Blow-up Solution And Energy Decay Of Wave Equations

In this paper,we are concerned with two problems.The first one is blow-up solution for coupled string equations with nonlinear principal parts.And the other one is the energy decay for the Cauchy problem of the variable-coefficient wave equation with nonlinear time-dependent and space-dependent damping.The method we use to study the first problem is energy function method and nonlinear analysis,and the main method employed to study the second problem is Riemannian geometry method.The paper includes three chapters.In Chapter 1,we give a brief introduction on the research advancement of blow-up and energy decay of wave equations.In Chapter 2,we discuss blow-up problem for the following coupled string equations where L is the length of the string,?,f1,f2 are some nonlinear functions satisfying some conditions.The two equations are coupled by the forces exerted in the interior of the string.The nonlinear damp is imposed on the right end of the string.We assume that the initial energy is nonnegative.The interaction among nonlinear boundary damping,nonlinear inte-rior source and nonlinear principal parts is analyzed.A sufficient condition under which the solutions of the system blow up is obtained.In Chapter 3,we discuss the energy decay for the following Cauchy problem of the wave equation with nonlinear time-dependent and space-dependent damping,where the damping?(t)?(x,ut)is localized in a bounded domain and near infinity,and the wave equation is of variable-coefficient principal part.We apply the multiplier method for variable-coefficient equations to obtain an energy decay.