Asymptotic Behavior Of Solutions Of Two Kinds Of Nonhomogeneous Boundary Wave Equations

Viscoelastic mechanics theory studies what regularity the stress and inotropic strain satisfy when viscoelastic materials are under the function of load. Viscoelastic mechanics has been discussed frequently in mathematics and physics. Viscoelastic mechanics theory, which not only has signi?cant theoretic means but also has been applied extensively in physics, engineering mechanics, celestial mechanics and so on, is a combination of mathematics and physics. Most equations of viscoelastic mechanics are partial differential equations which are the hot spot of applied partial differential equation. General decay results of solutions for viscoelastic problem are the hot spot which has been discussed in recent years.In this thesis, we mainly study general decay results of solutions for the wave equation with non-homogeneous boundary. It is divided into two chapters accord to contents:In Chapter 1, We consider the viscoelastic wave equations with boundary controlutt- μ?u-(μ + λ)?(divu) +∫t0g(t- s)?u(s)ds = 0 ? ×(0, ∞),u = 0 Γ0×(0, ∞),μ?u?ν-∫t0g(t- s)?u?ν(s)ds +(μ + λ)(divu)ν + h(ut) = 0 Γ1×(0, ∞),u(x, 0) = u0, ut(x, 0) = u1 x ∈ ?,where μ, λ are Lame moduli. Here, u =(u1, · · ·, un)Tis a n-dimensional vector function, divu = u1x1+ u2x2+ · · · + un xnis the divergence of u, ? is a bounded domain of Rnwith a smooth boundary ?? and ν is the unit outward normal to ??. We study the stability result of its solution when the relaxation function g satis?es g(t) >0, g′< 0, ?t ≥ 0. The relaxation function and boundary control function satisfy the assumptions(A1)-(A3). We consider the general decay result of the equations by constructing auxiliary functionals and using essential inequalities and obtain a general decay formula.In Chapter 2, we consider the following semi-linear wave equation with memory-type boundary?utt- k0?u + b(x)h(ut) = 0(x, t) ∈ ? ×(0, ∞),u = 0(x, t) ∈ Γ0×(0, ∞),u(x, t) =-k0∫t0g(t- s)?u?νds(x, t) ∈ Γ1×(0, ∞),u(x, 0) = u0(x), ut(x, 0) = u1(x) x ∈ ?,where k0 is a positive constant, ? is a bounded domain of Rnwith a smooth boundary?? with ?? = Γ0∪ Γ1, Γ0∩ Γ1= ? and Γ0, Γ1have positive measures. In this chapter, we consider more general resolvent kernels, which not decays polynomially or exponentially. If f satis?es f(t) =(f ? g)(t) + g(t), we say f is the resolvent kernel of g. Here ? denote revolution and(f ? g)(t) =∫t0f(t- s)g(s)ds. Under the assumptions(A4)-(A7) on the functions g, k, b, constructing auxiliary functionals and using essential inequalities and properties of convex function, we get a general decay formula which can be shown by convex function.