Figure 1.
Structure of the solution to the Riemann problem
-
Previous Article
On the mixing and Bernoulli properties for geodesic flows on rank 1 manifolds without focal points
- DCDS Home
- This Issue
-
Next Article
Classification of solutions to a nonlocal equation with doubly Hardy-Littlewood-Sobolev critical exponents
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 & Continuous Dynamical Systems, 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. 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. |
| [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. |
| [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. 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. |
| [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. |
| [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. |
| [1] |
Xiaopeng Zhao. Space-time decay estimates of solutions to liquid crystal system in $\mathbb{R}^3$. Communications on Pure & Applied Analysis, 2019, 18 (1) : 1-13. doi: 10.3934/cpaa.2019001 |
| [2] |
Beom-Seok Han, Kyeong-Hun Kim, Daehan Park. A weighted Sobolev space theory for the diffusion-wave equations with time-fractional derivatives on $ C^{1} $ domains. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3415-3445. doi: 10.3934/dcds.2021002 |
| [3] |
Linglong Du, Min Yang. Pointwise long time behavior for the mixed damped nonlinear wave equation in $ \mathbb{R}^n_+ $. Networks & Heterogeneous Media, 2021, 16 (1) : 49-67. doi: 10.3934/nhm.2020033 |
| [4] |
Linglong Du. Long time behavior for the visco-elastic damped wave equation in $\mathbb{R}^n_+$ and the boundary effect. Networks & Heterogeneous Media, 2018, 13 (4) : 549-565. doi: 10.3934/nhm.2018025 |
| [5] |
Montgomery Taylor. The diffusion phenomenon for damped wave equations with space-time dependent coefficients. Discrete & Continuous Dynamical Systems, 2018, 38 (11) : 5921-5941. doi: 10.3934/dcds.2018257 |
| [6] |
Burak Ordin, Adil Bagirov, Ehsan Mohebi. An incremental nonsmooth optimization algorithm for clustering using $ L_1 $ and $ L_\infty $ norms. Journal of Industrial & Management Optimization, 2020, 16 (6) : 2757-2779. doi: 10.3934/jimo.2019079 |
| [7] |
Abdeladim El Akri, Lahcen Maniar. Uniform indirect boundary controllability of semi-discrete $ 1 $-$ d $ coupled wave equations. Mathematical Control & Related Fields, 2020, 10 (4) : 669-698. doi: 10.3934/mcrf.2020015 |
| [8] |
Abdelwahab Bensouilah, Van Duong Dinh, Mohamed Majdoub. Scattering in the weighted $ L^2 $-space for a 2D nonlinear Schrödinger equation with inhomogeneous exponential nonlinearity. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2735-2755. doi: 10.3934/cpaa.2019122 |
| [9] |
Waixiang Cao, Lueling Jia, Zhimin Zhang. A $ C^1 $ Petrov-Galerkin method and Gauss collocation method for 1D general elliptic problems and superconvergence. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 81-105. doi: 10.3934/dcdsb.2020327 |
| [10] |
Weiwei Ao, Chao Liu. Asymptotic behavior of sign-changing radial solutions of a semilinear elliptic equation in $ \mathbb{R}^2 $ when exponent approaches $ +\infty $. Discrete & Continuous Dynamical Systems, 2020, 40 (8) : 5047-5077. doi: 10.3934/dcds.2020211 |
| [11] |
Hongyong Cui, Peter E. Kloeden, Wenqiang Zhao. Strong $ (L^2,L^\gamma\cap H_0^1) $-continuity in initial data of nonlinear reaction-diffusion equation in any space dimension. Electronic Research Archive, 2020, 28 (3) : 1357-1374. doi: 10.3934/era.2020072 |
| [12] |
Nicholas J. Kass, Mohammad A. Rammaha. Local and global existence of solutions to a strongly damped wave equation of the $ p $-Laplacian type. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1449-1478. doi: 10.3934/cpaa.2018070 |
| [13] |
Tuan Anh Dao, Michael Reissig. $ L^1 $ estimates for oscillating integrals and their applications to semi-linear models with $ \sigma $-evolution like structural damping. Discrete & Continuous Dynamical Systems, 2019, 39 (9) : 5431-5463. doi: 10.3934/dcds.2019222 |
| [14] |
Koya Nishimura. Global existence for the Boltzmann equation in $ L^r_v L^\infty_t L^\infty_x $ spaces. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1769-1782. doi: 10.3934/cpaa.2019083 |
| [15] |
Luca Battaglia, Francesca Gladiali, Massimo Grossi. Asymptotic behavior of minimal solutions of $ -\Delta u = \lambda f(u) $ as $ \lambda\to-\infty $. Discrete & Continuous Dynamical Systems, 2021, 41 (2) : 681-700. doi: 10.3934/dcds.2020293 |
| [16] |
Niklas Sapountzoglou, Aleksandra Zimmermann. Well-posedness of renormalized solutions for a stochastic $ p $-Laplace equation with $ L^1 $-initial data. Discrete & Continuous Dynamical Systems, 2021, 41 (5) : 2341-2376. doi: 10.3934/dcds.2020367 |
| [17] |
Jiangang Qi, Bing Xie. Extremum estimates of the $ L^1 $-norm of weights for eigenvalue problems of vibrating string equations based on critical equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3505-3516. doi: 10.3934/dcdsb.2020243 |
| [18] |
Linglong Du, Caixuan Ren. Pointwise wave behavior of the initial-boundary value problem for the nonlinear damped wave equation in $\mathbb{R}_{+}^{n} $. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3265-3280. doi: 10.3934/dcdsb.2018319 |
| [19] |
Aihua Fan, Jörg Schmeling, Weixiao Shen. $ L^\infty $-estimation of generalized Thue-Morse trigonometric polynomials and ergodic maximization. Discrete & Continuous Dynamical Systems, 2021, 41 (1) : 297-327. doi: 10.3934/dcds.2020363 |
| [20] |
Umberto De Maio, Peter I. Kogut, Gabriella Zecca. On optimal $ L^1 $-control in coefficients for quasi-linear Dirichlet boundary value problems with $ BMO $-anisotropic $ p $-Laplacian. Mathematical Control & Related Fields, 2020, 10 (4) : 827-854. doi: 10.3934/mcrf.2020021 |
2019 Impact Factor: 1.338
Tools
Metrics
Other articles
by authors
[Back to Top]