The Initial Boundary Value Problem For Quasi-linear Wave Equation With Viscous Damping

In this paper, we are concerned with the following initial boundary value problem:where δ> 0, μ > 0, p ≥ 1, q> 1 are constants, σ(s) is a given nonlinear function, φ {x) and ψ(x) are given initial value functions, (0,1) =Ω and subscripts x and t indicate the partial derivative with respect to x and t respectively. Equations of type of (1) are a class of nonlinear evolution equations governing the motion of a viscoelastic solid composed of the material of the rate type. It can also be seen as field equation governing the longitudinal motion of a viscoelastic bar obeying the nonlinear Voigt model.This paper consists of four chapters. The first chapter is the introduction. In the second chapter, we will study the existence and uniqueness of the local generalized solution and the local classical solution for the initial boundary value problem of the quasi-linear wave equation with viscous damping. In the third chapter, we will establish a new ordinary differential inequality. In the fourth chapter, first, we will apply the inequality established in the third chapter to give the sufficient conditions of blow-up of the solution for the problem (1) — (3), then we give an example. The main results are the following:Theorem 1 Suppose that(1) σ∈C^m(R), |σ{s)| ≤ K|s|^υ, |σ'(s)| ≤ K|s|^v-1 etc., where v≥2;(2) φ∈H^m(Ω) and ψ∈H^m-1(Ω)If 4 ≤ m≤ min{p + 1, q + 1}(if m is an odd number, then m ≤ min{p + 2, q 4- 2}), specifically, 4 ≤ m ≤ q + 1(if m is an odd number, then m≤q + 2)as p=1, then the...