Niveau: Supérieur, Master, Bac+4
MATHEMATICAL CONTROL AND doi:10.3934/mcrf.2011.1.251 RELATED FIELDS Volume 1, Number 2, June 2011 pp. 251–265 DECAY OF SOLUTIONS OF THE WAVE EQUATION WITH LOCALIZED NONLINEAR DAMPING AND TRAPPED RAYS Kim Dang Phung Yangtze Center of Mathematics, Sichuan University Chengdu 610064, China (Communicated by Sylvain Ervedoza) Abstract. We prove some decay estimates of the energy of the wave equation governed by localized nonlinear dissipations in a bounded domain in which trapped rays may occur. The approach is based on a comparison with the linear damped wave equation and an interpolation argument. Our result extends to the nonlinear damped wave equation the well-known optimal logarithmic decay rate for the linear damped wave equation with regular initial data. 1. Introduction. We study a nonlinear wave equation in a bounded connected open set ? of Rn, n > 1 with a C2 boundary ∂?. Let M = ( ?ij ) 1≤i,j≤n ? C∞ ( ?;Rn?n ) be a symmetric and uniformly positive definite matrix. Therefore ( ?ij ) 1≤i,j≤n =M1/2 is well defined. Denote ? = (∑n j=1 ? 1j∂xj , . . . , ∑n j=1 ? nj∂xj ) and by ∆ = ∑n i,j=1 ∂xi(? ij∂xj ) the “Laplacian” associated to the matrix M. We deal with the following damped wave equation ? ?? ?? ∂2t u?∆u+ ag (∂tu) = 0 in ?? (0,+∞)
- wave equation
- holds when
- well-known optimal
- preliminary estimates
- trapped rays
- polynomial decay
- optimal logarithmic
- localized nonlinear