The Research On Two Types Of Nonlinear Wave Equations

Nonlinear wave equation is an important mathematical model,which is often used to describe natural phenomena,and is one of the most advanced research topics in nonlinear mathematical physics science.Because of its important application background and the mathematical difficulties caused by nonlinear,it has aroused people's great interest in research,and has wide application and strong vitality.Through the study of nonlinear wave equation and qualitative analysis,it is helpful for people to understand the essential characteristics of the system,and greatly promote the development of related disciplines such as physics,mechanics and engineering technology.In this thesis,we study some properties of the solutions for two types of nonlinear wave equations.One is the damped Rosenau equation,the other is the modified b-family of equations.It mainly includes the following several parts:In chapter 1,we introduce the physical background,research significance,research status and development trend of the two types of nonlinear wave equations.In chapter 2,we study the Cauchy problem for a class of damped Rosenau equations.Firstly,using the contraction mapping principle,the existence and uniqueness of the solution are proved.And then the blow-up of the solution of the equation is obtained by the concavity method.At last,we consider the asymptotic behavior of the solution by utilizing an estimate for the uniform decay of solutions of the linearized version.In chapter 3,we discuss the persistence property of solutions to the modified b-family of equations.Firstly,we present some basic knowledge about persistence property.Then through the estimation of the solutions,we establish that certain decay properties of the initial data persist as long as the solution exists.