On the wave equation with hyperbolic dynamical boundary ... · On the wave equation with hyperbolic...

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On the wave equation with hyperbolic dynamicalboundary conditions, interior and boundary

damping and sources

Enzo Vitillaro

Dipartimento di Matematica ed InformaticaUniversita degli Studi di Perugia ITALY

XVII Italian Meeting on Hyperbolic EquationsUniversity of Pavia

September 8, 2017

Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 1 / 25

1. The problem.

E. Vitillaro, Arch. Rat. Mech. Anal. 2017

” , arXiv 2016

utt −∆u + α(x)P(ut) = f (u) in (0,∞)× Ω,

u = 0 on (0,∞)× Γ0,

utt + ∂νu −∆Γu + β(x)Q(ut) = g(u) on (0,∞)× Γ1,

u(0, x) = u0(x), ut(0, x) = u1(x) in Ω

Ω ⊂ RN , N ≥ 2, bounded, open C1, Γ := ∂Ω;

Γ = Γ0 ∪ Γ1, Γ0 ∩ Γ1 = ∅, Γ1 6= ∅ relatively open on Γ,σ(Γ0 ∩ Γ1) = 0, σ is the hypersurface measure on Γ,

u = u(t, x), t ≥ 0, x ∈ Ω, ∆ = ∆x , ∆Γ Laplace–Beltramioperator on the manifold Γ;

ν = outward normal to Ω;

α ∈ L∞(Ω), α ≥ 0, β ∈ L∞(Γ1), β ≥ 0.

1. The problem.

” , arXiv 2016

utt −∆u + α(x)P(ut) = f (u) in (0,∞)× Ω,

u = 0 on (0,∞)× Γ0,

u(0, x) = u0(x), ut(0, x) = u1(x) in Ω

α ∈ L∞(Ω), α ≥ 0, β ∈ L∞(Γ1), β ≥ 0.

1. The problem.

” , arXiv 2016

utt −∆u + α(x)P(ut) = f (u) in (0,∞)× Ω,

u = 0 on (0,∞)× Γ0,

u(0, x) = u0(x), ut(0, x) = u1(x) in Ω

α ∈ L∞(Ω), α ≥ 0, β ∈ L∞(Γ1), β ≥ 0.

The two–dimensional model: a bass drum

Lagrangian function (∇Γ Riemannian gradient):

L(u) =1

t − |∇u|2 − 2F(u)]

∫Γ1

t − |∇Γu|2 − 2G(u)]

The two–dimensional model: a bass drum

Lagrangian function (∇Γ Riemannian gradient):

L(u) =1

t − |∇u|2 − 2F(u)]

∫Γ1

t − |∇Γu|2 − 2G(u)]

2. References.

Wave equation with internal damping and source terms (Γ1 = ∅)

• Georgiev, Todorova J.D.E. ’94• Levine, Pucci, Serrin Contemp. Math. ’97• Levine, Serrin Arch. Rat. Mech. Anal. ’97• Pucci, Serrin J.D.E. ’98• Vitillaro Arch. Rat. Mech. Anal. ’99• Levine, Todorova Proc. A.M.S. ’01• Serrin, Todorova, Vitillaro Diff. Int. Eqs. ’03• Todorova, Vitillaro J.M.A.A. ’05• Radu Adv. Diff. Eqs. ’05• ” Appl. Math. ’08

Wentzell’s type (utt replaced by ∆u) boundary conditions

• Favini, R. Goldstein and others J. Evol. Equ.’02• Engel Arch. Mat. Basel ’03• Arendt and others Semigroup Forum ’03• Favini and others Appl. Anal ’03• Engel, Fragnelli Adv. Diff. Eqs. ’05• Mugnolo Math. Nachr. ’06• Ruiz– Goldstein Adv. Diff. Eqs. ’06

Wave equation with internal and/or boundary damping and sources

• Lasiecka, Tataru Diff. Int. Eqs. ’93• Vitillaro Rend. Mat. Trieste ’00• ” J.D.E. ’02• Chueshov, Eller, Lasiecka Comm. P.D.E. ’02• Cavalcanti, D. Cavalcanti, Martinez J.D.E. ’04• Cavalcanti, D. Cavalcanti, Lasiecka J.D.E. ’07• Bociu, Lasiecka Nonlinear Anal. ’08• ” D.C.D.S. A ’08• ” J.D.E. ’10• Bociu, Rammaha, Toundykov Math. Nachr. ’11• ” Math. Comp. Simul. ’12• Bociu, Toundykov J.D.E. ’12• ” Palest. J. Math. ’13

Kinetic and hyperbolic boundary conditions

• Kirane Hokkaido Math. J. ’92• Conrad, Morgul, Rao, I.E.E.E. Trans. Automat. Control ’94• Andrews, Kuttler, Shillor J. Math. Anal. Appl. ’96• Conrad, Morgul Siam J. Control Optim. ’98• Doronin, Lar’kin, Souza Electron. J.D.E. ’98• Guo, Xu I.E.E.E. Trans. Automat. Control ’00• Darmawijoyo, van Horssen Nonlin, Dynam. ’02• Doronin, Lar’kin Nonlinear Anal. ’02• Xiao, Liang J. D. E. & Trans. A.M.S. ’04• Vazquez, Vitillaro Math. Mod. Meth. Appl. Sci. ’08• Jameson Graber, Said–Houari Appl. Math. Optim. ’12• Jameson Graber J. Evol. Eqs. ’12• Vitillaro Contemp. Math. ’13• Lasiecka, Fourrier Evol. Equ. Control Theory ’13• Jameson Graber, Lasiecka Semigroup Forum ’14• Zahn Arxiv ’15• Figotin, Reyes J. Math. Phys. ’15

Kinetic and hyperbolic boundary conditions

• Kirane Hokkaido Math. J. ’92• Conrad, Morgul, Rao, I.E.E.E. Trans. Automat. Control ’94• Andrews, Kuttler, Shillor J. Math. Anal. Appl. ’96• Conrad, Morgul Siam J. Control Optim. ’98• Doronin, Lar’kin, Souza Electron. J.D.E. ’98• Guo, Xu I.E.E.E. Trans. Automat. Control ’00• Darmawijoyo, van Horssen Nonlin, Dynam. ’02• Doronin, Lar’kin Nonlinear Anal. ’02• Xiao, Liang J. D. E. & Trans. A.M.S. ’04• Vazquez, Vitillaro Math. Mod. Meth. Appl. Sci. ’08• Jameson Graber, Said–Houari Appl. Math. Optim. ’12• Jameson Graber J. Evol. Eqs. ’12• Vitillaro Contemp. Math. ’13• Lasiecka, Fourrier Evol. Equ. Control Theory ’13• Jameson Graber, Lasiecka Semigroup Forum ’14• Zahn Arxiv ’15• Figotin, Reyes J. Math. Phys. ’15

3. Preliminaries

Nonlinearities

P(v) ' a|v |m−2v + b|v |m−2v , 1 < m ≤ m, a ≥ 0, b > 0

Q(v) ' c|v |µ−2v + d |v |µ−2v , 1 < µ ≤ µ, c ≥ 0, d > 0

f (u) ' γ|u|p−2u + γ|u|p−2u + c1, 2 ≤ p ≤ p, γ, γ, c1 ∈ R

g(u) ' δ|u|q−2u + δ|u|q−2u + c2, 2 ≤ q ≤ q, δ, δ, c2 ∈ R

N−2 if N ≥ 3,

∞ if N = 2,r Γ =

2(N−1)N−3 if N ≥ 4,

∞ if N = 2, 3.

3. Preliminaries

Nonlinearities

P(v) ' a|v |m−2v + b|v |m−2v , 1 < m ≤ m, a ≥ 0, b > 0

Q(v) ' c|v |µ−2v + d |v |µ−2v , 1 < µ ≤ µ, c ≥ 0, d > 0

f (u) ' γ|u|p−2u + γ|u|p−2u + c1, 2 ≤ p ≤ p, γ, γ, c1 ∈ R

g(u) ' δ|u|q−2u + δ|u|q−2u + c2, 2 ≤ q ≤ q, δ, δ, c2 ∈ R

N−2 if N ≥ 3,

∞ if N = 2,r Γ =

2(N−1)N−3 if N ≥ 4,

∞ if N = 2, 3.

Sources classification

(i) f is subcritical when p ≤ 1 +rΩ2 , that is when the Nemitskii

operator u 7→ f (u) is locally Lischitz from H1(Ω) to L2(Ω)(in terms of Sobolev embedding it is sub–subcritical).

(ii) f is supercritical when 1 +rΩ2 < p ≤ rΩ, that is when u 7→ f (u)

is not anymore locally Lipschitz but it as a potential in H1(Ω)(in terms of Sobolev embedding it is subcritical–critical).

(iii) f is super–supercritical when p > rΩ, that is when u 7→ f (u)does not have any potential(in terms of Sobolev embedding it is supercritical).

Analogous classification for g .

The subcritical case: f and g are subcritical.

The supercritical case: f and g are subcritical or supercritical,but for the previous case.

The super–supercritical case: the remaining case.

In the sequel we shall distinguish between the subcritical case andthe others ones.

Main assumptions (to skip in the subcritical case)

p > 1 +rΩ2 ⇒ infΩ α > 0, q > 1 +

rΓ2 ⇒ infΓ1 β > 0;

2 ≤ p ≤ 1 +rΩ

m′, 2 ≤ q ≤ 1 +

µ′(ρ = maxρ, 2 for ρ > 1);

Regions when N=3

Regions when N=4

Ω r1+ /2=3

Ω r =4

Ω r +1=5 r1+ /2=4Γ r =6Γ r +1=7Γ

Ω r1+ /2=4

Ω r =6

Ω r +1=7

green: subcriticalyellow: supercriticalred: super-supercritical

if 1 +rΩ2 < p = 1 + rΩ/m

′ then N ≤ 4 and p > 3 or γ = 0;

if 1 +rΓ2 < q = 1 + r Γ/µ

′ then N ≤ 5 and q > 3 or δ = 0.

(dimension restrictions on the critical hyperbolas)

Add–on assumption (to skip in the subcritical case)

(S) if p > 1 +rΩ2 then N ≤ 4 and p > 3 or γ = 0;

if q > 1 +rΓ2 then N ≤ 5 and q > 3 or δ = 0.

(dimension restrictions for non–subcritical sources)

if 1 +rΩ2 < p = 1 + rΩ/m

′ then N ≤ 4 and p > 3 or γ = 0;

if 1 +rΓ2 < q = 1 + r Γ/µ

′ then N ≤ 5 and q > 3 or δ = 0.

(dimension restrictions on the critical hyperbolas)

Add–on assumption (to skip in the subcritical case)

(S) if p > 1 +rΩ2 then N ≤ 4 and p > 3 or γ = 0;

if q > 1 +rΓ2 then N ≤ 5 and q > 3 or δ = 0.

(dimension restrictions for non–subcritical sources)

Functional spaces

L2(Γ1) = u ∈ L2(Γ) : u = 0 a.e. on Γ0,H0 = L2(Ω)× L2(Γ1),

H1 = (u, v) ∈ [H1(Ω)× H1(Γ)] ∩ H0 : v = u|Γ,L2,ρα (Ω) = u ∈ L2(Ω) : α|u|ρ ∈ L1(Ω), 2 ≤ ρ <∞,

L2,θβ (Γ1) = u ∈ L2(Γ1) : β|u|θ ∈ L1(Γ1), 2 ≤ θ <∞,

Z(0,T ) = Lm(0,T ; L2,mα (Ω))× Lµ(0,T ; L2,µ

β (Γ1)),T > 0;

H1,ρ,θα,β = H1 ∩ [L2,ρ

α (Ω)× L2,θβ (Γ1)], H1,ρ,θ = H1,ρ,θ

By Sobolev embeddings and our main assumptions

H1,ρ,θα,β = H1,ρ,θ = H1 if ρ ≤ rΩ and θ ≤ r Γ

H1,ρ,θα,β = H1,ρ,θ if ρ ≤ rΩ or p > 1 + rΩ/2 and θ ≤ r Γ or q > 1 + r Γ/2

Functional spaces

L2(Γ1) = u ∈ L2(Γ) : u = 0 a.e. on Γ0,H0 = L2(Ω)× L2(Γ1),

H1 = (u, v) ∈ [H1(Ω)× H1(Γ)] ∩ H0 : v = u|Γ,L2,ρα (Ω) = u ∈ L2(Ω) : α|u|ρ ∈ L1(Ω), 2 ≤ ρ <∞,

L2,θβ (Γ1) = u ∈ L2(Γ1) : β|u|θ ∈ L1(Γ1), 2 ≤ θ <∞,

Z(0,T ) = Lm(0,T ; L2,mα (Ω))× Lµ(0,T ; L2,µ

β (Γ1)),T > 0;

H1,ρ,θα,β = H1 ∩ [L2,ρ

α (Ω)× L2,θβ (Γ1)], H1,ρ,θ = H1,ρ,θ

By Sobolev embeddings and our main assumptions

H1,ρ,θα,β = H1,ρ,θ = H1 if ρ ≤ rΩ and θ ≤ r Γ

H1,ρ,θα,β = H1,ρ,θ if ρ ≤ rΩ or p > 1 + rΩ/2 and θ ≤ r Γ or q > 1 + r Γ/2

Weak solutionsFor any (u0, u1) ∈ H1 × H0 a weak solution of problem (P) in[0,T ], T > 0, is u = (u, u|Γ) ∈ L∞(0,T ; H1) ∩W 1,∞(0,T ; H0),u′ = (ut , (u|Γ)t) ∈ Z(0,T ), satisfying (P) in a suitable distributionalsense. A weak solution in [0,T ) is a weak solution in [0,T ′] for anyT ′ ∈ (0,T ).

Any weak solution u of problem (P) satisfies

u ∈ C([0,T ]; H1) ∩ C1([0,T ]; H0), for all T ∈ dom u,

12‖u′‖2

H0 + 12‖∇u‖2

2 + 12‖∇Γu‖2

∣∣∣ts+

[∫ΩαP(ut)ut

∫ΓβQ((u|Γ)t)u|Γ)t −

f (u)ut −∫

Γg(u)(u|Γ)t

Weak solutionsFor any (u0, u1) ∈ H1 × H0 a weak solution of problem (P) in[0,T ], T > 0, is u = (u, u|Γ) ∈ L∞(0,T ; H1) ∩W 1,∞(0,T ; H0),u′ = (ut , (u|Γ)t) ∈ Z(0,T ), satisfying (P) in a suitable distributionalsense. A weak solution in [0,T ) is a weak solution in [0,T ′] for anyT ′ ∈ (0,T ).

Any weak solution u of problem (P) satisfies

u ∈ C([0,T ]; H1) ∩ C1([0,T ]; H0), for all T ∈ dom u,

12‖u′‖2

H0 + 12‖∇u‖2

2 + 12‖∇Γu‖2

∣∣∣ts+

[∫ΩαP(ut)ut

∫ΓβQ((u|Γ)t)u|Γ)t −

f (u)ut −∫

Γg(u)(u|Γ)t

4. The subcritical case: 2 ≤ p ≤ 1 + rΩ/2, 2 ≤ q ≤ 1 + r Γ/2

Theorem (Local Hadamard well-posedness)

Given U0 ∈ H1 and U1 ∈ H0 problem (P) has a unique maximalweak solution u in [0,Tmax). Moreover

limt→T−max

‖U(t)‖H1 + ‖U ′(t)‖H0 =∞ provided Tmax <∞.

Next u continuously depends on the initial data in H1 × H0.For any couple of exponents

(ρ, θ) ∈ [2,maxrΩ,m]× [2,maxr Γ, µ] ∩ R2

and any (u0, u1) ∈ H1,ρ,θα,β × H0, the weak solution u of problem (P)

enjoys the further regularity u ∈ C([0,Tmax); H1,ρ,θα,β ).

Finally u continuously depends on the initial data in H1,ρ,θα,β × H0.

Some regularity results...

limt→T−max

Finite time blow–up is possible!

Set F(u) =∫ u

0 f (s) ds and G(u) =∫ u

0 g(s) ds and

E(u0, u1) =1

2‖u1‖2

2‖∇u0‖2

2‖∇Γu0‖2

−∫

ΩF (u0)−

∫Γ1

G(u0) .

Theorem (Blow–up for linear damping terms)

Suppose that P(v) = v , Q(v) = v , p, q > 2 and

γ, γ ≥ 0, δ, δ ≥ 0, γ + γ + δ + δ > 0.

Then if E(u0, u1) < 0 (such data exist) we have Tmax <∞ and

limt→T−max

‖u(t)‖H1 + ‖u′(t)‖H0 = limt→T−max

‖u(t)‖pp + ‖u(t)‖qq,Γ1=∞.

Set F(u) =∫ u

0 g(s) ds and

E(u0, u1) =1

2‖u1‖2

2‖∇u0‖2

2‖∇Γu0‖2

−∫

ΩF (u0)−

∫Γ1

G(u0) .

γ, γ ≥ 0, δ, δ ≥ 0, γ + γ + δ + δ > 0.

limt→T−max

‖u(t)‖pp + ‖u(t)‖qq,Γ1=∞.

Set F(u) =∫ u

0 g(s) ds and

E(u0, u1) =1

2‖u1‖2

2‖∇u0‖2

2‖∇Γu0‖2

−∫

ΩF (u0)−

∫Γ1

G(u0) .

γ, γ ≥ 0, δ, δ ≥ 0, γ + γ + δ + δ > 0.

limt→T−max

‖u(t)‖pp + ‖u(t)‖qq,Γ1=∞.

Global existence assumption

(GE) One among the following cases occurs for f :1 γ > 0, infΩ α > 0 if p > 2 and p ≤ p ≤ max2,m;2 γ ≤ 0, γ > 0, infΩ α > 0 if p > 2 and p ≤ max2,m;3 γ, γ ≤ 0.

The analogous cases apply to g .

Theorem (Global existence and dynamical system generation)

Suppose that (GE) holds. Then for any (u0, u1) ∈ H1 × H0 theunique maximal weak solution of (P) is global in time, that isTmax =∞.Consequently the semi–flow generated by (P) is a dynamical system

in H1 × H0 and in H1,ρ,θα,β × H0 for

(ρ, θ) ∈ [2,maxrΩ,m]× [2,maxr Γ, µ] ∩ R2.

5. The supercritical and super–supercritical cases

We set

σΩ =

(p−2)rΩrΩ−2 if rΩ < p = 1 +

rΩm′ ,

2 otherwise,

σΓ =

(q−2)rΓrΓ−2 if r Γ < q = 1 +

rΓµ′ ,

2 otherwise,

lΩ :=

σΩ if rΩ < p = 1 +

rΩm′ ,

p if rΩ < p < 1 +rΩm′ ,

2 if p ≤ rΩ,

lΓ :=

σΓ if r Γ < q = 1 +

rΓµ′ ,

q if r Γ < q < 1 +rΓµ′ ,

2 if q ≤ r Γ.

ClearlyH1,lΩ ,lΓ → H1,σΩ ,σΓ → H1

and H1,lΩ ,lΓ = H1,σΩ ,σΓ = H1 provided 2 ≤ p ≤ rΩ, 2 ≤ q ≤ r Γ.

5. The supercritical and super–supercritical cases

We set

σΩ =

(p−2)rΩrΩ−2 if rΩ < p = 1 +

rΩm′ ,

2 otherwise,

σΓ =

(q−2)rΓrΓ−2 if r Γ < q = 1 +

rΓµ′ ,

2 otherwise,

lΩ :=

σΩ if rΩ < p = 1 +

rΩm′ ,

p if rΩ < p < 1 +rΩm′ ,

2 if p ≤ rΩ,

lΓ :=

σΓ if r Γ < q = 1 +

rΓµ′ ,

q if r Γ < q < 1 +rΓµ′ ,

2 if q ≤ r Γ.

ClearlyH1,lΩ ,lΓ → H1,σΩ ,σΓ → H1

and H1,lΩ ,lΓ = H1,σΩ ,σΓ = H1 provided 2 ≤ p ≤ rΩ, 2 ≤ q ≤ r Γ.

Theorem (Local existence, continuation, global existence)

1 For any u0 ∈ H1,σΩ ,σΓ , u1 ∈ H0 problem (P) has a maximalweak solution u ∈ C([0,Tmax); H1,σΩ ,σΓ ).Moreover

limt→T−max

‖u(t)‖H1 + ‖u′(t)‖H0 =∞ provided Tmax <∞.

2 If (GE) holds for any u0 ∈ H1,lΩ ,lΓ we have Tmax =∞ andu ∈ C([0,∞); H1,lΩ ,lΓ ).

3 Previous conclusions hold if H1,σΩ ,σΓ and H1,lΩ ,lΓ are replaced byH1,ρ,θ provided

ρ ≤ rΩ or p > 1 + rΩ/2 , θ ≤ r Γ or q > 1 + r Γ/2

and respectively:1 (ρ, θ) ∈ [σΩ ,maxrΩ ,m]× [σΓ ,maxr Γ , µ] ∩ R2,

2 (ρ, θ) ∈ [lΩ ,maxrΩ ,m]× [lΓ ,maxr Γ , µ] ∩ R2.

limt→T−max

sΩ :=

(p−2)rΩrΩ−2 p > rΩ,

2, p ≤ rΩ,sΓ :=

(q−2)rΓrΓ−2 q > r Γ,

2, q ≤ r Γ,

ClearlyH1,sΩ ,sΓ → H1,lΩ ,lΓ → H1,σΩ ,σΓ → H1

and H1,sΩ ,sΓ = H1 provided 2 ≤ p ≤ rΩ, 2 ≤ q ≤ r Γ.

sΩ :=

(p−2)rΩrΩ−2 p > rΩ,

2, p ≤ rΩ,sΓ :=

(q−2)rΓrΓ−2 q > r Γ,

2, q ≤ r Γ,

ClearlyH1,sΩ ,sΓ → H1,lΩ ,lΓ → H1,σΩ ,σΓ → H1

and H1,sΩ ,sΓ = H1 provided 2 ≤ p ≤ rΩ, 2 ≤ q ≤ r Γ.

Theorem (Existence and uniqueness)

Under the assumption (S) the following conclusions hold:

1 for any u0 ∈ H1,sΩ,sΓ, u1 ∈ H0 problem (1) has a uniquemaximal weak solution u in [0,Tmax).Moreover

u ∈ C([0,Tmax); H1,sΩ ,sΓ ),

limt→T−max

‖u(t)‖H1 +∥∥u′(t)

∥∥H0 =∞,

limt→T−max

‖u(t)‖H1,sΩ ,sΓ +∥∥u′(t)

∥∥H0 =∞

provided Tmax <∞;

2 if also (GE) holds for any u0 ∈ H1,sΩ ,sΓ we have Tmax =∞;

3 previous conclusions hold if H1,sΩ ,sΓ is replaced by H1,ρ,θ

provided ρ ≤ rΩ or p > 1 + rΩ/2 , θ ≤ r Γ or q > 1 + r Γ/2 and

(ρ, θ) ∈ [sΩ,maxrΩ,m]× [sΓ,maxr Γ, µ] ∩ R2.

u ∈ C([0,Tmax); H1,sΩ ,sΓ ),

limt→T−max

‖u(t)‖H1 +∥∥u′(t)

∥∥H0 =∞,

limt→T−max

‖u(t)‖H1,sΩ ,sΓ +∥∥u′(t)

∥∥H0 =∞

provided Tmax <∞;

u ∈ C([0,Tmax); H1,sΩ ,sΓ ),

limt→T−max

‖u(t)‖H1 +∥∥u′(t)

∥∥H0 =∞,

limt→T−max

‖u(t)‖H1,sΩ ,sΓ +∥∥u′(t)

∥∥H0 =∞

provided Tmax <∞;

u ∈ C([0,Tmax); H1,sΩ ,sΓ ),

limt→T−max

‖u(t)‖H1 +∥∥u′(t)

∥∥H0 =∞,

limt→T−max

‖u(t)‖H1,sΩ ,sΓ +∥∥u′(t)

∥∥H0 =∞

provided Tmax <∞;

We now restrict to

(p,m) 6= (rΩ, rΩ), (q, µ) 6= (r Γ, r Γ).

In this case

p ≥ rΩ ⇒ rΩ ≤ sΩ < m, and q ≥ r Γ ⇒ r Γ ≤ sΓ < µ,

and then there exist exponents

s1 ∈

(sΩ, rΩ], if p < rΩ,

(sΩ,m], otherwise,s2 ∈

(sΓ, r Γ], if q < r Γ,

(sΓ, µ], otherwise.

Theorem (Hadamard well–posedness )

Let (S) hold and p,m, q, µ, s1, s2 satisfy previous relations. Then

1 problem (P) is locally well–posed in H1,s1,s2 × H0;

2 if also (GE) holds problem (P) generates a dynamical system init;

3 In particular when 2 ≤ p < rΩ, 2 ≤ q < r Γ, previousstatements hold in H1 × H0.

We now restrict to

(p,m) 6= (rΩ, rΩ), (q, µ) 6= (r Γ, r Γ).

In this case

s1 ∈

We now restrict to

(p,m) 6= (rΩ, rΩ), (q, µ) 6= (r Γ, r Γ).

In this case

s1 ∈

We now restrict to

(p,m) 6= (rΩ, rΩ), (q, µ) 6= (r Γ, r Γ).

In this case

s1 ∈

We now restrict to

(p,m) 6= (rΩ, rΩ), (q, µ) 6= (r Γ, r Γ).

In this case

s1 ∈

Summaries when N = 3, 4

6. Conclusion

Further developments

1 Blow–up results when the damping terms are nonlinear.

2 An optimal potential well theory.

3 When the internal border of the drum is metallic one has toreplace −∆Γ with (−∆Γ)2.

Open problems

1 All the uncovered cases.

2 Remove the assumption

2 ≤ p ≤ 1 +rΩ

m′, 2 ≤ q ≤ 1 +

µ′.

6. Conclusion

Open problems

2 ≤ p ≤ 1 +rΩ

m′, 2 ≤ q ≤ 1 +

µ′.

6. Conclusion

Open problems

2 ≤ p ≤ 1 +rΩ

m′, 2 ≤ q ≤ 1 +

µ′.

6. Conclusion

Open problems

2 ≤ p ≤ 1 +rΩ

m′, 2 ≤ q ≤ 1 +

µ′.

6. Conclusion

Open problems

2 ≤ p ≤ 1 +rΩ

m′, 2 ≤ q ≤ 1 +

µ′.

Thank you very much for your attention!

On the wave equation with hyperbolic dynamical boundary ... · On the wave equation with hyperbolic...

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