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

and Applied Analysis 3 an energy condition. Section 4 is devoted to long time behaviors of global solutions, including the existence of global fast solution and slow solution. 2. Local Existence and Uniqueness In this section, we prove the following local existence and uniqueness of the solution to 1.2 by contraction mapping principle. Theorem 2.1. For any given u0 satisfying u0 ∈ C1 α 0, s0 with α ∈ 0, 1 , u0 0 u0 s0 0 and u0 > 0 in 0, s0 , there is a T > 0 such that problem 1.2 admits a unique solution u, s ∈ C1 α, 1 α /2 0, s t × 0, T × C1 α/2 0, T . 2.1 Furthermore, ‖u x, t ‖C1 α, 1 α /2 0,s t × 0,T ‖s t ‖C1 α/2 0,T C, 2.2 where C and T depend only on α, s0 and ‖u0‖C1 α 0,s0 . Proof. We first make a change of variable to straighten the free boundary. Let ξ x s t , u x, t v ξ, t . 2.3 Then the problem 1.2 is reduced to vt − s ′ t s t ξvξ − d s2 t vξξ Kav2 − bv, 0 < ξ < 1, 0 < t < T, v 1, t 0, 0 < t < T, vξ 0, t 0, 0 < t < T, s 0 s0 > 0, v ξ, 0 v0 ξ : u0 s0ξ 0, 0 ξ 1, s′ t − μ s t vξ 1, t , 0 < t < T. 2.4 This transformation changes the free boundary x s t to the fixed line ξ 1 at the expense of making the equation more complicated. In the first equation of 2.4 , the coefficients contain the unknown s t . Now we denote s∗ −μv′ 0 1 and set ST { s ∈ C1 0, T : s 0 s0, s′ 0 s0 s∗, 0 s′ t s t s∗ 1 } , UT { v ∈ C 0, 1 × 0, T : v ξ, 0 v0 ξ , ‖v − v0‖C 0,1 × 0,T 1 } . 2.5 4 Abstract and Applied Analysis It is easy to see that ΣT : UT × ST is a complete metric space with the metric D v1, s1 , v2, s2 ‖v1 − v2‖C 0,1 × 0,T ∥s′1s1 − s2s2 ∥∥ C 0,T . 2.6 Next applying standard L theory and the Sobolev imbedding theorem see 29 , we then find that for any v, s ∈ ΣT , the following initial boundary value problem: ṽt − s ′ t s t ξṽξ − d s2 t ṽξξ Kav2 − bv, 0 < ξ < 1, 0 < t < T, ṽ 1, t 0, 0 < t < T, ṽξ 0, t 0, 0 < t < T, ṽ ξ, 0 v0 ξ 0, 0 ξ 1 2.7 admits a unique solution ṽ ∈ C1 α, 1 α /2 0, 1 × 0, T and ‖ṽ‖C1 α, 1 α /2 0,1 × 0,T C‖ṽ‖W2,1,p 0,1 × 0,T C1, 2.8 where p 3/ 1 − α , C1 is a constant dependent on α, s0 and ‖u0‖C1 α 0,s0 . Defining s̃ by using the last equation of 2.4