Stabilization for the semilinear wave equation with geometric control condition

Joly, Romain; Laurent, Camille

doi:10.2140/apde.2013.6.1089

Cited by 49 publications

(70 citation statements)

References 63 publications

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“…This seems hard to prove for finite times, since the propagation of analytic microlocal regularity is often more complicated to obtain for nonlinear equations. To overcome this, in a forthcoming article with R. Joly [18], we prove such analyticity in time for a solution satisfying ∂ t u = 0 on R × ω with a subcritical analytic nonlinearity. This would then give some unique continuation result in infinite time, which is sufficient for stabilization.…”

Section: A Few Comments About Unique Continuationmentioning

confidence: 99%

On stabilization and control for the critical Klein-Gordon equation on a 3-D compact manifold

Laurent

2012

Journées Équations Aux Dérivées Partielles

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RésuméOn étudie la stabilisation et le contrôle interne de l'équation de KleinGordon critique sur des variétés de dimension 3. Sous des conditions géomé-triques légèrement plus fortes que la condition de contrôle géométrique classique, on prouve la décroissance exponentielle de solutions bornées dans l'espace d'énergie mais petites dans des normes plus faibles. La preuve combine la décomposition en profils et des arguments microlocaux. Cette décomposition, analogue à celle de Bahouri-Gérard [2] sur R 3 , nécessite l'analyse de certains effets dus à la géométrie. Elle utilise des résultats de S. Ibrahim [16] sur le comportement d'ondes de concentration sur les variétés. AbstractWe study the internal stabilization and control of the critical nonlinear Klein-Gordon equation on 3-D compact manifolds. Under a geometric assumption slightly stronger than the classical geometric control condition, we prove exponential decay for some solutions bounded in the energy space but small in a lower norm. The proof combines profile decomposition and microlocal arguments. This profile decomposition, analogous to the one of Bahouri-Gérard [2] on R 3 , is performed by taking care of possible geometric effects. It uses some results of S. Ibrahim [16] on the behavior of concentrating waves on manifolds.

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Section: A Few Comments About Unique Continuationmentioning

confidence: 99%

On stabilization and control for the critical Klein-Gordon equation on a 3-D compact manifold

Laurent

2012

Journées Équations Aux Dérivées Partielles

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“…They have shown that the local energy decays to zero in the sense that, under suitable assumptions, the energy of any solution escapes away from any compact set, see [18], [26] and [2] and the references therein. Secondly, several works have studied the damped wave equation in an unbounded manifold and with a non-linearity, but assuming that the damping satisfies γ(x) ≥ α > 0 outside a compact set, see [33], [10], [9] and [16].…”

Section: Some Previous Workmentioning

confidence: 99%

“…Another related problem is the asymptotic behaviour of the non-linear equation as studied in [33], [10], [9] or [16]. One considers the non-linear equation To each solution of (6.5), one can associate the energy…”

Section: Global Attractor and Stabilisation For The Non-linear Equationmentioning

confidence: 99%

Exponential decay for the damped wave equation in unbounded domains

Burq

Joly

2016

Commun. Contemp. Math.

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We study the decay of the semigroup generated by the damped wave equation in an unbounded domain. We first prove under the natural geometric control condition the exponential decay of the semigroup. Then we prove under a weaker condition the logarithmic decay of the solutions (assuming that the initial data are smoother). As corollaries, we obtain several extensions of previous results of stabilisation and control.Onétudie la décroissance du semi-groupe des ondes amorties dans un domaine non borné. Notre premier résultat est que, sous une hypothèse naturelle de contrôle géométrique, le semigroupe décroît exponentiellement vite. On démontre ensuite sous une hypothèse plus faible la décroissance logarithmique des solutions associéesà des données initiales plus régulières. On obtient en corollaire plusieurs généralisations de résultats de stabilisation et de contrôle.

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“…We refer to to the linear case and for the case of nonlinearities of order at most

p \leq 3

. The case

p \in [3, 5)

has been studied by Dehman, Lebeau and Zuazua in and Joly and Laurent in . The critical case

p = 5

has been investigated in .…”

Section: Introductionmentioning

confidence: 99%

Local uniform stability for the semilinear wave equation in inhomogeneous media with locally distributed Kelvin–Voigt damping

Astudillo

Cavalcanti

Fukuoka

et al. 2018

Mathematische Nachrichten

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We consider the semilinear wave equation posed in an inhomogeneous medium Ω with smooth boundary ∂Ω subject to a local viscoelastic damping distributed around a neighborhood ω of the boundary according to the Geometric Control Condition. We show that the energy of the wave equation goes uniformly and exponentially to zero for all initial data of finite energy taken in bounded sets of finite energy phase‐space. As far as we know, this is the first stabilization result for a semilinear wave equation with localized Kelvin–Voigt damping.

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Stabilization for the semilinear wave equation with geometric control condition

Cited by 49 publications

References 63 publications

On stabilization and control for the critical Klein-Gordon equation on a 3-D compact manifold

On stabilization and control for the critical Klein-Gordon equation on a 3-D compact manifold

Exponential decay for the damped wave equation in unbounded domains

Local uniform stability for the semilinear wave equation in inhomogeneous media with locally distributed Kelvin–Voigt damping

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