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

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.