Global Existence And Blow-Up For The Solutions To Nonlinear Parabolic-Elliptic System Modeling Chemotaxis

A basic quality of all living organisms is that they sense the environment in which they live and respond to it. The mechanism for the response is called taxis involving an external signal and the response of the orgnism to the signal. The response frequently involves movement toward or away from an external stimulus. The purposes of taxis range from movement toward food and avoidance of noxious substances to large-scale aggregation for the purpose of survival. Many different types of taxis are known, including aerotaxis, chemotaxis, geotaxis, haptotaxis, and others.The classical mathematical models which described the chemotactic pheonomenon were at first introduced by Keller and Segel as a model to describe the agrregation of slime mold amoebae due to an attractive chemical substance. A great many mathematicans and biologists have devoted themselves to this model, and obtained many perfect achievements. For example, Childress, Percus and Nagai have furthur studied the Keller-Segel model and suggesting the following.(i) Chemotactic collapse is not possible in one-dimensional case.(ii) In higher dimension cases, chemotactic collapse can occur, which in the two-dimensional case, chemotactic collapse can occur if the total cell number onΩis the larger than a critical number, but cannot occur for a total cell number onΩless than the critical number. Othmer and Stevens established the Othmer-Stevens model based on Keller-Segel model. There are no diffusion items in the control equations of the Othmer-Stevens model, which is much different from the classical Keller-Segel type. Othmer and Stevens, as well as Sleeman and Levine considered the Othmer-Stevens model and their execellent works show that this Othmer-Stevens model indeed reseals different biological phenomenon. However, there are very few results for there types of chemotaxis. The other types of model were also considered when the control equations are hypebolic type.In this paper we will study the following parabolic-elliptic system which is a kind of simplified Keller-Segel type. We will use some usefull mathmatical methods such as Lyapunov function, Moser iteration to investigate the problem. We proved that: (1) There are both global and non-global solutions in two dimension space under different conditions; (2) In higher dimension, the solution will blow-up in finite time under some initial condition.