The Research On Well-posedness For Some Classes Of Evolution Equations With Exponential Source And General Source

This thesis aims to reveal the relationship between the initial data and the well-posedness of solutions for evolution equations with the exponential source and general source by using potential well theory,concavity method and functional analysis together.What's more,while considering the evolution equations with exponential source,this thesis tries to figure out the behavior of solutions under different initial energy level(sub-critical energy level,critical energy level and sup-critical energy level),which clearly show the influence of the initial data on the dynamic behavior of the solutions.Chapter 2 undertakes a comprehensively study on the global well-posedness of solution-s for 2-D Klein-Gordon equation with exponential nonlinearity at full initial energy level,and gives the relationship between nonlinear exponential source and the depth of potential well.By using Galerkin method,the related linear problem is constructed,then the local well-posedness is given by combing with the contracting mapping principle.The structure of potential well is introduced,based on this,the properties and relationship of different function-als is obtained,then with the help of variational theory the depth of potential well is defined.By using boundedness theorem,concavity method and the relationship between the depth of potential well and initial energy,the global existence and finite time blow-up of solutions at different initial energy level are proved.Furthermore,this thesis makes a first try to investigate the behavior of solution at sup-critical initial energy level.Chapter 3 focuses on the global existence and finite time blow-up of solutions at different initial energy for 2-D wave equation with exponential source and nonlinear damping.By employing fixed point theorem,the local existence of solution is given,then by using potential well method and concavity method,the sharp condition of global existence and nonexistence is obtained.Under three different initial energy level,this chapter analysis the relationship between the initial data and solutions.This thesis firstly consider the wave equation with exponential source and nonlinear damping at different initial energy level.Chapter 4 considers the global existence and finite time blow-up of solutions for three classes of evolution equations.A new assumption to the general source is introduced in this thesis,which can include more classical nonlinearities in former literatures.Then for every equation,the potential well structure is constructed,therefore the stable sets and unstable sets are obtained by the definition of the depth of potential well and contradiction method,then by undertaking the compactness theorem and the concavity method respectively,the global existence and nonexistence of solutions at sub-critical are proved.This thesis makes the first attempt to study the dynamic behavior of nonlinear wave equation,nonlinear heat equation and NLS equation in the same frame which further extend the potential well theory.