The Research On Well-poseness For Two Classes Of Fractional Laplacian Parabolic Equations

The thesis considers the well-posedness of solutions for two classes of fractional Laplacian parabolic equations combining the potential well theory and functional analysis,along with the Galerkin method and the concave method.This kind of equation is derived from the modeling of the anomalous diffusion phenomenon of long-range correlation in space and long-term memory effect in time.The main purpose of this thesis is to study the global well-posedness of solutions at subcritical initial energy,critical initial energy and supercritical initial energy,respectively,and to seek the effect of the Kirhhoff function and the fractional Laplacian on the nature of solutions.The second chapter studies the initial boundary value problem of a class of fractional parabolic equation with Kirchhoff term and nonlinear polynomial source term.In this chapter,by giving the potential energy functional,Nehari functional,unstable set,potential well depth,and some related lemmas,along with the potential well theory and the concave method,the conclusions of finite time blow up of the solution at three different energy levels(i.e.subcritical energy level,critical energy level and supercritical energy level)are proved,and also the upper bounds of the finite time blow up of the solution at the subcritical and supercritical energy levels are estimated,which solves the blow up problem for this considered equation left in [14].The third chapter investigates the initial boundary value problem for a class of fractional parabolic equation with Kirchhoff term,fractional damping and nonlinear polynomial source term.In this chapter,the existence and uniqueness of local solution is proved by the Banach's fixed point theorem firstly.Subsequently,in the framework of potential well,the Galerkin method is used to prove the global existence of the solution and derive that the solution decays exponentially and in polynomial form via the Gronwall's inequality and the method of separation of variables,respectively.It is shown that the concave method allows the finite time blow up along with the upper and lower bounds of blow up time at subcritical initial energy level.At the same time,with the aid of scale transform,the global existence,the finite time blow up and the asymptotic behavior of the solution are generalized from the subcritical initial energy level to the critical initial energy level.Finally,the finite time blow up of the solution and the upper bound of the blow up time are obtained with the help of the control inequality and an improved concave method.The fourth chapter is concerned with the initial boundary value problem for a class of fractional parabolic equation with Kirchhoff term and exponential source term.Based on the potential well method,the Galerkin method,an improved concave method and the fractional Moser-Trudinger inequality,the global existence and the finite time blow up of the solution at subcritical energy level and critical energy level are obtained in this chapter,respectively.