Some Properties Of The Solutions For Several Nonlocal Diffusion Systems

In this thesis, we will discuss the second critical exponent for a nonlocal diffusion system、global solutions for a nonlocal diffusion systems with a reaction term and a Dirichlet boundary condition、the blow-up analysis for a nonlocal porous medium diffusion system with a reaction term and a absorption term.This thesis consists of five chapters:the first two chapters are introduction and some basic knowledge. In the introduction, the background and current research for the three parabolic problems are given. In the second chapter,some basic knowledge to be used in this thesis is given.In the third, fourth and fifth chapter, we separately consider the following systemIn the third chapter of this thesis, we consider the second critical exponent of solution for the nonlocal diffusion system (3.1).In this chapter, firstly we give the related properties of the first eigenvalue and the first characteristic function for the system (3.1).Then this chapter proves the two lemma. And we obtain the blow-up exponent according to the initial value function to meet the blow-up condition of the solution in a finite time.By establishing the global super-solution and using comparison principle, we obtain the exponent of the global solution.Above all, the second critical exponent is established.In the fourth chapter, we discuss the global solution of system (4.1)with Dirichlet boundary condition. We applied the method of Banach fixed point theorem to prove the existence of local solutions. By making positive lemma,we proved the comparison principle of system (4.1).Finally we prove the existence of global solution by means of super-solution to the problems mentioned above.In the fifth chapter, we study the properties of the solution for the nonlocal porous medium diffusion system (5.1).In this chapter, we establish the existence-uniqueness of local solution and comparison principle.Then by using the method of super-solution and sub-solution, the global existence and finite time blow-up of the solution for the system are established.At last,we not only get the blow-up rate of the solution, but also the blow-up set are obtained.