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

Abstract

<p style='text-indent:20px;'>In this paper we study a <inline-formula><tex-math id="M2">\begin{document}$ 2\times2 $\end{document}</tex-math></inline-formula> semilinear hyperbolic system of partial differential equations, which is related to a semilinear wave equation with nonlinear, time-dependent damping in one space dimension. For this problem, we prove a well-posedness result in <inline-formula><tex-math id="M3">\begin{document}$ L^\infty $\end{document}</tex-math></inline-formula> in the space-time domain <inline-formula><tex-math id="M4">\begin{document}$ (0,1)\times [0,+\infty) $\end{document}</tex-math></inline-formula>. Then we address the problem of the time-asymptotic stability of the zero solution and show that, under appropriate conditions, the solution decays to zero at an exponential rate in the space <inline-formula><tex-math id="M5">\begin{document}$ L^{\infty} $\end{document}</tex-math></inline-formula>. The proofs are based on the analysis of the invariant domain of the unknowns, for which we show a contractive property. These results can yield a decay property in <inline-formula><tex-math id="M6">\begin{document}$ W^{1,\infty} $\end{document}</tex-math></inline-formula> for the corresponding solution to the semilinear wave equation.</p>