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In this paper we study a semilinear wave equation with nonlinear,time-dependent damping in one space dimension. For this problem, we prove awell-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 ofthe zero solution and show that, under appropriate conditions, the solutiondecays to zero at an exponential rate in the space $W^{1,\infty}$. The proofsare based on the analysis of the corresponding semilinear system for the firstorder derivatives, for which we show a contractive property of the invariantdomain.

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