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.