Uniform stabilization for the semi-linear wave equation with nonlinear Kelvin-Voigt damping

TL;DR

This paper establishes uniform energy decay estimates for solutions to a semilinear wave equation with localized nonlinear Kelvin-Voigt damping and frictional damping, using microlocal analysis and observability inequalities.

Contribution

It introduces a novel approach to prove uniform decay rates for semilinear wave equations with mixed damping types, relaxing geometric control conditions.

Findings

Proved uniform decay rates for the energy of solutions.

Established observability inequalities combining unique continuation and microlocal analysis.

Demonstrated decay results under less restrictive geometric conditions.

Abstract

This paper is concerned with the decay estimate of solutions to the semilinear wave equation subject to two localized dampings in a bounded domain. The first one is of the nonlinear Kelvin-Voigt type and is distributed around a neighborhood of the boundary according to the Geometric Control Condition. While the second one is a frictional damping and we consider it hurting the geometric condition of control. We show uniform decay rate results of the corresponding energy for all initial data taken in bounded sets of finite energy phase-space. The proof is based on obtaining an observability inequality which combines unique continuation properties and the tools of the Microlocal Analysis Theory.