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


Introduction
As we know, mathematical aspects of biological population have been considered widely.Most of the authors have studied growth and diffusions of biological population in a homogeneous or heterogeneous fixed environment 1, 2 , and the nonlinear differential equations are described such as Logistic equation and Fisher equation.
In this paper, we consider the following single bacillus population model: The present paper aims to investigate the parabolic equation with a moving boundary in one-dimensional space.
As in 5 , assumed that the amount of the species flowing across the free boundary is increasing with respect to the moving length, the condition on the interface free boundary is s t −μu x s t , t by using the Taylor expansion.Here μ is a positive constant and measures the ability of the bacillus disperse in a new area.The free boundary is regarded as the moving front, the detailed biological implication see Section 6 of 6 for the logistic model; the authors also 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 u t − du xx Kau 2 − bu, 0 < x < s t , 0 < t < T, u s t , t 0, 0 < t < T, u x 0, t 0, 0 < t < T, s 0 s 0 > 0, u x, 0 u 0 x 0, 0 x s 0 , s t −μu x s t , t , 0 < t < T, where the condition u x 0, t 0 indicates that the habitat is semiunbounded domain and there is no migration cross the left boundary.
When Ka b 0, the problem is reduced to one phase Stefan problem, which accounts for phase transitions between solid and fluid states such as the melting of ice in contact with water 7 .Stefan problems have been studied by many authors.For example, the weak solution was considered by Oleȋnik in 8 , and the existence of a classical solution was given by Kinderlehrer and Nirenberg in 9 .For the two-phase Stefan problem, the local classical solution was obtained in 10, 11 , and the global classical solution was given by Borodin in 12 .
The free boundary problems have been investigated in many areas, for example, the decrease of oxygen in a muscle in the vicinity of a clotted bloodvessel 13 , the etching problem 14 , the combustion process 15 , the American option pricing problem 16, 17 , chemical vapor deposition in hot wall reactor 18 , image processing 19 , wound healing 20 , tumor growth 21-24 and the dynamics of population 5, 25, 26 .In this paper, we consider the free boundary problem 1.2 and focus on studying the blowup behavior of the solution and asymptotic behavior of the global solutions.We will give sufficient conditions to ensure the existence of fast solution and slow solution.Here if T ∞, we say the solution exists globally whereas if the solution ceases to exist for some finite time, that is, T < ∞ and lim t → T u x, t L ∞ 0,s t × 0,t → ∞, we say that the solution blows up.If T ∞ and lim t → ∞ s t < ∞, the solution is called fast solution since that the solution decays uniformly to 0 at an exponential rate, while if T ∞ and lim t → ∞ s t ∞, it is called slow solution, see 27, 28 in detail.
The remainder of this paper is organized as follows.In Section 2, local existence and uniqueness will be given.Section 3 deals with the result of blowup behavior by constructing an energy condition.Section 4 is devoted to long time behaviors of global solutions, including the existence of global fast solution and slow solution.

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.
Proof.We first make a change of variable to straighten the free boundary.Let Then the problem 1.2 is reduced to

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

2.5
It is easy to see that Σ T : U T × S T is a complete metric space with the metric Next applying standard L p theory and the Sobolev imbedding theorem see 29 , we then find that for any v, s ∈ Σ T , the following initial boundary value problem: where p 3/ 1 − α , C 1 is a constant dependent on α, s 0 and u 0 C 1 α 0,s 0 .Defining s by using the last equation of 2.4 we have and hence s ∈ C α/2 0, T with

2.13
Therefore if we take To prove that F is a contraction mapping on Σ T for T > 0 sufficiently small, we take v i , s i ∈ Σ T i 1, 2 and denote v i , s i F v i , s i .Then it follows form 2.8 and 2.11 that

2.15
Using the W 2,1,p estimates for parabolic equations and Sobolev's imbedding yields where C 3 is independent of T .Taking the difference of the equations for s 1 s 1 , s 2 s 2 results in Combining inequalities 2.16 and 2.17 , we obtain where C 4 is independent of T .Using the property of norm 19 and the fact s 1 0 s 2 0 , s i t s 0 and s 2 t s * 1 /s 0 give that Hence, for

2.23
Thus for this T , F is a contraction.Now using the contraction mapping theorem gives the conclusion that there is a v ξ, t , s t in Σ T such that F v ξ, t , s t v ξ, t , s t .In other words, v ξ, t , s t is the solution of the problem 2.4 and therefore u x, t , s t is the solution of the problem 1.2 .Moreover, by using the Schauder estimates, we have additional regularity of the solution, s t ∈ C 1 α/2 0, T and u ∈ C 2 α,1 α/2 0, s t × 0, T .Thus u x, t , s t is the classical solution of the problem 1.2 .Now we give the monotone behavior of the free boundary s t .
Theorem 2.2.The free boundary for the problem 1.2 is strictly monotone increasing, that is, for any solution in 0, T , one has s t > 0 for 0 < t T.

2.24
Proof.Using the Hopf lemma to the equation 1.2 yields that u x s t , t < 0 for 0 < t T.

2.25
Thus, combining this inequality with the Stefan condition gives the result.

Finite Time Blowup
In this section we discuss the blowup behavior.First we present the following lemma.
Lemma 3.1.The solution of the problem 1.2 exists and is unique, and it can be extended to 0, T max where T max ∞.Moreover, if T max < ∞, one has Proof.It follows from the uniqueness and Zorn's lemma that there is a number T max such that 0, T max is the maximal time interval in which the solution exists.In order to prove the present lemma, it suffices to show that, when T max < ∞,

3.2
In what follows we use the contradiction argument.Assume that T max < ∞ and u L ∞ 0,s t × 0,t < ∞.Since s t is bounded in 0, T max by Theorem 2.1, using a bootstrap argument and Schauder's estimate yields a priori bound of u t, x C 1 α 0,s t for all t ∈ 0, T max .Let the bound be M * .It follows from the proof of Theorem 2.1 that there exists a τ > 0 depending only on M * such that the solution of the problem 1.2 with the initial time T max − τ/2 can be extended uniquely to the time T max − τ/2 τ, which contradicts the assumption.This completes the proof.
In order to investigate the behavior of the free boundary, we introduce the energy of the solution u at t by Proof.We see the following auxiliary free boundary problem:

3.13
By the same argument as in Theorem 2.1, the solution of the above problem exists for all t > 0 since the solution is bounded.Moreover, one can deduce from the maximum principle that u v 0 and s t h t s 0 on 0, T max .Similarly as in Lemma 3.2, denoting |v t | 1 h t 0 v t, x dx, we easily obtains Using H ölder's inequality and the fact that s t h t s 0 yields that for all t 0,

10 Abstract and Applied Analysis
On the other hand, by the maximum principle, we have v w, where w is the solution of the following Cauchy problem:

3.17
By the L 1 − L ∞ estimate for the heat equation, we have hence, by 3.14 ,

3.19
Therefore, we have

3.29
This implies that G is concave, decreasing, and positive for t t 1 , which is impossible.The contradiction shows that T max < ∞, which gives the blowup result.
Remark 3.5.The above theorem shows that the solution of the free boundary problem 1.2 blows up if the death rate b is sufficiently small and the initial datum u 0 is sufficiently large.

Global Fast Solution and Slow Solution
In this section, we study the asymptotic behavior of the global solutions of 1.2 .We first give the following existence of fast solution.
Theorem 4.1 fast solution .Let u be a solution of problem 1.2 .If u 0 is small in the following sense: where γ, β and ε > 0 to be chosen later.

Abstract and Applied Analysis 13
An easy computation yields for all t > 0 and 0 < x < ϑ t .
By using the maximum principle, one then sees that s t < ϑ t and u x, t < v x, t for 0 x s t , as long as u exists.In particular, it follows from the continuation property 3.1 that u exists globally.The proof is complete.
Remark 4.2.The above result shows that the free boundary converges to a finite limit and that the solution u t decays uniformly to 0 at an exponential rate.Compared to the case see Theorem 4.5 , the free boundary grows up to infinity and the decay rate of the solution is at most polynomial, the former solution is therefore called fast solution.
Before we give the existence result of slow solution, we need the following uniform a priori estimate for all global solutions of problem 1.2 .Proposition 4.3.Let u be a solution of the problem 1.2 with T max ∞.Then there is a constant where C remains bounded for u 0 C 1 α , s 0 , and 1/s 0 bounded.
Proof.First from the local theory for problem 1.2 , for each M > 1 there exists σ > 0 such that, if u 0 C 1 α < M and 1/M < s 0 < M, then u x, t L ∞ < 2M on 0, σ .
Proof.Assume that l : lim sup t → ∞ u x, t L ∞ 0,s t > 0 by contradiction.It follows from Proposition 4.3 that l < ∞.Let t 0 > 0 be such that sup t 0 , ∞ u x, t L ∞ 0,s t 3/2 l.Then there exists a sequence t n → ∞ such that u x, t n L ∞ 0,s t n 3/4 l.Now pick x n ∈ 0, s n t n such that u x, t n L ∞ 0,s t n u x n , t n .

4.15
Therefore the function v n satisfies v n 0, 0 1, 0 v n 2, lim y → ∞ v n y, τ 0 and Theorem 4.5 slow solution .Let φ ∈ C 1 0, s 0 satisfy φ 0, φ / ≡ 0 with φ x 0 φ s 0 0, and b satisfy the same condition as in Theorem 3.4.Then there exists λ > 0 such that the solution of 1.2 with initial data u 0 λφ is a global slow solution, which satisfies that s ∞ ∞ and s t O t 2/3 as t −→ ∞. 4.17 Proof.Denote the solution of 1.2 as u u 0 ; • to emphasize the dependence on the initial function u 0 when necessary.So as to s t , s ∞ and the maximal existence time T max .Similarly as in 27 , define Σ λ > 0; T max λφ ∞ and s ∞ λφ < ∞ .

4.18
According to the Theorem 4.1, Σ / ∅ since that the solution is global if λ is sufficiently small.

u t − dΔu Kau 2
− bu, 1.1 which was first proposed by Verhulst see 3 .Parameters a, b, d and K are positive constants.Ecologically, a represents the net birth rate, b is the death rate, d denotes the diffusion coefficient, and K measures the living resource for bacillus.In 4 , Jin et al. considered the model and established a time-dependent dynamic basis to quantitatively clarify the biological wave behavior of the popular growth and propagation.

16 where b n λ 2
n b.Note that b n b 3l/4 −1 , therefore there exists a subsequence {b n k } and b * b 3l/4 −1 such that b n k → b * as k → ∞.Similarly as Lemmas 2.1-2.3 in 27 , we have obtained a function w y 0, bounded and continuous on 0, ∞ and satisfies that −w yy Kaw 2 − b * w.Therefore w ≡ 0 or w ≡ b * /Ka.If w ≡ 0, this is a contradiction to the fact that w 0 1 since v n 0, 0 1.If w ≡ b * /Ka, this is also a contradiction to the fact that lim y → ∞ w y 0.

Theorem 3.4. Let
u be the solution of the problem 1.2 , if b < πd 3 /8 2ds 0 μ|u 0 | 1 2 , then one has and there exist real numbers C, β > 0 depending on u 0 such that −1/2.We extend u •, t by 0 on s t , ∞ and define the rescaled function . But this implies that s ∞ λφ < ∞ by Theorem 4.1, which is a contradiction with the definition of λ * .On the other hand, as a consequence of the blowup result of Theorem 3.4, we know that T max ∞ implies E t 0 for all t 0, hence by using 3.4 we have * sufficiently close to λ * t 0 s τ 3 dτ 1/3 t 2/3 .The proof is complete.