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

  • * Corresponding author: Debora Amadori

    * 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"

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  • 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.


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  • 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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    [2] D. Amadori and L. Gosse, Error estimates for well-balanced and time-split schemes on a locally damped semilinear wave equation, Math. Comp., 85 (2016), 601-633.  doi: 10.1090/mcom/3043.
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