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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 326386, 17 pages
http://dx.doi.org/10.1155/2011/326386
Research Article

Global Existence and Blowup Analysis to Single-Species Bacillus System with Free Boundary

School of Mathematical Science, Yangzhou University, Yangzhou 225002, China

Received 28 January 2011; Accepted 26 May 2011

Academic Editor: Gaston Mandata N'Guerekata

Copyright © 2011 Zhi Ling and Zhigui Lin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper is concerned with a reaction-diffusion equation which describes the dynamics of single bacillus population with free boundary. The local existence and uniqueness of the solution are first obtained by using the contraction mapping theorem. Then we exhibit an energy condition, involving the initial data, under which the solution blows up in finite time. Finally we examine the long time behavior of global solutions; the global fast solution and slow solution are given. Our results show that blowup occurs if the death rate is small and the initial value is large enough. If the initial value is small the solution is global and fast, which decays at an exponential rate while there is a global slow solution provided that the death rate is small and the initial value is suitably large.