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.