On the decay in W^1,∞ for the 1D semilinear damped wave equation on a bounded domain

2020/09/01 by Amadori, Debora, Aqel, Fatima Al-Zahrà
#35B40 #35L20 #35L50 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2009.00731

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,∞ 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 W^1,∞. 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.

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