Blow-up Of Solutions For Some Wave Equations With Damping Terms

This paper studies blow-up of solution to a wave equation with the combined nonlinearities,blow-up of solution for coupled system of wave equations with variable coefficients,blow-up of solution for a class of coupled wave equations with damping terms.Applying the test function method and iteration method verifies blow-up of solutions to several types of wave equations.The specific contents are started as follows.The first chapter mainly describes the physical background and significance of nonlinear wave equation,the research status in China and abroad,and introduces the main research contents and results of this paper.The second chapter is concerned with the initial boundary value problem of the variable coefficient wave equation with scattering damping term,potential term and combined nonlinearities on exterior domain.In this chapter,by constructing multipliers,using the test function method and iterative method,we prove that a solution of the problem will blow up in a finite time,and then the lifespan of solution is estimated.The innovation is that when V（x）0,we extend the problem discussed in[13]to the variable coefficient problem with potential term on exterior domain.When V（x）=0,the problem studied in[13]is extended to the variable coefficient problem on exterior domain.WhenV（x）K₀,the problem studied in[14]is extended to the variable coefficient problem with negative potential term on exterior domain.In the third chapter,we study the initial boundary value problem of coupled wave equation with dampings and negative mass terms.And we use the test function method and iterative method to obtain the blow-up result and lifespan estimates of solutions in the sub-critical case.In the critical case,Blow-up results of solutions are analyzed by introducing the auxiliary function.The upper bound lifespan estimates of solutions are obtained.In the fourth chapter,we study the initial boundary value problem of coupled wave equations with damping terms.The innovation lies in extending the coupled system with weak damping terms in the whole space to the coupled wave equations with scattering damping term and weak damping term on exterior domain.Using the iterative method and test function method,we study blow-up of solutions to the problem in different dimensions.And we also establish lifespan estimates of solutions to the problem.