The Cauchy Problem For Some Semi-Linear Wave Equation With Weak Dissipation

From the 1980s, a new way dealing with existence of global solutions of non-linear evolution equations had been given. The method is strongly depended on the energy estimates and decay estimates of the solution of the linear homogenous equation. Thus we can get the existence and decay estimates of global solution of the non-linear evolution equation under some assumpitions on nonlinear term and its initial data are enough small.This method can be used to deal with a large number of non-linear evolution equations. Many authors have studied linear wave equations with dissipative terms. If the weak dissipative term has relations with the time, it can be regarded as telegraph equation.In Chapter 1, the background and current development of non-linear wave equations are introduced, and some results related are given.In Chapter 2 and Chapter 3, with the help of decay estimates of solution in suitable Sobolev space for a linear wave equation with weak dissipative term, we get the global existence and decay estimates to the corresponding Cauchy problem for the semi-linear wave equation with weak dissipative term which was with small initial data and under some conditions on nonlinear term F by global iterative method and contract mapping. Especially, in Chapter 2 we study the problem which F depends on Du, and in Chapter 3 we study the problem which F contains u.