The Research On Well-posedness Of Nonlinear Wave Equation With Logarithmic Source

Based on potential well theory,concavity method,energy perturbation method and functional analysis together,this thesis aims to reveal the relationship between the initial data and the well-posedness of solutions for the initial boundary value problem of the nonlinear wave equations with the logarithmic source under three different initial energy levels(sub-critical energy level,critical energy level and sup-critical energy level).Particularly,this thesis derives that the solution for the initial boundary value problem of the nonlinear wave equations with the logarithmic source under appropriate conditions will blow up at infinity time,which clearly indicates the influence of the logarithmic source on the dynamic behavior of the solutions.Chapter 2 focuses on the initial boundary value problem of the nonlinear wave equation with logarithmic source term,strong damping term and weak damping term at three possible energy levels.The local existence and uniqueness are proved by using the Galerkin method to construct the system of linear differential equations,and contract mapping principle.Further for the problem with the logarithmic source term,the proper potential wells are introduced to build up the variational structure and then give the depth of potential well also the properties of the corresponding functionals and manifolds.Then this work proves the global existence and infinite time blow up of the solution at both sub-critical initial energy level and the critical initial energy level.Also the exponential decay in time of the energy is obtained due to the presence of the damping terms after the global existence of solution.Finally,the infinite time blow up of the solution at the sup-critical energy level is obtained.Chapter 3 deals with the global existence and infinite time blow-up of solutions at different initial energy levels for the initial boundary value problem of the nonlinear fourth order damped wave equation with logarithmic source and nonlinear damping term.Firstly based on Galerkin method and boundedness principle it gives the global existence and infinite time blow up of solution at subcritical energy level.With the help of the scaling method,the global existence and infinite time blow up of solution are obtained at critical energy level.Finally,we gain the result of infinite time blow up at sup-critical energy level.Chapter 4 considers the global existence and finite time blow-up of solutions for Cauchy problem to the sixth order Boussinesq equations at subcritical energy level and the critical energy level.By taking the Fourier transform of the sixth order Boussinesq equation,the potential well is constructed,therefore the stable sets and unstable sets are obtained by the definition of the depth of potential well and contradictory arguments.Then by taking advantage of the compactness theorem and the concavity method respectively,the global existence and infinite time blow up of solutions at at subcritical energy level and the critical energy level are proved.