Existence And Blow Up Of Solutions For A Class Quasi-linear Wave Equations

This dissertation mainly investigates the properties of solutions of a class of quasi-linear wave equations with a viscosity , a nonlinear perturbation (source term) and a dissipative term (damping term), which arises from the longitudinal motion of a visco-elastic material. It contains the problems of existence and uniqueness of solutions, global existence , asymptotic behaviour and finite time blow up, etc.This thesis consists of four parts. The former three chapters considers the initial-boundary value problem for the nonlinear wave equationin a bounded domain fi C Rn.Chapter 2 investigates the problem with a,g and / like (v2) = (1 + v2)-1/2, g(v) = - |v| pv, p > 0 and f(v) = |v|mv, m > 0, respectively. We first give a proof of the global existence and uniqueness of solution for the problem by useing of a "stable set " method. Moreover, we treat the so-called H2-solutions instead of usual energy finite solutions and derive a precise decay estimate for the energy.In Chapter 3, (1 + , p >1 and |ut|m-1, m>1 are place of (| u|2), g(u) and f(ut), respectively. By the energy function we make use of two kinds of different techniques to show that under certain conditions the global solution blows up in finite time.In Chapter 4, | u|k-2, k 2 is place of (| u| 2) again. We use the same techniques in Chapter 3 to prove that under certain conditions the solution blows up in finite time.In fact, the methods adopted here can be applied to some wave problems.Finally, Chapter 5 is concerned with the initial-boundary value problem for a class of quasi-linear hyperbolic wave equationsWe use a Lemma to gain the sufficient condition of nonexistence of global solution of the initial-boundary value problem for the second-order quasi-linear hyperbolic equation.