General Stability for Kirchhoff Equation with a Non Constant Delay Term

The aim of this paper is study a Kirchhoff equation with a non-constant delay term: {█(u_tt+M(|(|∇u| )|^2 )∆u-∆u+u_t (x,t-τ(t) )+u_t=0 x∈Ω,t>0@u(x,t)=0 x∈∂Ω ,t>0 @u(x,0)=u_0 (x), u_t (x,0)=u_1 (x)x∈Ω)┤ Where Ω is a bounded domain with smooth boundary∂Ω, and M(s)=β_1+β_2 s^γ,γ>0,β_1>1,β_2>0 We first establish the existence of solutions of the problem by means Galerkin method, and proved a general stability results of the energy. using lyapunov functional and the logarithmic Sobolev Inequality. Keywords - Kirchhoff Equation, Time Delay, Existence, General Decay, Lyapunov Function.