Electronic Journal of Differential Equations,
Vol. 2008(2008), No. 109, pp. 1-22.

Title: Energy decay for wave equations of
$\phi$-Laplacian type with weakly nonlinear dissipation
  
Authors: Abbes Benaissa (Djillali Liabes Univ., Algeria)
         Aissa Guesmia  (Univ. Paul Verlaine - Metz, France)
 
Abstract: 
 In this paper, first we prove the existence of global
 solutions in Sobolev spaces for the initial boundary value problem
 of the wave equation of $\phi$-Laplacian with a general
 dissipation of the form
 $$
 (|u'|^{l-2}u')'-\Delta_{\phi}u+\sigma(t) g(u')=0 \quad\text{in }
 \Omega\times \mathbb{R}_+ ,
 $$
 where $\Delta_{\phi}=\sum_{i=1}^n \partial_{x_i}\bigl(\phi
 (|\partial_{x_i}|^2)\partial_{x_i}\bigr)$. Then we prove general
 stability estimates using multiplier method and general weighted
 integral inequalities proved by the second author in [18].
 Without imposing any growth condition at the
 origin on $g$ and $\phi$, we show that the energy of the system is
 bounded above by a quantity, depending on $\phi$, $\sigma$ and
 $g$, which tends to zero (as time approaches infinity). These
 estimates allows us to consider large class of functions $g$ and
 $\phi$ with general growth at the origin. We give some
 examples to illustrate how to derive from our general estimates
 the polynomial, exponential or logarithmic decay. The results of
 this paper improve and generalize many existing results in the
 literature, and generate some interesting open problems.

Submitted September 19, 2007. Published August 11, 2008.
Math Subject Classifications: 35B40, 35L70.
Key Words: Wave equation; global existence; general dissipative term; 
           rate of decay; multiplier method; integral inequalities.