Article Contents
Article Contents

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

• * Corresponding author: Debora Amadori

Partially supported by Miur-PRIN 2015, "Hyperbolic Systems of Conservation Laws and Fluid Dynamics: Analysis and Applications", # Grant No. 2015YCJY3A_003, and by 2018 INdAM-GNAMPA Project "Equazioni iperboliche e applicazioni"

• In this paper we study a $2\times2$ 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 $(0,1)\times [0,+\infty)$. 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.

Mathematics Subject Classification: Primary: 35L50, 35B40; Secondary: 35L20.

 Citation:

• Figure 1.  Structure of the solution to the Riemann problem

Figure 2.  Multiple interaction, time-dependent case

Figure 3.  Illustration of the polygonals $y_j(t)$ and of the wave strengths $\sigma_j(t)$

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