Decay Of Wave Equations With Memory

In this paper, we study the decay of wave equations with memory.The one is where Ω an open bounded set of Rn with smooth boundary Γ.The above model can be used to describe the evolution of a system consisting of two elastic membranes subject to an elastic force that attracts one membrane to the other with coefficient a>0.Note that the term∫0∞g(s)Δu(t-s)dsand∫∞0g(s)Δv(t-s)ds,acts on the membrane as a stabilizer.This paper will show that the system above is dissipative but the system is not exponentially stable.In addition,I will show that the solution of system decays polynomially to zero,but does not have exponential decay.The main contribution of the paper is to use the term of memory g instead of the damping ut used in the papers mentioned above.The other is Ω is an open bounded domain of Rn with the smooth boundary (?)Ω.=Γ0∪Γ1,v is the unit outer normal vector.Thepartition Γ0and Γ1are closed,disjoint, with meas(Γ0)>0and satisfying Where m(x)=x-x0,x0∈Rn.where the kernel k satisfy k(0)>0, k(t)≥0, k’(t)≤0. k"(t)≥r(t)·(-k’(t)), Here r is function on R+â†'R+and satisfy r(t)>0, r’(t)≤0and∫0∞r(t)dt=+∞.The above model we consider a system of wave equation,in a bounded domain,where the memory-type damping is acting on a part of the boundary.We find a general decay result from the exponential and polynomial decay result,which are special cases. I will show the existence of the solutions of the system first,then,I will prove general decay of solutions by introducing the Lyapunov function.