))) + D1 − D2]e−C∗ssig+

0 (u1 − u2) dx ds,

a.e. t ∈ (0, T ). Taking into account that

F [p(b(u1)) − p(b(u2))]e−C∗ssig+0 (v1 − v2)�C∗[b(u1) − b(u2)]e−C∗ssig+

0 (v1 − v2)

= C∗[v1 − v2]+ a.e. s ∈ (0, T ),

we have that∫�[v1(t) − v2(t)]+ dx�

∫�[v0,1 − v0,2]+ dx − C∗

∫ t

0

∫�[v1 − v2]+ dx ds

+ C∗∫ t

0

∫�[v1 − v2]+ dx ds +

∫ t

0

∫�

e−C∗s |D1 − D2| dx ds a.e. t ∈ (0, T ),

from which we derive∫�

e−C∗t [b(u1) − b(u2)]+(t) dx�∫�[b(u0,1) − b(u0,2)]+ dx

+∫ t

0

∫�

e−C∗s |D1 − D2| dx ds a.e. t ∈ (0, T ),

therefore∫�[b(u1) − b(u2)]+(t) dx�eC∗t

(∫�[b(u0,1) − b(u0,2)]+ dx +

∫ ∫�t

e−C∗s |D1 − D2| dx ds

),

a.e. t ∈ (0, T ) and finally we get (41). �

Corollary 4.1. Let u1 and u2 be solutions to the problems P1 and P2, respectively. Let us assume that u0,1 �u0,2, a.e.x ∈ �, and that D1 = D2, then u1 �u2, a.e. (x, t) ∈ �T .

Proof (End of the proof of the uniqueness part of Lemma 4.1). Let w1 and w2, be solutions to the problem P(wj ) forthe same data, then

w1(x, t) = w2(x, t) a.e. (x, t) ∈ �T ,

taking into consideration that P(wj ) satisfies the generality mentioned in definition of Pi , and that, therefore, we canapply Lemma 4.4 to it, considering w1 = u1, w2 = u2, u0,1 = u0,2 and D1 = D2. �

4.2. Decoupled problem for the ice thickness

In this section, we prove the existence and uniqueness of a solution to (Sj )4 and (Sj )7 departing from the assumptionthat we have already proved the existence and uniqueness of a triple (wj−1, hj−1, �j−1) ∈ V solution to the systemSj−1 and the existence and uniqueness of solution to problem P(wj ).


Definition 4.7. Let h0 and the positive constants L, M, T be admissible data in the sense given in Definition 3.1 andlet the coefficient function Aj satisfy Properties 3.1. For j ∈ N, we shall consider the problem P(hj ) which consistsof (Sj )4 and (Sj )7:

P(hj )

{h′

j = −M1/2h−3/2j

(∫� Aj(x, ·) dx

)−1/2 in �T ,

hj (0) = h0.

Then, we have the following result regarding the existence and uniqueness of solution to the problem P(hj ).

Lemma 4.5. There exists a unique strong solution hj ∈ W 1,∞(0, T ) ⊂ C([0, T ]), strictly positive and decreasingfunction ∀t ∈ (0, T ), to the problem P(hj ), given by

hj (t) =[h

5/20 − 5

2M1/2

∫ t

0

(∫�

Aj(x, r) dx

)−1/2

dr

]2/5

∀t ∈ [0, T ].

Proof. The equation of the problem P(hj ) can be considered as one of separated variables because it can be expressedin the form

h′j (t) = −M1/2h

−3/2j (t)X(t), (46)

where X(t) := (∫� Aj(x, t) dx)−1/2 and X ∈ C([0, T ]) as Aj(x, t) > (2�)1/2 a.e. (x, t) ∈ �T and Aj ∈ C([0, T ];

L2(�)). Integrating in (46) we obtain the explicit expression

hj (t) =[h

5/20 − 5

2M1/2

∫ t

0X(s) ds

]2/5

a.e. t ∈ (0, T ).

Due to the fact that h0 is assumed to be an admissible data it holds the following estimate:

hj (t) =[h

5/20 − 5

2M1/2

∫ t

0

(∫�

Aj(x, r) dx

)−1/2

dr

]2/5

�mh > 0 a.e. t ∈ (0, T ).

Note that h′j < 0, a.e. t ∈ (0, T ) as Aj > 0, a.e. (x, t) ∈ �T . Hence, we deduce that

‖hj‖L∞(0,T ) �h0 and ‖h′j‖L∞(0,T ) �

(M/L)1/2

m3/2h (2�)1/4

a.e. t ∈ (0, T ) ∀j ∈ N.

From the previous estimates we have that hj ∈ W 1,∞(0, T ) and in particular, as W 1,∞(0, T ) has compact embeddingin C([0, T ]) (see [5]), then hj ∈ C([0, T ]). As a consequence of being the function hj continuous, we have that theestimates and expressions relative to hj hold ∀t ∈ [0, T ], instead of a.e. t ∈ (0, T ). �

Note that the family {‖hj‖W 1,∞(0,T )} is uniformly bounded in j ∈ N. Therefore there will be a subsequence of {hj },that will be labelled in the same way, converging to a function h ∈ C([0, T ]). It is enough to take into considerationthat the sequence {‖hj‖W 1,∞(0,T )} is uniformly bounded with respect to j ∈ N and that W 1,∞(0, T ) has compactembedding in C([0, T ]). Besides, we get that the functions Bj , Dj , Ej belong to the space L∞(0, T ) and there existsa positive constant C, such that

‖Bj‖L∞(0,T ), ‖Dj‖L∞(0,T ), ‖Ej‖L∞(0,T ) �C ∀j ∈ N. (47)

Remark 4.2. According to (17), being the functions Dj uniformly bounded with respect to the norm of the spaceL∞(�), then we obtain that there exists a positive constant W such that

‖wj‖L∞(�T ), ‖Aj‖L∞(�T )�W ∀j ∈ N. (48)


4.3. Decoupled problem for the accumulated velocity

In this section, we prove the existence and uniqueness of a solution to (Sj )5 and (Sj )7 departing from the assumptionthat we have already proved the existence and uniqueness of solutions to problems P(wj ) and P(hj ).

Definition 4.8. For j ∈ N, let �0 be an admissible data in the sense given in Definition 3.1 and let the coefficientfunctions Aj and Ej satisfy Properties 3.1. We consider the problem P(�j ) consisting of (Sj )4 and (Sj )7:

P(�j )

{�t�j = Aj(x, t)Ej (t) a.e. t ∈ (0, T ) a.e. x ∈ �,

�j (x, 0) = �0 in �.

Then, regarding the existence and uniqueness of solution to the problem P(�j ), we have the following result:

Lemma 4.6. There is a unique strong solution to P(�j ), �j ∈ W 1,∞(0, T ; L∞(�)) ∩ L2(0, T ; H 1(�)), given by

�j (x, t) = �0 +∫ t

0Aj(x, s)Ej (s) ds a.e. t ∈ (0, T ) a.e. x ∈ �. (49)

Proof. Note that the product AjEj is integrable in �T due to the fact that Aj and Ej ∈ L∞(�T ). In particular,Aj(x, ·)Ej (·) is integrable in (0, t), a.e. x ∈ �. Therefore by integrating in (0, t) equation P(�j )1, we derive theexplicit formula (49). The uniqueness is a consequence of the application of the fundamental theorem of the variationalcalculus. We also deduce from the fact that Aj and Ej are globally bounded that �j ∈ W 1,∞(0, T ; L∞(�)). Note thatAj and Ej are positive in a.e. t ∈ (0, T ) and a.e. x ∈ � and hence �j > 0 and �t�j > 0, a.e. t ∈ (0, T ), a.e. x ∈ �. �

Remark 4.3. As a consequence of (47) and (48), we can deduce the existence of a positive constant C, such that

‖Cj‖L∞(�T ), ‖Fj‖L∞(�T ) �C ∀j ∈ N. (50)

End of Proof of Theorem 4.1. Finally, we note that the triple (wj , hj , �j ), with wj , hj and �j , solutions to P(wj ),P(hj ) and P(�j ), respectively (see Lemmas 4.1, 4.5 and 4.6) is the unique solution to the decoupled system Sj in thesense given in Definition 4.2. �

5. Proof of Theorem 3.1: Convergence

In this section, we shall undertake the proof of the existence of a unique bounded weak solution of a bounded weaksolution to the coupled system S. To be precise, we shall prove the existence result by proving that there is a subsequenceof the sequence of solutions {(wj , hj , �j )} to the problems Sj , j ∈ N, which converges to a weak solution to thesystem S.

We begin by observing that {‖hj‖W 1,∞(0,T )} is uniformly bounded and therefore we can find a subsequence of {hj }that converges to a function h in the space C([0, T ]). Note also that thanks to (48), we have that for a subsequence

Aj = (2wj)1/2 → A = (2w)1/2 strongly in Lp(�T ), 1�p < ∞,

moreover A�(2�)1/2. As a consequence of this,

limj→∞

∫�

Aj(x, t) dx =∫�

A(x, t) dx a.e. t ∈ (0, T ).

To summarize, we have that

limj→∞ hj (t) = lim

j→∞

[h

5/20 − [5/2]M1/2

∫ t

0

(∫�

A(x, r) dx

)−1/2

dr

]2/5

a.e. t ∈ (0, T ),


and therefore

h(t) =[h

5/20 − [5/2]M1/2

∫ t

0

(∫�

A(x, r) dx

)−1/2

dr

]2/5

∀t ∈ [0, T ].

Hence, we deduce that h�mh > 0, ∀t ∈ [0, T ], and from the estimates performed in Section 3.1 we get that h′j → h′

strongly in Lp(0, T ), 1�p < ∞,

h′ = −M1/2h−3/2(∫

�A(x, r) dx

)−1/2

, h′ ∈ L∞ and h′ < 0 a.e. t ∈ (0, T ).

Then h ∈ Vh. Note that the sequences {Aj }, {Bj }, {Dj } and {Ej }, verify the following convergence results:

Bj → B = (h|h′|)3 and Ej → E = (hh′)2 strongly in Lp(0, T ), 1�p < ∞,

Dj → D = � − �

hstrongly in C([0, T ]), D′

j → D′ strongly in Lp(0, T ),

1�p < ∞, and also that B, E and D satisfy Properties 3.1. Moreover, taking into account the estimates (47), (50) and(48), we deduce that the positive constants appearing in estimates (35)–(39), can be chosen in a way that they are valid∀j ∈ N. This fact implies that ‖wj‖L2(0,T ;H 1(�)) and ‖�t b(wj )‖L2(�T ) are uniformly bounded with respect to j andthat wj(t) ∈ K, a.e. t ∈ [0, T ], ∀j ∈ N. Then, we can deduce the existence of a subsequence of {wj }, that we label inthe same way, such that

wj ⇀ w in L2(0, T ; H 1(�)) and �t b(wj ) ⇀ �t b(w) in L2(�T ).

Moreover, we can derive that w ∈ Vw. Since Aj converges strongly in L2(�T ) to A and Ej converges strongly in L2(�T )

to E, we have that AjEj converges strongly in L2(�T ) to AE. By Lemma 4.6, we have that �j ∈ L∞((0, T ); W 1,1(�)),and that �t�j ∈ L∞((0, T ); L1(�)) (recall that �t�j ∈ L∞(�T )) and the bounds of such a function in such a space areuniform in j ∈ N. The uniform bound of �j and �t�j in the mentioned functional spaces allows us to apply a resultpresented in [20], and so, to deduce the existence of a subsequence of {�j }, that will be labelled in the same way, thatconverges strongly to � in C([0, T ]; Lp(�)), for 1�p < ∞ (note that W 1,1(�) has a compact embedding in the spaceLp(�) for 1�p < ∞, �j and �t�j are inL∞). Then, passing to the limit

�(x, t) = �0 +∫ t

0A(x, s)E(s) ds a.e. t ∈ (0, T ) a.e. x ∈ �.

Moreover, � is a positive, non-decreasing with respect to t function. Thanks to the regularity of the functions A and E,we deduce that � ∈ V�. Moreover, we get that there exists a subsequence {Cj }, such that

Cj → C = (h|h′|)2�−1/2 strongly in Lp(�T ), 1�p < ∞,

and therefore, C, satisfies Properties 3.1. In addition,

Fj = Bj − Cj → F = B − C strongly in Lp(�T ), 1�p < ∞.

In particular, Fj → F , strongly in L2(�T ) and F satisfies Properties 3.1 and hence Fjp(b(wj )) ⇀ Fp(b(w)) inL2(�T ).

To summarize, we have found a subsequence {wj }, such that

wj ⇀ w in L2(0, T ; H 1(�)), wj → w a.e. (x, t) ∈ �T ,

w(t) ∈ K a.e. t ∈ [0, T ], �t b(wj ) ⇀ �t b(w) in L2(�T ),

p(b(wj )) → p(b(w)) in L2(�T ) and Fjp(b(wj )) ⇀ Fp(b(w)) in L2(�T ).

The estimates and convergence results just obtained allow for passing to the limit (in a weak sense) with respect to theparameter j in (15) and (16), which leads to the fact that w also verifies (11) and (12). In conclusion, it results that thetriple given by (w, h, �) is a weak solution to the system (S) as (w, h, �) ∈ V and (11)–(14) are verified. �


References

[1] H.W. Alt, S. Luckhaus, Quasilinear elliptic–parabolic differential equations, Math. Z. 183 (1993) 311–341.[2] P. Benilan, Equations d’evolution dans un espace de Banach quelconque et applications, Thése, Orsay, 1972.[3] P. Benilan, Evolution equations and accretive operators, Lecture Notes, University of Kentucky, 1972.[4] P. Benilan, P. Wittbould, On mild and weak solutions of elliptic–parabolic problems, Adv. Differential Equations 1 (1996) 1053–1073.[5] H. Brezis, Analyse Fonctionelle, Mason, Paris, 1983.[6] N. Calvo, J. Durany, A.I. Muñoz, E. Schiavi, C. Vázquez, A coupled multivalued model for ice streams and its numerical simulations. IMA J.

Appl. Math. 1–30 (2005), Advanced Access Published on May 17, 2005.[7] J.I. Díaz, E. Schiavi, On a degenerate parabolic/hyperbolic system in glaciology giving rise to a free boundary, Nonlinear Anal. 38 (1999)

787–814.[8] A.C. Fowler, Sliding with cavity formation, J. Glaciol. 33 (1987) 255–267.[9] A.C. Fowler, Ice sheet surging and ice stream formation, Ann. Glaciol. 23 (1995) 68–73.

[10] A.C. Fowler, C. Johnson, Hydraulic runaway: a mechanism for thermally regulated surges of ice sheets, J. Glaciol. 41 (1995) 554–561.[11] A.C. Fowler, E. Schiavi, A theory of ice sheet surges, J. Glaciol. 44 (146) (1998) 104–118.[12] C. Johnson, The mathematical and numerical modelling of antartic ice streams, Ph.D. Thesis, University of Oxford, 1996.[14] L.A. Lliboutry, Traité de Glaciologie, 1, Mason, Paris, 1964.[15] A.I. Muñoz, Modelado, análisis y resolución numérica de un problema de obstáculo en glaciología, Ph.D. Thesis, Universidad Rey Juan Carlos,

2003.[16] A.I. Muñoz, E. Schiavi, U. Kindelán, Non linear phenomena in glaciology: ice-surging and streaming, Revista de la Real Academia de Ciencias

Exactas, Físicas y Naturales, Serie A: Matemáticas. Racsam, vol. 95, Clave A, 2002.[17] A.I. Muñoz, J.I. Tello, Uniqueness and collapse of solution for a mathematical model with nonlocal terms arising in glaciology, Math. Models

Meth. Appl. Sci. 15 (4) (2005) 623–642.[18] W.R. Peltier, S. Marshall, Coupled energy-balance/ice-sheet model simulations of the glacial cycle: a possible connection between terminations

and terrigemous dust, J. Geophys. Res. 100 (1995) 14269–14289.[19] R.L. Pfeffer, Dynamics of Climate, vol. 137, Pergamon Press, New York, 1960.[20] J. Simon, Compacts sets in the space Lp(0, t; B), Ann. Mat. Pura. Appl. 4 (CXLVI) (1987) 65–96.[21] I. Vrabie, Compactness Methods for Nonlinear Evolutions, Pitman Monograph and Surveys in Pure and Applied Mathematics, vol. 32, 1987.

Further reading

[13] J.L. Lions, Quelques méthodes de resolution des problémes aux limites non linéaires, Dunod, Paris, 1969.

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