Existence and uniform decay rates of solutions to a degenerate system with memory conditions at the boundary

摘要

In this article we study the degenerate system (ρ1,ρ2⩾0) subject to memory conditions on the boundary given byρ1(x)utt−Δu+α(u−v)=0inΩ×]0,+∞[,ρ2(x)vtt−Δv−α(u−v)=0inΩ×]0,+∞[,u=0onΓ0,u+∫0tg1(t−s)∂u∂ν(s)ds=0onΓ1×]0,+∞[,v=0onΓ0,v+∫0tg2(t−s)∂v∂ν(s)ds=0onΓ1×]0,+∞[,(u(0),v(0))=(u0,v0)(ρ1ut(0),ρ2vt(0))=(ρ1u1,ρ2v1)inΩ,where Ω is a bounded region in Rn whose boundary is partitioned into disjoint sets Γ0, Γ1. We prove that the dissipations given by the memory terms are strong enough to guarantee exponential (or polynomial) decay provided the relaxation functions also decay exponentially (or polynomially) and with the same rate of decay.