
Previous Article
Wellposedness, 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) : 211238. doi: 10.3934/dcds.2004.10.211 
[2] 
Pengyan Ding, Zhijian Yang. Wellposedness and attractor for a strongly damped wave equation with supercritical nonlinearity on $ \mathbb{R}^{N} $. Communications on Pure & Applied Analysis, , () : . doi: 10.3934/cpaa.2021006 
[3] 
Dandan Li. Asymptotics of singularly perturbed damped wave equations with supercubic exponent. Discrete & Continuous Dynamical Systems  B, 2021 doi: 10.3934/dcdsb.2021056 
[4] 
Kaixuan Zhu, Yongqin Xie, Xinyu Mei. Pullback attractors for a weakly damped wave equation with delays and supcubic nonlinearity. Discrete & Continuous Dynamical Systems  B, 2020 doi: 10.3934/dcdsb.2020294 
[5] 
Zhijian Yang, Zhiming Liu. Global attractor for a strongly damped wave equation with fully supercritical nonlinearities. Discrete & Continuous Dynamical Systems  A, 2017, 37 (4) : 21812205. doi: 10.3934/dcds.2017094 
[6] 
Xinyu Mei, Yangmin Xiong, Chunyou Sun. Pullback attractor for a weakly damped wave equation with supcubic nonlinearity. Discrete & Continuous Dynamical Systems  A, 2021, 41 (2) : 569600. doi: 10.3934/dcds.2020270 
[7] 
Michele Coti Zelati. Global and exponential attractors for the singularly perturbed extensible beam. Discrete & Continuous Dynamical Systems  A, 2009, 25 (3) : 10411060. doi: 10.3934/dcds.2009.25.1041 
[8] 
Pengyan Ding, Zhijian Yang. Attractors of the strongly damped Kirchhoff wave equation on $\mathbb{R}^{N}$. Communications on Pure & Applied Analysis, 2019, 18 (2) : 825843. doi: 10.3934/cpaa.2019040 
[9] 
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) : 13511358. doi: 10.3934/cpaa.2019065 
[10] 
Ahmad Z. Fino, Wenhui Chen. A global existence result for twodimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 53875411. doi: 10.3934/cpaa.2020243 
[11] 
V. Pata, Sergey Zelik. A remark on the damped wave equation. Communications on Pure & Applied Analysis, 2006, 5 (3) : 611616. doi: 10.3934/cpaa.2006.5.611 
[12] 
Xinyu Mei, Chunyou Sun. Attractors for A supcubic weakly damped wave equation in $ \mathbb{R}^{3} $. Discrete & Continuous Dynamical Systems  B, 2019, 24 (8) : 41174143. doi: 10.3934/dcdsb.2019053 
[13] 
Qingquan Chang, Dandan Li, Chunyou Sun. Random attractors for stochastic timedependent damped wave equation with critical exponents. Discrete & Continuous Dynamical Systems  B, 2020, 25 (7) : 27932824. doi: 10.3934/dcdsb.2020033 
[14] 
Zhaojuan Wang, Shengfan Zhou. Existence and upper semicontinuity of random attractors for nonautonomous stochastic strongly damped wave equation with multiplicative noise. Discrete & Continuous Dynamical Systems  A, 2017, 37 (5) : 27872812. doi: 10.3934/dcds.2017120 
[15] 
John M. Ball. Global attractors for damped semilinear wave equations. Discrete & Continuous Dynamical Systems  A, 2004, 10 (1&2) : 3152. doi: 10.3934/dcds.2004.10.31 
[16] 
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 
[17] 
Changming Song, Hong Li, Jina Li. Initial boundary value problem for the singularly perturbed Boussinesqtype equation. Conference Publications, 2013, 2013 (special) : 709717. doi: 10.3934/proc.2013.2013.709 
[18] 
Alexandre Mouton. Expansion of a singularly perturbed equation with a twoscale converging convection term. Discrete & Continuous Dynamical Systems  S, 2016, 9 (5) : 14471473. doi: 10.3934/dcdss.2016058 
[19] 
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) : 551593. doi: 10.3934/mcrf.2016016 
[20] 
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) : 46134626. doi: 10.3934/dcds.2013.33.4613 
2019 Impact Factor: 1.338
Tools
Metrics
Other articles
by authors
[Back to Top]