On the decay in $W^{1,\infty}$ for the 1D semilinear damped wave equation on a bounded domain

TL;DR

This paper investigates the decay behavior of solutions to a 1D semilinear damped wave equation with nonlinear damping, establishing well-posedness and exponential decay in the $W^{1,inity}$ norm.

Contribution

It proves well-posedness and exponential decay in $W^{1,inity}$ for the 1D semilinear damped wave equation with time-dependent damping, a novel result in this setting.

Findings

Solutions decay exponentially in $W^{1,inity}$ norm.

Well-posedness in $W^{1,inity}$ established.

Contractive property of the derivative system demonstrated.

Abstract

In this paper we study a semilinear wave equation with nonlinear, time-dependent damping in one space dimension. For this problem, we prove a well-posedness result in $W^{1,\infty}$ in the space-time domain $(0,1)\times [0,+\infty)$. Then we address the problem of the time-asymptotic stability of the zero solution and show that, under appropriate conditions, the solution decays to zero at an exponential rate in the space $W^{1,\infty}$. The proofs are based on the analysis of the corresponding semilinear system for the first order derivatives, for which we show a contractive property of the invariant domain.