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

In this paper we study a $2 imes2$ semilinear hyperbolic system of partial differential equations, which is related to a semilinear wave equation with nonlinear, time-dependent damping in one space dimension. For this problem, we prove a well-posedness result in $L^infty$ in the space-time domain $(0,1) imes [0,+infty)$. 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 $L^{infty}$. The proofs are based on the analysis of the invariant domain of the unknowns, for which we show a contractive property. These results can yield a decay property in $W^{1,infty}$ for the corresponding solution to the semilinear wave equation.