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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
| 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 |
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 and Continuous Dynamical Systems,
2021, 41
(11)
: 5359-5396.
doi: 10.3934/dcds.2021080
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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. |
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. |
| [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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