# American Institute of Mathematical Sciences

November  2021, 41(11): 5359-5396. doi: 10.3934/dcds.2021080

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

 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

Received  September 2020 Revised  March 2021 Published  November 2021 Early access  May 2021

Fund Project: 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.

Citation: Debora Amadori, Fatima Al-Zahrà Aqel. On the decay in $W^{1,\infty}$ for the 1D semilinear damped wave equation on a bounded domain. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5359-5396. doi: 10.3934/dcds.2021080
##### References:
 [1] D. Amadori, F. A.-Z. Aqel and E. Dal Santo, Decay of approximate solutions for the damped semilinear wave equation on a bounded 1d domain, J. Math. Pures Appl., 132 (2019), 166-206.  doi: 10.1016/j.matpur.2019.05.010. [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. [3] D. Amadori and L. Gosse, Error Estimates for Well-Balanced Schemes on Simple Balance Laws. One-Dimensional Position-Dependent Models, SpringerBriefs in Mathematics, Springer International Publishing, 2015. doi: 10.1007/978-3-319-24785-4. [4] D. Amadori and L. Gosse, Stringent error estimates for one-dimensional, space-dependent $2\times2$ relaxation systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 621-654.  doi: 10.1016/j.anihpc.2015.01.001. [5] A. Bressan, Hyperbolic Systems of Conservation Laws - The One-Dimensional Cauchy Problem, Oxford Lecture Series in Mathematics and its Applications 20, Oxford University Press, 2000. [6] Y. Chitour, S. Marx and C. Prieur, $L^p$–asymptotic stability analysis of a 1D wave equation with a nonlinear damping, J. Differential Equations, 269 (2020), 8107-8131.  doi: 10.1016/j.jde.2020.06.007. [7] L. Gosse, Computing Qualitatively Correct Approximations of Balance Laws, SIMAI Springer Series, Vol. 2, Springer, 2013. doi: 10.1007/978-88-470-2892-0. [8] L. Gosse and G. Toscani, An asymptotic-preserving well-balanced scheme for the hyperbolic heat equations, C. R. Math. Acad. Sci. Paris, 334 (2002), 337-342.  doi: 10.1016/S1631-073X(02)02257-4. [9] A. Haraux, $L^p$ estimates of solutions to some non-linear wave equations in one space dimension, Int. J. Mathematical Modelling and Numerical Optimisation, 1 (2009), 146-152. [10] A. Haraux, P. Martinez and J. Vancostenoble, Asymptotic stability for intermittently controlled second-order evolution equations, SIAM J. Control Optim., 43 (2005), 2089-2108.  doi: 10.1137/S0363012903436569. [11] R. A. Horn and C. R. Johnson, Matrix Analysis, $2^{nd}$ edition, Cambridge University Press, 2013. [12] P. Martinez and J. Vancostenoble, Stabilization of the wave equation by on-off and positive-negative feedbacks, ESAIM Control Optim. Calc. Var., 7 (2002), 335-377.  doi: 10.1051/cocv:2002015. [13] R. Natalini and B. Hanouzet, Weakly coupled systems of quasilinear hyperbolic equations, Differential Integral Equations, 9 (1996), 1279-1292. [14] E. Zuazua, Propagation, observation, and control of waves approximated by finite difference methods, SIAM Rev., 47 (2005), 197-243.  doi: 10.1137/S0036144503432862.

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##### References:
 [1] D. Amadori, F. A.-Z. Aqel and E. Dal Santo, Decay of approximate solutions for the damped semilinear wave equation on a bounded 1d domain, J. Math. Pures Appl., 132 (2019), 166-206.  doi: 10.1016/j.matpur.2019.05.010. [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. [3] D. Amadori and L. Gosse, Error Estimates for Well-Balanced Schemes on Simple Balance Laws. One-Dimensional Position-Dependent Models, SpringerBriefs in Mathematics, Springer International Publishing, 2015. doi: 10.1007/978-3-319-24785-4. [4] D. Amadori and L. Gosse, Stringent error estimates for one-dimensional, space-dependent $2\times2$ relaxation systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 621-654.  doi: 10.1016/j.anihpc.2015.01.001. [5] A. Bressan, Hyperbolic Systems of Conservation Laws - The One-Dimensional Cauchy Problem, Oxford Lecture Series in Mathematics and its Applications 20, Oxford University Press, 2000. [6] Y. Chitour, S. Marx and C. Prieur, $L^p$–asymptotic stability analysis of a 1D wave equation with a nonlinear damping, J. Differential Equations, 269 (2020), 8107-8131.  doi: 10.1016/j.jde.2020.06.007. [7] L. Gosse, Computing Qualitatively Correct Approximations of Balance Laws, SIMAI Springer Series, Vol. 2, Springer, 2013. doi: 10.1007/978-88-470-2892-0. [8] L. Gosse and G. Toscani, An asymptotic-preserving well-balanced scheme for the hyperbolic heat equations, C. R. Math. Acad. Sci. Paris, 334 (2002), 337-342.  doi: 10.1016/S1631-073X(02)02257-4. [9] A. Haraux, $L^p$ estimates of solutions to some non-linear wave equations in one space dimension, Int. J. Mathematical Modelling and Numerical Optimisation, 1 (2009), 146-152. [10] A. Haraux, P. Martinez and J. Vancostenoble, Asymptotic stability for intermittently controlled second-order evolution equations, SIAM J. Control Optim., 43 (2005), 2089-2108.  doi: 10.1137/S0363012903436569. [11] R. A. Horn and C. R. Johnson, Matrix Analysis, $2^{nd}$ edition, Cambridge University Press, 2013. [12] P. Martinez and J. Vancostenoble, Stabilization of the wave equation by on-off and positive-negative feedbacks, ESAIM Control Optim. Calc. Var., 7 (2002), 335-377.  doi: 10.1051/cocv:2002015. [13] R. Natalini and B. Hanouzet, Weakly coupled systems of quasilinear hyperbolic equations, Differential Integral Equations, 9 (1996), 1279-1292. [14] E. Zuazua, Propagation, observation, and control of waves approximated by finite difference methods, SIAM Rev., 47 (2005), 197-243.  doi: 10.1137/S0036144503432862.
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