-
Previous Article
Well-posedness, blowup, and global existence for an integrable shallow water equation
- DCDS Home
- This Issue
-
Next Article
Some notes on periodic systems with linear part at resonance
Asymptotic regularity of solutions of singularly perturbed damped wave equations with supercritical nonlinearities
| 1. | Université de Poitiers, Laboratoire d'Applications des Mathématiques - SP2MI, Boulevard Marie et Pierre Curie - Téléport 2, 86962 Chasseneuil Futuroscope Cedex |
$\varepsilon\partial_t^2u+\gamma\partial_t u-\Delta_x u+f(u)=g,\quad u|_{\partial\Omega}=0,$
where $\gamma$ is a positive constant and $\varepsilon>0$
is a small parameter.
We do not make any
growth restrictions on the nonlinearity $f$
and, consequently, we do not have the
uniqueness of weak solutions for this problem.
We prove
that the trajectory dynamical system acting on the space
of all properly defined weak energy solutions of this equation possesses a global
attractor $\mathcal A_\varepsilon^{tr}$ and verify that this attractor consists
of global strong regular solutions, if $\varepsilon>0$ is small enough.
Moreover, we also establish that, generically, any weak energy
solution converges exponentially to the attractor $\mathcal A_\varepsilon^{tr}$.
| [1] |
Pierre Fabrie, Cedric Galusinski, A. Miranville, Sergey Zelik. Uniform exponential attractors for a singularly perturbed damped wave equation. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 211-238. doi: 10.3934/dcds.2004.10.211 |
| [2] |
Zhijian Yang, Zhiming Liu. Global attractor for a strongly damped wave equation with fully supercritical nonlinearities. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2181-2205. doi: 10.3934/dcds.2017094 |
| [3] |
Michele Coti Zelati. Global and exponential attractors for the singularly perturbed extensible beam. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 1041-1060. doi: 10.3934/dcds.2009.25.1041 |
| [4] |
Pengyan Ding, Zhijian Yang. Attractors of the strongly damped Kirchhoff wave equation on $\mathbb{R}^{N}$. Communications on Pure & Applied Analysis, 2019, 18 (2) : 825-843. doi: 10.3934/cpaa.2019040 |
| [5] |
Yanbing Yang, Runzhang Xu. Nonlinear wave equation with both strongly and weakly damped terms: Supercritical initial energy finite time blow up. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1351-1358. doi: 10.3934/cpaa.2019065 |
| [6] |
V. Pata, Sergey Zelik. A remark on the damped wave equation. Communications on Pure & Applied Analysis, 2006, 5 (3) : 611-616. doi: 10.3934/cpaa.2006.5.611 |
| [7] |
Zhaojuan Wang, Shengfan Zhou. Existence and upper semicontinuity of random attractors for non-autonomous stochastic strongly damped wave equation with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2787-2812. doi: 10.3934/dcds.2017120 |
| [8] |
Xinyu Mei, Chunyou Sun. Attractors for A sup-cubic weakly damped wave equation in $ \mathbb{R}^{3} $. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4117-4143. doi: 10.3934/dcdsb.2019053 |
| [9] |
John M. Ball. Global attractors for damped semilinear wave equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 31-52. doi: 10.3934/dcds.2004.10.31 |
| [10] |
Changming Song, Hong Li, Jina Li. Initial boundary value problem for the singularly perturbed Boussinesq-type equation. Conference Publications, 2013, 2013 (special) : 709-717. doi: 10.3934/proc.2013.2013.709 |
| [11] |
Alexandre Mouton. Expansion of a singularly perturbed equation with a two-scale converging convection term. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1447-1473. doi: 10.3934/dcdss.2016058 |
| [12] |
Fouad Hadj Selem, Hiroaki Kikuchi, Juncheng Wei. Existence and uniqueness of singular solution to stationary Schrödinger equation with supercritical nonlinearity. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4613-4626. doi: 10.3934/dcds.2013.33.4613 |
| [13] |
Yi He, Gongbao Li. Concentrating solitary waves for a class of singularly perturbed quasilinear Schrödinger equations with a general nonlinearity. Mathematical Control & Related Fields, 2016, 6 (4) : 551-593. doi: 10.3934/mcrf.2016016 |
| [14] |
Q-Heung Choi, Tacksun Jung. A nonlinear wave equation with jumping nonlinearity. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 797-802. doi: 10.3934/dcds.2000.6.797 |
| [15] |
Ciprian G. Gal. Robust exponential attractors for a conserved Cahn-Hilliard model with singularly perturbed boundary conditions. Communications on Pure & Applied Analysis, 2008, 7 (4) : 819-836. doi: 10.3934/cpaa.2008.7.819 |
| [16] |
Filippo Dell'Oro. Global attractors for strongly damped wave equations with subcritical-critical nonlinearities. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1015-1027. doi: 10.3934/cpaa.2013.12.1015 |
| [17] |
Veronica Belleri, Vittorino Pata. Attractors for semilinear strongly damped wave equations on $\mathbb R^3$. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 719-735. doi: 10.3934/dcds.2001.7.719 |
| [18] |
Stéphane Gerbi, Belkacem Said-Houari. Exponential decay for solutions to semilinear damped wave equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 559-566. doi: 10.3934/dcdss.2012.5.559 |
| [19] |
Maurizio Grasselli, Vittorino Pata. On the damped semilinear wave equation with critical exponent. Conference Publications, 2003, 2003 (Special) : 351-358. doi: 10.3934/proc.2003.2003.351 |
| [20] |
Cedric Galusinski, Serguei Zelik. Uniform Gevrey regularity for the attractor of a damped wave equation. Conference Publications, 2003, 2003 (Special) : 305-312. doi: 10.3934/proc.2003.2003.305 |
2018 Impact Factor: 1.143
Tools
Metrics
Other articles
by authors
[Back to Top]