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

Zhi Ling,Zhigui Lin

doi:10.1155/2011/326386

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.

Highlights

As we know, mathematical aspects of biological population have been considered widely
D s2 t vξξ 0 < ξ < 1, 0 < t < T, v 1, t 0, 0 < t < T, 2.7 vξ 0, t 0, 0 < t < T, v ξ, 0 v0 ξ 0, 0 ξ 1 admits a unique solution v ∈ C1 α, 1 α /2 0, 1 × 0, T and v C1 α, 1 α /2 0,1 × 0,T
In order to investigate the behavior of the free boundary, we introduce the energy of the solution u at t by

Summary

Introduction

Mathematical aspects of biological population have been considered widely. The free boundary is regarded as the moving front, the detailed biological implication see Section 6 of 6 for the logistic model; the authors compared their results in biological terms with some documented ecological observations there In this way, we have the following problem for u x, t and a free boundary x s t such that ut − duxx Kau2 − bu, 0 < x < s t , 0 < t < T, u s t , t 0, 0 < t < T, ux 0, t 0, 0 < t < T, 1.2 s 0 s0 > 0, u x, 0 u0 x 0, 0 x s 0 , s t −μux s t , t , 0 < t < T, where the condition ux 0, t 0 indicates that the habitat is semiunbounded domain and there is no migration cross the left boundary.

Local Existence and Uniqueness

Finite Time Blowup

Global Fast Solution and Slow Solution