The Research On The Well-Posedness For High Order Wave Equation With Logarithmic Type Source

In this thesis,we study the initial boundary value problem of fourth order wave equation with nonlinear strain and logarithmic nonlinearity,the Cauchy problem of sixth order Boussinesq equation with logarithmic nonlinearity and the Cauchy problem of fourth order Boussinesq equation with strong damping term and logarithmic nonlinearity by the potential well method,the concavity mothod and the functional analysis.This study aims to reveal the affects of the initial data and the well-posedness of solutions for evolution equations with logarithmic nonlinearity.Chapter 2 undertakes a comprehensively study on the global well-posedness of solutions for fourth order wave equation with nonlinear strain and logarithmic nonlinearity at three initial energy levels.The local well-posedness is given first by Galerkin method and contracting mapping principle.Based on the local existence,we obtain the global existence and infinite time blow up of solutions at sub-critical and critical energy levels.Furthermore,this chapter makes the first try to obtain the infinite time blow up of the solution of the problem at the sup-critical energy level.Chapter 3 considers global well-posedness of solutions for Cauchy problem of a class of sixth-order Boussinesq equations with logarithmic nonlinearity at sub-critical and critical energy levels.This chapter gives exact value of the depth of potential well in the frame of potential well theory.Then,we obtain the global existence of the solutions at the sub-critical and critical energy levels.For the infinite blow up of sub-critical and critical energy levels,the auxiliary functions are constructed to prove the infinite time blow up.Chapter 4 focuses on global existence and infinite time blow up of solutions for Cauchy problem of fourth order Boussinesq equation with strong damping term and logarithmic nonlinearity at sub-critical and critical energy levels.This chapter gives exact value of the depth of potential well in the frame of potential well theory.Since this problem contains strong damping terms,the non-increasing energy is exhibited.Then,the global existence of solutions at sub-critical and critical energy levels are proved.For the infinite blow up of sub-critical and critical energy,the auxiliary functions are introduced to prove the infinite time blow up of the problem.