Global And Blow-up Of Solution For A Class Of Nonlinear Wave Equations

Since nonlinear wave equations have enormous potential in wide varieties of applica-tions, many specialists and scholars have devoted themselves to the research of the theoryand have achieved many perfect productions. In this paper, we perform researches of theglobal existence and blow up of solutions to the following nonlinear wave equations.The main contents in this paper can be summarized as follows:1. In the first section, we introduce the developmental process and significance ofnonlinear evolution equations. And we introduce the research results and some recentadvances for the above nonlinear wave equations and the organization of this paper isgiven.2. In the second section, we mainly discuss the existence and uniqueness of local solu-tions for the above nonlinear wave equations by using successive approximation methodsand energy estimate methods on the base of the existence and uniqueness of solutions oflinear wave equations.3. In the third section, we shall prove the existence of global solutions for the abovenonlinear wave equations by the energy functional and local continuity method. Especiallywe also show the global existence and uniqueness of the trivial solution for the abovenonlinear wave equations when the initial data are zero functions.4. In the fourth section, we shall prove the existence and uniqueness of weak solutionsfor the above nonlinear wave equations under the growth conditions on the nonlinearterms.5. In the fifth section, we shall prove the blow up of the solutions for the abovenonlinear wave equations by using the energy estimate methods and di?erential inequality. Moreover, we get the estimate of the lifespan.6. In the sixth section, some conclusions of the paper are given.