American Institute of Mathematical Sciences

Advanced
2009, 2009(Special): 592-601. doi: 10.3934/proc.2009.2009.592

Global solutions to the Cauchy problem for the weakly coupled system of damped wave equations

Takashi Narazaki 1,

1. 

Hiratsuka, Kanagawa, 259-1292, Japan

Received  August 2008 Revised  August 2009 Published  September 2009

In this paper we study the Cauchy problem for the weakly coupled system of damped wave equations. Recently Sun and Wang [12] have shown the existence and nonexistence of the Cauchy problem for the weakly coupled system of damped wave equations, provided that the space dimension $n=1, 3$. In this paper we will generalize their existence result to the case where $n=1,2.3$, and we improve time decay estimates when $n=3$. Moreover, the Cauchy problem with slow decaying initial data is treated.
Keywords: Cauchy problem, Damped wave equation, Weakly coupled system.
Mathematics Subject Classification: Primary: 35L55, 35L15, 35L70.
Citation: Takashi Narazaki. Global solutions to the Cauchy problem for the weakly coupled system of damped wave equations. Conference Publications, 2009, 2009 (Special) : 592-601. doi: 10.3934/proc.2009.2009.592
[1]

Marcello D'Abbicco. A note on a weakly coupled system of structurally damped waves. Conference Publications, 2015, 2015 (special) : 320-329. doi: 10.3934/proc.2015.0320
[2]

Felipe Linares, M. Panthee. On the Cauchy problem for a coupled system of KdV equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 417-431. doi: 10.3934/cpaa.2004.3.417
[3]

Bopeng Rao, Zhuangyi Liu. A spectral approach to the indirect boundary control of a system of weakly coupled wave equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 399-414. doi: 10.3934/dcds.2009.23.399
[4]

Farah Abdallah, Mouhammad Ghader, Ali Wehbe, Yacine Chitour. Optimal indirect stability of a weakly damped elastic abstract system of second order equations coupled by velocities. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2789-2818. doi: 10.3934/cpaa.2019125
[5]

Yang Han. On the cauchy problem for the coupled Klein Gordon Schrödinger system with rough data. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 233-242. doi: 10.3934/dcds.2005.12.233
[6]

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

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]

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]

V. Varlamov, Yue Liu. Cauchy problem for the Ostrovsky equation. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 731-753. doi: 10.3934/dcds.2004.10.731
[11]

Adrien Dekkers, Anna Rozanova-Pierrat. Cauchy problem for the Kuznetsov equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 277-307. doi: 10.3934/dcds.2019012
[12]

Renjun Duan, Shuangqian Liu. Cauchy problem on the Vlasov-Fokker-Planck equation coupled with the compressible Euler equations through the friction force. Kinetic & Related Models, 2013, 6 (4) : 687-700. doi: 10.3934/krm.2013.6.687
[13]

Jeong Ja Bae, Mitsuhiro Nakao. Existence problem for the Kirchhoff type wave equation with a localized weakly nonlinear dissipation in exterior domains. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 731-743. doi: 10.3934/dcds.2004.11.731
[14]

Huijun He, Zhaoyang Yin. On the Cauchy problem for a generalized two-component shallow water wave system with fractional higher-order inertia operators. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1509-1537. doi: 10.3934/dcds.2017062
[15]

Olivier Goubet, Ezzeddine Zahrouni. On a time discretization of a weakly damped forced nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1429-1442. doi: 10.3934/cpaa.2008.7.1429
[16]

Lorena Bociu, Petronela Radu. Existence of weak solutions to the Cauchy problem of a semilinear wave equation with supercritical interior source and damping. Conference Publications, 2009, 2009 (Special) : 60-71. doi: 10.3934/proc.2009.2009.60
[17]

Masahiro Ikeda, Takahisa Inui, Mamoru Okamoto, Yuta Wakasugi. $ L^p $-$ L^q $ estimates for the damped wave equation and the critical exponent for the nonlinear problem with slowly decaying data. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1967-2008. doi: 10.3934/cpaa.2019090
[18]

Rudong Zheng, Zhaoyang Yin. The Cauchy problem for a generalized Novikov equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3503-3519. doi: 10.3934/dcds.2017149
[19]

Belkacem Said-Houari, Salim A. Messaoudi. General decay estimates for a Cauchy viscoelastic wave problem. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1541-1551. doi: 10.3934/cpaa.2014.13.1541
[20]

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

 Impact Factor: 

Tools

Metrics

  • PDF downloads (16)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences