Blow Up And Global Solutions For A Nonlocal Equation Of Parabolic Type

This paper mainly studies the global existence and blow up properties for the following nonlocal parabolic equation with Dirichlet boundary condition in a smooth bounded region whereParabolic equation with nonlocal term plays a role in extensive fields such as the study of the behavior of a population in ecology, nonlocal heat equation. Many mathematicians such as Sattinger, Ikehata, Suzuki and Payne and so on had extensively studied the equations and obtained a wealth of conclusions. We will state some related results in the first part of this paper.The second part of this paper is the core of our research work, we introduce some basic knowledge to be used later and make a brief introduction at first. Then we use the scaling method, contradiction method to devote to the proof of our main results as follow.Defining the energy functional E(u): the Nehari functional and Nehari manifold N: Finding two invariant sets W and V by using a powerful method called potential well method. where(1) When u0∈C(Ω) (?) W01.p(Ω), and1 If u0 ∈ W, then the problem (0.1) admits a global solution, that is Tmax(u0)=∞(2) When u0 ∈C(Ω) (?)W01.p(Ω), and If u0 ∈V, then the solution of problem (0.1) blows up in finite time, that is Tmax(u0)<∞In the third part, we make a summary and give some new problems.