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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

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

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.

Keywords: Damped wave equation in 1d, space-time dependent relaxation model, time-asymptotic stability, $ L^\infty $ estimates.
Mathematics Subject Classification: Primary: 35L50, 35B40; Secondary: 35L20.
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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[11] R. A. Horn and C. R. Johnson, Matrix Analysis, $2^{nd}$ edition, Cambridge University Press, 2013.   Google Scholar
[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.  Google Scholar

[13]

R. Natalini and B. Hanouzet, Weakly coupled systems of quasilinear hyperbolic equations, Differential Integral Equations, 9 (1996), 1279-1292.   Google Scholar

[14]

E. Zuazua, Propagation, observation, and control of waves approximated by finite difference methods, SIAM Rev., 47 (2005), 197-243.  doi: 10.1137/S0036144503432862.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[11] R. A. Horn and C. R. Johnson, Matrix Analysis, $2^{nd}$ edition, Cambridge University Press, 2013.   Google Scholar
[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.  Google Scholar

[13]

R. Natalini and B. Hanouzet, Weakly coupled systems of quasilinear hyperbolic equations, Differential Integral Equations, 9 (1996), 1279-1292.   Google Scholar

[14]

E. Zuazua, Propagation, observation, and control of waves approximated by finite difference methods, SIAM Rev., 47 (2005), 197-243.  doi: 10.1137/S0036144503432862.  Google Scholar

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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