Electronic Journal of Differential Equations,
Vol. 2007(2007), No. 128, pp. 1-18.

Title: On the wave equations with memory in noncylindrical domains

Author: Mauro L. Santos (Univ. Federal do Para, Guama, Brazil)

Abstract: 
 In this paper we prove the exponential and polynomial decays rates
 in the case $n > 2$, as time approaches infinity of regular solutions
 of the wave equations with memory
 $$
 u_{tt}-\Delta u+\int^{t}_{0}g(t-s)\Delta u(s)ds=0 \quad \mbox{in } \widehat{Q}
 $$
 where $\widehat{Q}$ is a non cylindrical domains of $\mathbb{R}^{n+1}$, 
 $(n\ge1)$.  We show that the dissipation produced by memory effect is
 strong enough to produce exponential decay of solution provided
 the relaxation function $g$ also decays exponentially. When the
 relaxation function decay polynomially, we show that the solution
 decays polynomially with the same rate. For this we introduced a
 new multiplier that makes an important role in the obtaining of
 the exponential and polynomial decays of the energy of the system.
 Existence, uniqueness and regularity of solutions for any
 $n \ge 1$ are investigated. The obtained result extends known
 results from cylindrical to  non-cylindrical domains.
 
Submitted March 8, 2007. Published October 02, 2007.
Math Subject Classifications: 35K55, 35F30, 34B15.
Key Words: Wave equation; noncylindrical domain; memory dissipation.