Optimal decay rates for solutions of a nonlinear wave equation with localized nonlinear dissipation of unrestricted growth and critical exponent source terms under mixed boundary conditions

Abstract

This article investigates optimal decay rates for solutions to a semilinear hyperbolic equation with localized interior damping and a source term. Both dissipation and the source are fully nonlinear and the growth rate of the source map may include critical exponents (for Sobolev’s embedding H 1 → L 2 ). Besides continuity and monotonicity, no growth or regularity assumptions are imposed on the damping. We analyze the system in the presence of Neumann-type boundary conditions including the mixed cases: Dirichlet–Neumann–Robin. The damping affects a thin layer (a collar) near a portion of the boundary. To cope with the lack of control on the remaining section we develop a special method that accounts for propagation of the energy estimates from the dissipative region onto the entire domain. The Neumann system does not satisfy the Lopatinski condition in higher dimensions, hence the study of energy propagation in the absence of damping near the Neumann segment requires special geometric considerations.