Decay Rate Estimates Of Solutions For A Viscoelastic Kirchhoff-type Equations With Strong Damping And Acoustic Boundary Conditions

Recently,many authors are concerned about the equation with acoustic boundary conditions.On the basis of predecessors,our paper focuses on research the initial boundary value problem for a nonlinear Viscoelastic Kirchhoff-type equation with memory term,strong damping and acoustic boundary conditions(?) where Ω is a bounded domain of RN,N≥1,with a smooth boundary Γ of class C2.Assume that Γ consists of two disjoint parts,that is Γ＝Γ0∪Γ1 and we denote meas(Γ0∩Γ1)=0.v denotes the unit outward normal to Γ.f represents the kernel of the memory term.φ(x),ψ(x):Γ1→R+are essentially bounded functions(see assumption B1).The parameters α,b>0,a,β,σ≥0,m≥2,k>2 are constants.z is the normal displacement to the boundary at moment t with the boundary point x.u0(x),u1(x):Ω→R and z0(x):Γ1→R are given functions.Unlike the cases of σ=0,σ≥1 in some equations or τ=0 in[17],we prove the energy decay of solutions under the assumption of σ≥0 and we shall use the method introduced by Martine[1]and construct an auxiliary function with the help of[14](Lemma 3.1)to obtain the decay rate estimates of solution for the dissipative systems with the assumptions of σ≥0 and τ≥0 under the suitable assumptions on the initial data and boundary conditions.And other people’s papers are strongly dependent on the non-increasing of φ(t)in(B2).In this paper,our results have no this restriction.