On the wave equations with memory in noncylindrical domains

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 utt - Δu + ∫t0 g(t - s) Δu(s)ds = 0 in Q̂ where Q̂ is a non cylindrical domains of ℝn+1, (n ≥ 1). 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 ≥ 1 are investigated. The obtained result extends known results from cylindrical to non-cylindrical domains.