Study On Properties Of Solutions For Nonlinear Sixth-order Wave Equation With Strong Damping

In recent decades,nonlinear partial differential equations(e.g.wave equations,plate equations,thermal equations)have been used in physics,materials Science,water wave problems and other fields are becoming more and more widely used,and many experts pay more and more attention to such equations.In nonlinear partial differential equations,the existence and asymptotic behavior of solutions to wave equations are a major research direction.Particularly in the last few years,more and more scholars have invested in sixthorder wave equations.This thesis focuses on the properties of solutions for nonlinear sixthorder wave equations with strong damping.In section 1,This paper describes the current research status of the generalized form of nonlinear sixth-order wave equations and nonlinear wave equations with strong damping at home and abroad.In section 2,we consider the Cauchy problem for a class of nonlinear sixth-order wave equation with source.By giving some mild conditions on the initial values,we using Levine's convexity method,to establish the blow up result in finite time of solution at supercritical initial energy level.In section 3,we research a cauchy problem for a class nonlinear sixth-order wave equation with strong damping.By giving suitable conditions on the nonlinear term and the initial values,we prove the existence of local solution.We use the generalized concavity method of Levine,to establish the blow up result in finite time of solution with nonnegative initial energy.We also obtain the global existence and asymptotic behavior of solution by introducing the potential well method and constructing a Lyapunov functional.