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 2 × 2 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 ∞ in the space-time domain (0, 1) × [0, +∞). 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 ∞ . 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,∞ for the corresponding solution to the semilinear wave equation.