February & March  2004, 11(2&3): 351-392. doi: 10.3934/dcds.2004.11.351

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

Received  November 2002 Revised  February 2004 Published  June 2004

We study the asymptotic behavior of weak energy solutions of the following damped hyperbolic equation in a bounded domain $\Omega\subset\R^3$:

$\varepsilon\partial_t^2u+\gamma\partial_t u-\Delta_x u+f(u)=g,\quad u|_{\partial\Omega}=0,$

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 $\mathcal A_\varepsilon^{tr}$.

Citation: Sergey Zelik. Asymptotic regularity of solutions of singularly perturbed damped wave equations with supercritical nonlinearities. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 351-392. doi: 10.3934/dcds.2004.11.351
[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]

Kaixuan Zhu, Yongqin Xie, Xinyu Mei. Pullback attractors for a weakly damped wave equation with delays and sup-cubic nonlinearity. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020294

[3]

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

[4]

Xinyu Mei, Yangmin Xiong, Chunyou Sun. Pullback attractor for a weakly damped wave equation with sup-cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020270

[5]

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

[6]

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

[7]

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

[8]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020243

[9]

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

[10]

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

[11]

Qingquan Chang, Dandan Li, Chunyou Sun. Random attractors for stochastic time-dependent damped wave equation with critical exponents. Discrete & Continuous Dynamical Systems - B, 2020, 25 (7) : 2793-2824. doi: 10.3934/dcdsb.2020033

[12]

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

[13]

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

[14]

Yuzhu Han, Qi Li. Lifespan of solutions to a damped plate equation with logarithmic nonlinearity. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020101

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (31)
  • HTML views (0)
  • Cited by (23)

Other articles
by authors

[Back to Top]