Study On The Dynamic Properties Of Solutions For A Class Of Nonlocal Kirchhoff Models

This dissertation mainly investigates the dynamic properties of solutions for a class of nonlocal Kirchhoff models,such as the well-posedness,global existence and finite time blow-up of solutions.Firstly,we consider a Kirchhoff-type parabolic equation with fractional operator.We first extend the local existence results of solutions,then after making some suitable assumptions on the Kirchhoff function,we get the conditions on global existence and finite time blow-up of solutions with subcritical initial energy and critical initial energy.Secondly,through the further study of this Kirchhoff-type parabolic equation with fractional operator,we weaken the assumptions on the Kirchhoff function and we obtain the global existence and finite time blow-up of solutions with subcritical initial energy,critical initial energy and supercritical initial energy.Finally,we study a Kirchhoff-type parabolic equation with logarithmic nonlinearity.Many literatures use the logarithmic Sobolev inequality to deal with the logarithmic nonlinearity,but this inequality is not valid for the model studied in this dissertation,so we develop a new method to replace the logarithmic Sobolev inequality and we get the conditions on global existence and finite(infinite)time blow-up of solutions with subcritical initial energy and critical initial energy.More specifically,this dissertation is mainly divided into the following four chapters:In the first chapter,we first make a brief introduction to the Kirchhoff-type problems and potential well methods.Then we give the research background and purpose of this dissertation.Finally,we give the innovations and methods of this dissertation.In the second chapter,we investigate a Kirchhoff-type parabolic equation with fractional operator.Firstly,for the local existence of solutions,we extend existing results and obtain the global existence result of solutions.Secondly,after making some suitable assumptions on the Kirchhoff function,we get the conditions on global existence and finite time blow-up of solutions with subcritical initial energy and critical initial energy.In the third chapter,based on the second chapter,the conditions on global existence and finite time blow-up of solutions are further studied.By weakening the assumptions on the Kirchhoff function in the second chapter,we get the conditions on global existence and finite time blow-up of solutions with subcritical initial energy,critical initial energy and supercritical initial energy.Moreover,we also obtain the decay estimates for global solutions,the growth estimates for blow-up solutions and the upper and lower bounds of the blow-up time.Furthermore,we get a blowup condition which is independent of the depth of the potential well,and we study the upper bound of the blow-up time and the growth estimates for blow-up solutions.Finally,at subcritical initial energy and critical initial energy,we give some equivalent conditions for the solutions existing globally or blowing up in finite time.In the fourth chapter,we investigate a Kirchhoff-type parabolic equation with logarithmic nonlinearity.Firstly,by developing a new method to replace the logarithmic Sobolev inequality,we get the conditions on global existence and finite(infinite)time blow-up of solutions with subcritical initial energy and critical initial energy.Moreover,we also study the decay estimates for global solutions,the growth estimates for blow-up solutions and the upper and lower bounds of the blow-up time.Secondly,we obtain a blow-up condition which is independent of the depth of the potential well and study the upper bound of the blow-up time.Finally,by using the Lagrange multiplier method,we consider the existence of the ground state solutions and study the asymptotic behavior of the general global solutions.