Global Existence And Blow-up Of The Solutions To Some Classes Of Hyperbolic Equations

Nonlinear partial differential equations (NPDE) is one of the main research fields in nonlinear science. An intensive study of NPDE will accelerate the development of nonlinear analysis. This dis-sertation focus on the study of the global solutions, blow-up solutions of some nonlinear hyperbolic equations.The thesis is composed of three chapters.In chapter 1, the investigation history and the study status of the problems which are discussed in this thesis are listed, and the preliminaries, some important differential inequalities will be stated.In section 2, we deal with the solutions of initial-boundary value problem of the nonlinear hyperbolic equations By constructing suitable auxiliary functions and using differential inequalities, we prove that under some conditions on the functionσ(s), the nonlinear damping term and the nonlinear source term, the system has a global solution and a blow up solution, respectively. We can get the estimate for the blow-up time.In section 3, we deal with the solutions of initial-boundary value problem of the nonlinear viscoelastic hyperbolic equations with nonlinear damping By constructing suitable auxiliary functions and using differential inequalities.we prove that under some conditions on the functionσ(s), the nonlinear damping term and the nonlinear source term, the system has a blow-up solution. We can get the estimate for the blow-up time.