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

Debora Amadori; Fatima Al-Zahrà Aqel

doi:10.3934/dcds.2021080

Article Contents

2021, Volume 41, Issue 11: 5359-5396. Doi: 10.3934/dcds.2021080

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$ \mathbb{H}^{n+1} $

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

Debora Amadori^1, , and
Fatima Al-Zahrà Aqel^2,

1.
Dipartimento di Ingegneria e Scienze dell'Informazione e Matematica (DISIM), University of L'Aquila, L'Aquila, Italy
2.
Department of Mathematics, An-Najah National University, Nablus, Palestine

^* Corresponding author: Debora Amadori
^* Corresponding author: Debora Amadori

Received: September 2020

Revised: March 2021

Early access: May 2021

Published: November 2021

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"

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Abstract

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.

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

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Figure 1. Structure of the solution to the Riemann problem

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Figure 2. Multiple interaction, time-dependent case

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

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