2007, 2007(Special): 531-540. doi: 10.3934/proc.2007.2007.531

On the dynamics of a degenerate damped semilinear wave equation in \mathbb{R}^N : the non-compact case

1. 

Department of Statistics and Actuarial Science, University of the Aegean, Karlovassi 83200, Samos, Greece

2. 

Department of Mathematics, University of Aegean, Karlovassi, GR 83200, Samos, Greece

Received  October 2006 Revised  February 2007 Published  September 2007

We show the existence of a global attractor for a degenerate, linearly damped, semilinear wave equation in $mathbb{R}^N$ under a new condition concerning a variable non-negative diffusivity. In particular, we show the asymptotic compactness of the induced semiflow by combining the energy equation method with appropriate tail estimates.
Citation: Nikos I. Karachalios, Athanasios N Lyberopoulos. On the dynamics of a degenerate damped semilinear wave equation in \mathbb{R}^N : the non-compact case. Conference Publications, 2007, 2007 (Special) : 531-540. doi: 10.3934/proc.2007.2007.531
[1]

Nikos I. Karachalios, Nikos M. Stavrakakis. Estimates on the dimension of a global attractor for a semilinear dissipative wave equation on $\mathbb R^N$. Discrete and Continuous Dynamical Systems, 2002, 8 (4) : 939-951. doi: 10.3934/dcds.2002.8.939

[2]

Biyue Chen, Chunxiang Zhao, Chengkui Zhong. The global attractor for the wave equation with nonlocal strong damping. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6207-6228. doi: 10.3934/dcdsb.2021015

[3]

Kazuhiro Ishige, Michinori Ishiwata. Global solutions for a semilinear heat equation in the exterior domain of a compact set. Discrete and Continuous Dynamical Systems, 2012, 32 (3) : 847-865. doi: 10.3934/dcds.2012.32.847

[4]

Jiacheng Wang, Peng-Fei Yao. On the attractor for a semilinear wave equation with variable coefficients and nonlinear boundary dissipation. Communications on Pure and Applied Analysis, 2022, 21 (6) : 1857-1871. doi: 10.3934/cpaa.2021043

[5]

Zhijian Yang, Zhiming Liu. Global attractor for a strongly damped wave equation with fully supercritical nonlinearities. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 2181-2205. doi: 10.3934/dcds.2017094

[6]

Brahim Alouini. Global attractor for a one dimensional weakly damped half-wave equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (8) : 2655-2670. doi: 10.3934/dcdss.2020410

[7]

Mohammad A. Rammaha, Daniel Toundykov, Zahava Wilstein. Global existence and decay of energy for a nonlinear wave equation with $p$-Laplacian damping. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4361-4390. doi: 10.3934/dcds.2012.32.4361

[8]

Delin Wu and Chengkui Zhong. Estimates on the dimension of an attractor for a nonclassical hyperbolic equation. Electronic Research Announcements, 2006, 12: 63-70.

[9]

Joachim Krieger, Kenji Nakanishi, Wilhelm Schlag. Global dynamics of the nonradial energy-critical wave equation above the ground state energy. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 2423-2450. doi: 10.3934/dcds.2013.33.2423

[10]

Sun-Ho Choi. Weighted energy method and long wave short wave decomposition on the linearized compressible Navier-Stokes equation. Networks and Heterogeneous Media, 2013, 8 (2) : 465-479. doi: 10.3934/nhm.2013.8.465

[11]

Zhiming Liu, Zhijian Yang. Global attractor of multi-valued operators with applications to a strongly damped nonlinear wave equation without uniqueness. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 223-240. doi: 10.3934/dcdsb.2019179

[12]

Fengjuan Meng, Chengkui Zhong. Multiple equilibrium points in global attractor for the weakly damped wave equation with critical exponent. Discrete and Continuous Dynamical Systems - B, 2014, 19 (1) : 217-230. doi: 10.3934/dcdsb.2014.19.217

[13]

Abdelghafour Atlas. Regularity of the attractor for symmetric regularized wave equation. Communications on Pure and Applied Analysis, 2005, 4 (4) : 695-704. doi: 10.3934/cpaa.2005.4.695

[14]

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

[15]

Dalibor Pražák. On the dimension of the attractor for the wave equation with nonlinear damping. Communications on Pure and Applied Analysis, 2005, 4 (1) : 165-174. doi: 10.3934/cpaa.2005.4.165

[16]

Henri Berestycki, Juncheng Wei. On least energy solutions to a semilinear elliptic equation in a strip. Discrete and Continuous Dynamical Systems, 2010, 28 (3) : 1083-1099. doi: 10.3934/dcds.2010.28.1083

[17]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5321-5335. doi: 10.3934/dcdsb.2020345

[18]

Vanessa Barros, Carlos Nonato, Carlos Raposo. Global existence and energy decay of solutions for a wave equation with non-constant delay and nonlinear weights. Electronic Research Archive, 2020, 28 (1) : 205-220. doi: 10.3934/era.2020014

[19]

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 and Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243

[20]

Rachid Assel, Mohamed Ghazel. Energy decay for the damped wave equation on an unbounded network. Evolution Equations and Control Theory, 2018, 7 (3) : 335-351. doi: 10.3934/eect.2018017

 Impact Factor: 

Metrics

  • PDF downloads (33)
  • HTML views (0)
  • Cited by (0)

[Back to Top]