Decay of solutions of the wave equation with a local highly degenerate dissipation

Article type: Research Article

Authors: Martinez, Patrick

Affiliations: Institut de Recherche Mathématique Avancée, Université Louis Pasteur, 7, rue René Descartes, 67084 Strasbourg Cédex, France E‐mail: [email protected]‐strasbg.fr

Abstract: We consider the system of the wave equation with the Dirichlet boundary condition damped with a local linear dissipation of the type a(x)u'. In order to simplify, we assume that the domain is the open ball of \mathbf{R}^N centered in O and of radius R and that the function a is radial near the boundary. When a goes quickly to zero at the boundary, we obtain a precise decay rate estimate of the energy of the solutions. This extends some results of Zuazua and Nakao. The proof is based on a new nonlinear integral inequality and on the asymptotic behavior of the function a at the boundary.

Journal: Asymptotic Analysis, vol. 19, no. 1, pp. 1-17, 1999