September  2009, 2(3): 583-608. doi: 10.3934/dcdss.2009.2.583

On existence, uniform decay rates and blow up for solutions of systems of nonlinear wave equations with damping and source terms

1. 

Department of Mathematics and Statistics, Federal University of Campina Grande, 58109-970, Campina Grande, PB, Brazil

2. 

Department of Mathematics - State University of Maringá, 87020-900 Maringá, PR, Brazil

3. 

Department of Mathematics, State University of Maringá, Maringá, PR, 87020-900, Brazil

4. 

Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE, 68588-0130, United States

5. 

Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588

Received  August 2008 Revised  February 2009 Published  June 2009

This paper is concerned with the study of the nonlinearly damped system of wave equations with Dirichlét boundary conditions:

$u_{t t}$ $- \Delta u + |u_t|^{m-1}u_t = F_u(u,v) \text{ in }\Omega\times ( 0,\infty )$,
$v_{t t}$$ - \Delta v + |v_t|^{r-1}v_t = F_v(u,v) \text{ in }\Omega\times( 0,\infty )$,

where $\Omega$ is a bounded domain in $\mathbb{R}^n$, $n=1,2,3$ with a smooth boundary $\partial\Omega=\Gamma$ and $F$ is a $C^1$ function given by

$ F(u,v)=\alpha|u+v|^{p+1}+ 2\beta |uv|^{\frac{p+1}{2}}. $

Under some conditions on the parameters in the system and with careful analysis involving the Nehari Manifold, we obtain several results on the global existence, uniform decay rates, and blow up of solutions in finite time when the initial energy is nonnegative.

Citation: Claudianor O. Alves, M. M. Cavalcanti, Valeria N. Domingos Cavalcanti, Mohammad A. Rammaha, Daniel Toundykov. On existence, uniform decay rates and blow up for solutions of systems of nonlinear wave equations with damping and source terms. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 583-608. doi: 10.3934/dcdss.2009.2.583
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