Research On The Properties Of Solutions For Two Types Of Nonlinear Wave Equations

As a kind of partial differential equation,nonlinear wave equation is an important mathematical model often used to describe natural phenomena.In the process of solving and qualitative analysis of nonlinear wave equation,it is helpful for us to grasp the essential characteristics of the system,and it also promotes the progress of other disciplines and the development of the scientific community to a large extent.Thesis mainly studies the related properties of the solutions of the following two kinds of nonlinear wave equations:One is the Boussinesq equation with double damping;Another is Camassa-Holm equation with nonlinear dissipative term;In order to elaborate the relevant properties of solutions,the following aspects are developed in thesis.1.Introduce the physical background,research significance,research status and de-velopment trend of nonlinear wave equation at home and abroad.2.The Cauchy problem of the sixth-order Boussinesq equation with double damp-ing is studied.Firstly,the well-posedness of the local solution is derived by using the compression mapping principle,and the existence of the global solution is obtained by establishing a prior estimate of the local solution.Finally,the asymptotic property of the solution is proved by using the multiplier method.3.The Camassa-Holm equation with high order nonlinear dissipative term is dis-cussed.Firstly,the blow up criterion of Cauchy problem of the equation is established by the characteristic method and Ricatti type differential inequality,and the wave breaking phenomenon is obtained.Due to the existence of higher order nonlinear term,the conser-vation law E=_Ru~2+u_x~2dx does not hold.This difficulty can be solved by establishing energy inequalities.Finally,the persistence property of the solution and some unique continuation properties are established in the weighted L~pspace.