In this paper, we study a parabolic-elliptic system defined on a bounded domain of R3, which comes from a chemotactic model. We first prove the existence and uniqueness of local in time solution to this problem in the Sobolev spaces framework, then we study the norm behavior of solution, which may help us to determine the blow-up norm of the maximal solution.

In this paper we study the global in-time and blow-up solutions for the simplified Keller-Segel system modelling chemotaxis. We prove that there is a critical number which determines the occurrence of blow-up in two dimensional case for 1＜p＜2. In three or higher dimensional cases, we show that the radial symmetrical solution will be blow up if 1＜p＜N/N-2 (N≥3) for nonnegative initial value.

In this paper we investigate the properties of the solution for reaction-diffusion systems due to Othmer-Stevens which arise from 1-dimensional chemotaxis model, and prove some results for the solution to be blow-up in finite-time.

In this paper we investigate the properties of the solutions for some general reaction-diffusion systems due to Othmer-Stevens which arise in modelling chemotaxis, and prove some results about collapse and finite-time action of certain local modifications of the environment.

In this paper, we consider some classes of non linear partial differential equations with regular singularity and irregular singularity with respect to t=0 and x=0 respectively. Our purpose is to establish a result similar to the k-summability known in the case of singular ordinary differential equations. It's shown that, under some generic type conditions (holomorphic to t), all formal solutions are Borel summable or k-summable with respect to xin all directions except at most a countable number.