Semi-classical defect measure and internal stabilization for the semilinear wave equation subject to Zarembaboundary conditions

TL;DR

This paper extends the uniform decay results from linear to nonlinear semilinear wave equations with Zaremba boundary conditions, using a contradiction argument based on an observation estimate, under a uniqueness assumption.

Contribution

It introduces a method to analyze internal stabilization of nonlinear wave equations with Zaremba boundary conditions, building on previous linear decay results.

Findings

Established decay estimates for nonlinear wave equations with Zaremba boundary conditions

Demonstrated the effectiveness of contradiction arguments in nonlinear stabilization

Extended linear decay results to nonlinear contexts under a uniqueness assumption

Abstract

In this article we exploite the uniform decay for damped linear wave equation with Zaremba boundary condition, obtained in a previous work, to treat the same problem in nonlinear context. We need a uniqueness assumption, usual for this type of nonlinear problem. The result is deduced from an observation estimate for nonlinear problem proved by a contradiction argument.