Stabilization and control for the subcritical semilinear wave equation

Abstract

In this paper, we analyze the exponential decay property of solutions of the semilinear wave equation in R 3 with a damping term which is effective on the exterior of a ball. Under suitable and natural assumptions on the nonlinearity we prove that the exponential decay holds locally uniformly for finite energy solutions provided the nonlinearity is subcritical at infinity. Subcriticality means, roughly speaking, that the nonlinearity grows at infinity at most as a power p < 5. The method of proof combines classical energy estimates for the linear wave equation allowing to estimate the total energy of solutions in terms of the energy localized in the exterior of a ball, Strichartz's estimates and results by P. Gérard on microlocal defect measures and linearizable sequences. We also give an application to the stabilization and controllability of the semilinear wave equation in a bounded domain under the same growth condition on the nonlinearity but provided the nonlinearity has been cutoff away from the boundary.  2003 Éditions scientifiques et médicales Elsevier SAS RÉSUMÉ.-Nous étudions dans cet article la décroissance exponentielle de l'énergie pour une équation d'ondes semi-linéaire dans R 3 , avec un terme d'amortissement effectif à l'extérieur d'une boule. En supposant la non linéarité sous critique et vérifiant certaines conditions naturelles, nous obtenons un résultat de stabilisation locale, c'est-à-dire une décroissance exponentielle de l'énergie, uniforme sur les boules de l'espace d'énergie où sont choisies les données initiales. La démonstration repose sur des inégalités d'énergie classiques qui estiment l'énergie totale en fonction de l'énergie localisée à l'extérieur d'une boule. Elle utilise aussi les estimations de Strichartz et les résultats de P. Gérard sur les mesures de défaut microlocales et les suites linéarisables. Nous donnons aussi, en application, un résultat de stabilisation et de contrôle pour l'équation des ondes semi-linéaire sur un ouvert borné, avec une non linéarité sous critique, tronquée loin du bord.  2003 Éditions scientifiques et médicales Elsevier SAS 1 Supported by grant BFM2002-03345 of the MCYT (Spain) and the Networks "Homogeneization and Multiple Scales" and "New materials, adaptive systems and their nonlinearities: modelling, control and numerical simulation (HPRN-CT-2002-00284)" of the EU.

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

“…In some sense, it is a high frequency controllability result and expresses in a rough physical way that we can control some "small noisy data". In the subcritical case, two similar kind of results were proved : in Dehman-Lebeau-Zuazua [11] similar results were proved for the nonlinear wave equation but without the smallness assumption in L 2 ×H −1 while in Dehman-Lebeau [10], they obtain similar high frequency controllability results for the subcritical equation but in a uniform time which is actually the time of linear controllability (see also the work of the author [21] for the Schrödinger equation). Actually, this smallness assumption is made necessary in our proof because we are not able to prove in general the following unique continuation result.…”

“…It mainly describes the proof contained in Dehman-Lebeau-Zuazua [11]. We also emphasize which points fail in the critical case, that we will describe in the next section.…”

“…Section 2 is devoted to a description of the proof of Dehman-Lebeau-Zuazua [11] of the exponential decay in the subcritical case, stressing the new difficulties that will appear in the critical case. Section 3 is devoted to a sketch of the proof of Theorem 1.2 and of the profile decomposition that it requires.…”

“…There are many results in that direction. The more advanced in the subcritical case is certainly the work of Dehman-Lebeau-Zuazua [11]. We will describe its proof in section 2 below.…”

A Carleman estimates based approach for the stabilization of some locally damped semilinear hyperbolic equations

“…Later on, Dehman [7] reduced the two proofs in [45] to a single proof, but for initial data that are bounded in the energy space. Subsequently, using Strichartz dispersive inequalities, the results of [7,45,46] were improved to include all the subcritical nonlinearities f in the three dimensional setting, meaning that q < 4 in (1.1), by Dehman, Lebeau and Zuazua in [8]. It is also of interest to mention Nakao's papers [29,30], where the authors discuss the same type of questions for systems involving nonlinearities of the form f (x, s) -that are bounded in x -and nonlinear damping locally distributed on a neighborhood of a suitable subset of the boundary; they establish polynomial and exponential energy decay estimates for small enough initial data.…”