Study On The Properties Of The Solutions To A Nonlocal Parabolic Equation With Singular Potential

In this dissertation,we study a nonlocal parabolic equation with singular poten-tial on a bounded smooth domain with homogeneous Neumann boundary condition.And the global existence,blow-up and vacuum isolation of the solutions are consid-ered.First of all,we find a threshold of global existence and blow-up to the solutions of the equation by constructing Nehari manifold?energy functional and a series of new functionals.We also consider the existence of solutions when the initial data at the low energy level,i.e.,J(u0)?d,where J(u0)is the initial energy and d is the potential well depth.Secondly,when the initial energy is less than the critical initial energy,the vacuum isolating behavior of the solutions is also discussed and a vacuum isolation region is obtained.In the end,we prove that there exist solutions of the problem with arbitrary initial energy that blow up in finite time and two sufficient conditions for the finite time blow up are obtained.We also obtain the upper bounds of the blow-up time for blow-up solutions.This dissertation is divided into three parts:In the first part,we mainly give the current researches about the properties(such as global existence?finite time blow-up?vacuum isolation,etc.)of solutions to nonlinear parabolic equations,as well as the purposes,methods and innovations of this dissertation.In the second chapter,we study the properties of the solutions to a nonlocal parabolic equation with singular potential.We explain the existence space of solu-tions by using the embedding theorem and some inequalities of the Sobolev space.The global existence?blow-up and vacuum isolating behavior of the solutions with low initial energy are discussed by means of potential well method and disproof method.In the third chapter,on the basis of the second chapter,the properties of blowing up in finite time are further studied.By constructing some new energy functionals and a series of sets,we prove the existence of blow-up solutions with arbitrary initial energy,find out two sufficient conditions of blow-up by using concavity method.And we also obtain the upper bounds of the blow-up time for blow-up solutions.