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

Abstract : In this Ph.D. thesis, we study a semilinear wave equation with nonlinear and time-dependent damping term. After rewriting the equation as a first order system, we define a class of approximate solutions employing typical tools of hyperbolic systems of conservation laws, such as the Riemann problem. We prove that the initial-boundary value problem is well-posed for initial data in $L^\infty$ space. By recasting the problem as a discrete-time nonhomogeneous system, which is related to a probabilistic interpretation of the solution, we provide a strategy to study its long-time behavior uniformly with respect to the mesh size parameter $\DX=1/N\to 0$. The proof makes use of the Birkhoff decomposition of doubly stochastic matrices and of accurate estimates on the iteration system as $N\to\infty$. Under appropriate assumptions on the nonlinearity, we prove the exponential convergence in $L^\infty$ of the solution to the first order system towards a stationary solution, as $t\to+\infty$, as well as uniform error estimates for the approximate solutions.
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https://hal.archives-ouvertes.fr/tel-02910377
Contributor : Fatima Alzahra Aqel <>
Submitted on : Wednesday, August 5, 2020 - 11:20:52 AM
Last modification on : Thursday, August 6, 2020 - 3:13:24 AM

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Fatima Alzahra A. N. Aqel, Dottoranda Fatima, Alzahra Aqel. Decay in $W^{1,\infty}$ for the 1D semilinear damped wave equation on a bounded domain. Analysis of PDEs [math.AP]. Universita Degli Studi di L'Aquila, 2020. English. ⟨tel-02910377⟩

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