Uniform decay rate estimates for the coupled semilinear wave system in inhomogeneous media with locally distributed nonlinear damping

Article type: Research Article

Authors: Almeida, A.F.^{a; *} | Cavalcanti, M.M.^b | Gonzalez, R.B.^b | Gonzalez Martinez, V.H.^b | Zanchetta, J.P.^b

Affiliations: [a] Federal University of Latin American Integration, 85866-000, Foz do Iguaçu, PR, Brazil. E-mail: [email protected] | [b] Department of Mathematics, State University of Maringá, 87020-900, Maringá, PR, Brazil. E-mails: [email protected], [email protected], [email protected], [email protected]

Correspondence: [*] Corresponding author. E-mail: [email protected].

Abstract: We consider a coupled semilinear wave system posed in an inhomogeneous medium, with smooth boundary, subject to a nonlinear damping distributed around a neighborhood of the boundary according to the Geometric Control Condition. We show that the energy of the coupled system goes uniformly to zero, for all initial data of finite energy taken in bounded sets of finite energy phase-space. The approach involves refined techniques of microlocal analysis and follows ideas due to Burq and Gérard given in (Burq and Gérard (2001)).

Keywords: Coupled system, observability, uniform decay rate

DOI: 10.3233/ASY-191547

Journal: Asymptotic Analysis, vol. 117, no. 1-2, pp. 67-111, 2020