Upload
others
View
6
Download
0
Embed Size (px)
Citation preview
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
(P)
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.
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 2 / 25
1. The problem.
E. Vitillaro, Arch. Rat. Mech. Anal. 2017
” , arXiv 2016
(P)
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.
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 2 / 25
1. The problem.
E. Vitillaro, Arch. Rat. Mech. Anal. 2017
” , arXiv 2016
(P)
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.
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 2 / 25
The two–dimensional model: a bass drum
Lagrangian function (∇Γ Riemannian gradient):
L(u) =1
2
∫Ω
[u2
t − |∇u|2 − 2F(u)]
+1
2
∫Γ1
[u2
t − |∇Γu|2 − 2G(u)]
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 3 / 25
The two–dimensional model: a bass drum
Lagrangian function (∇Γ Riemannian gradient):
L(u) =1
2
∫Ω
[u2
t − |∇u|2 − 2F(u)]
+1
2
∫Γ1
[u2
t − |∇Γu|2 − 2G(u)]
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 3 / 25
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
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 4 / 25
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
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 5 / 25
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
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 6 / 25
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
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 7 / 25
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
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 7 / 25
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
Set
rΩ =
2N
N−2 if N ≥ 3,
∞ if N = 2,r Γ =
2(N−1)N−3 if N ≥ 4,
∞ if N = 2, 3.
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 8 / 25
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
Set
rΩ =
2N
N−2 if N ≥ 3,
∞ if N = 2,r Γ =
2(N−1)N−3 if N ≥ 4,
∞ if N = 2, 3.
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 8 / 25
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.
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 9 / 25
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.
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 9 / 25
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.
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 9 / 25
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.
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 9 / 25
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 +
r Γ
µ′(ρ = maxρ, 2 for ρ > 1);
p
m
2
1
2
4
q
μ
Regions when N=3
Regions when N=4
2
3
2
1
2q
μ
1
2
4
2
1
m
p
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
q2
1
μ
2
1
p
m
green: subcriticalyellow: supercriticalred: super-supercritical
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 10 / 25
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)
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 11 / 25
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)
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 11 / 25
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
1,1 ,
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
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 12 / 25
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
1,1 ,
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
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 12 / 25
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 ).
Lemma
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
2,Γ
∣∣∣ts+
∫ t
s
[∫ΩαP(ut)ut
+
∫ΓβQ((u|Γ)t)u|Γ)t −
∫Ω
f (u)ut −∫
Γg(u)(u|Γ)t
]= 0.
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 13 / 25
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 ).
Lemma
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
2,Γ
∣∣∣ts+
∫ t
s
[∫ΩαP(ut)ut
+
∫ΓβQ((u|Γ)t)u|Γ)t −
∫Ω
f (u)ut −∫
Γg(u)(u|Γ)t
]= 0.
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 13 / 25
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...
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 14 / 25
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...
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 14 / 25
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...
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 14 / 25
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...
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 14 / 25
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...
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 14 / 25
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...
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 14 / 25
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
H1 +1
2‖∇u0‖2
2 +1
2‖∇Γu0‖2
2,Γ+
−∫
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=∞.
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 15 / 25
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
H1 +1
2‖∇u0‖2
2 +1
2‖∇Γu0‖2
2,Γ+
−∫
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=∞.
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 15 / 25
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
H1 +1
2‖∇u0‖2
2 +1
2‖∇Γu0‖2
2,Γ+
−∫
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=∞.
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 15 / 25
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.
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 16 / 25
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.
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 16 / 25
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.
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 16 / 25
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.
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 16 / 25
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.
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 16 / 25
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.
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 16 / 25
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.
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 17 / 25
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.
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 17 / 25
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.
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 18 / 25
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.
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 18 / 25
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.
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 18 / 25
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.
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 18 / 25
Set
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.
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 19 / 25
Set
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.
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 19 / 25
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.
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 20 / 25
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.
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 20 / 25
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.
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 20 / 25
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.
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 20 / 25
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.
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 21 / 25
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.
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 21 / 25
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.
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 21 / 25
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.
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 21 / 25
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.
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 21 / 25
Summaries when N = 3, 4
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 22 / 25
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 23 / 25
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 +
r Γ
µ′.
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 24 / 25
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 +
r Γ
µ′.
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 24 / 25
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 +
r Γ
µ′.
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 24 / 25
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 +
r Γ
µ′.
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 24 / 25
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 +
r Γ
µ′.
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 24 / 25
Thank you very much for your attention!
Enzo Vitillaro (Univ. Perugia) On the wave equation with hyperbolic ... September 8, 2017 25 / 25