Global Existence and Blowup Analysis to Single-Species Bacillus System with Free Boundary
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.
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: which was first proposed by Verhulst see . Parameters and are positive constants. Ecologically, represents the net birth rate, is the death rate, denotes the diffusion coefficient, and measures the living resource for bacillus. In , 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.
The present paper aims to investigate the parabolic equation with a moving boundary in one-dimensional space.
As in , 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 by using the Taylor expansion. Here 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  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 and a free boundary such that where the condition indicates that the habitat is semiunbounded domain and there is no migration cross the left boundary.
When , 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 . Stefan problems have been studied by many authors. For example, the weak solution was considered by Oleĭnik in , and the existence of a classical solution was given by Kinderlehrer and Nirenberg in . 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 .
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 , the etching problem , the combustion process , the American option pricing problem [16, 17], chemical vapor deposition in hot wall reactor , image processing , wound healing , 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 , we say the solution exists globally whereas if the solution ceases to exist for some finite time, that is, and , we say that the solution blows up. If and , the solution is called fast solution since that the solution decays uniformly to 0 at an exponential rate, while if and , 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.
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 satisfying with , and in , there is a such that problem (1.2) admits a unique solution Furthermore, where and depend only on and .
Proof. We first make a change of variable to straighten the free boundary. Let
Then the problem (1.2) is reduced to
This transformation changes the free boundary to the fixed line at the expense of making the equation more complicated. In the first equation of (2.4), the coefficients contain the unknown .
Now we denote and set It is easy to see that is a complete metric space with the metric
Next applying standard theory and the Sobolev imbedding theorem (see ), we then find that for any , the following initial boundary value problem: admits a unique solution and where , is a constant dependent on and .
Defining by using the last equation of (2.4) we have and hence with
Define map : by It is clear that is a fixed point of if and only if it solves (2.4).
By (2.10) and (2.11), we have Therefore if we take , then maps into itself.
To prove that is a contraction mapping on for sufficiently small, we take and denote . Then it follows form (2.8) and (2.11) that
By setting it follows that satisfies
Using the estimates for parabolic equations and Sobolev’s imbedding yields where is independent of . Taking the difference of the equations for results in Combining inequalities (2.16) and (2.17), we obtain where is independent of . Using the property of norm and the fact and give that if . Further, Hence, for we have Thus for this , is a contraction. Now using the contraction mapping theorem gives the conclusion that there is a in such that . In other words, is the solution of the problem (2.4) and therefore is the solution of the problem (1.2). Moreover, by using the Schauder estimates, we have additional regularity of the solution, and . Thus is the classical solution of the problem (1.2).
Now we give the monotone behavior of the free boundary .
Theorem 2.2. The free boundary for the problem (1.2) is strictly monotone increasing, that is, for any solution in , one has
Proof. Using the Hopf lemma to the equation (1.2) yields that Thus, combining this inequality with the Stefan condition gives the result.
Remark 2.3. If the initial function is smooth and satisfies the consistency condition: then the solution .
3. 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 where . Moreover, if , one has
Proof. It follows from the uniqueness and Zorn’s lemma that there is a number such that is the maximal time interval in which the solution exists. In order to prove the present lemma, it suffices to show that, when ,
In what follows we use the contradiction argument. Assume that and . Since is bounded in by Theorem 2.1, using a bootstrap argument and Schauder's estimate yields a priori bound of for all . Let the bound be . It follows from the proof of Theorem 2.1 that there exists a depending only on such that the solution of the problem (1.2) with the initial time can be extended uniquely to the time , 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 at by and its -norm by . Then we have the following lemma.
Lemma 3.2. Let be the solution of the problem (1.2). Then one has the relations
Proof. It is easy to see that
Integrating by parts and using yield
Differentiating the second equation of (1.2) with respect to , we have
which implies that
By substitution, we get
that is (3.4).
Now we show (3.5). It is obvious that Integrating the equation above, we get This completes the proof.
Lemma 3.3. Assume , and let . If , then one has .
Proof. We see the following auxiliary free boundary problem:
By the same argument as in Theorem 2.1, the solution of the above problem exists for all since the solution is bounded. Moreover, one can deduce from the maximum principle that and on . Similarly as in Lemma 3.2, denoting , we easily obtains
Using Hölder's inequality and the fact that yields that for all ,
so we have
On the other hand, by the maximum principle, we have , where is the solution of the following Cauchy problem: By the estimate for the heat equation, we have hence, by (3.14), Therefore, we have for . If , we get . Taking in the inequality (3.16) yields the desired estimate.
Theorem 3.4. Let be the solution of the problem (1.2), if , then one has whenever
Proof. As in , define the function
Direct calculations show that and
It follows from the identity (3.4) that
Now assume by contradiction. The assumption (3.20), together with Lemma 3.3, implies that
for all sufficiently large, we then have
Applying the Cauchy-Schwarz inequality yields
since by (3.22).
On the other hand, (3.25) implies that so that . We then obtain for some large .
Defining for , it follows that This implies that is concave, decreasing, and positive for , which is impossible. The contradiction shows that , 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 () is sufficiently small and the initial datum () is sufficiently large.
4. 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 be a solution of problem (1.2). If is small in the following sense: then . Moreover, and there exist real numbers , depending on such that
Proof. It suffices to construct a suitable global supersolution. Inspired by , we define
where and to be chosen later.
An easy computation yields for all and .
On the other hand, we have and . Setting , and , it follows that Assume that and choose , we also get for .
By using the maximum principle, one then sees that and for , as long as exists. In particular, it follows from the continuation property (3.1) that 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 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 be a solution of the problem (1.2) with . Then there is a constant , such that where remains bounded for , , and bounded.
Proof. First from the local theory for problem (1.2), for each there exists such that, if and , then on .
Assume that the result is false. Then there exists a and a sequence of global solutions of (1.2), such that For all large there exist and such that We define , then it is evident to see that as . We extend by 0 on and define the rescaled function for . Also, we denote which corresponds to the domain . The function satisfies and where . Note that as . Similarly as Lemma 2.1 in , there exists a subsequence of such that converges in to a function and converges in to a function , which satisfies . Moreover, similarly as Lemmas 2.2 and 2.3 in , in and there is a function which is bounded, continuous on and satisfies that , hence is concave. Therefore , which leads to a contradiction to the fact that . This completes the proof of Proposition 4.3.
The above proposition shows that all global solutions are uniformly bounded and the coming result implies that all global solutions decay uniformly to 0.
Proposition 4.4. Let be a solution of the problem (1.2) with . Then it holds that
Proof. Assume that by contradiction. It follows from Proposition 4.3 that . Let be such that . Then there exists a sequence such that .
Now pick such that As above, we define and then . We extend by 0 on and define the rescaled function for . Also, we denote Therefore the function satisfies , and where . Note that , therefore there exists a subsequence and such that as . Similarly as Lemmas 2.1–2.3 in , we have obtained a function , bounded and continuous on and satisfies that . Therefore or . If , this is a contradiction to the fact that since . If , this is also a contradiction to the fact that .
Theorem 4.5 (slow solution). Let satisfy with , and satisfy the same condition as in Theorem 3.4. Then there exists such that the solution of (1.2) with initial data is a global slow solution, which satisfies that and
Proof. Denote the solution of (1.2) as to emphasize the dependence on the initial function when necessary. So as to and the maximal existence time .
Similarly as in , define According to the Theorem 4.1, since that the solution is global if is sufficiently small.
If , then for sufficiently large, we have , which implies . Therefore is bounded.
Let and .
We first claim that . In fact, by continuous dependence, when , we know that approaches in and for each fixed . Since for all , it follows from Proposition 4.3 that for all . Therefore, since nonglobal solutions must satisfy .
Next we claim that . Assume for contradiction. If follows from Proposition 4.4 that as , which implies for some large . By continuous dependence, we have for sufficiently close to . But this implies that by Theorem 4.1, which is a contradiction with the definition of .
On the other hand, as a consequence of the blowup result of Theorem 3.4, we know that implies for all , hence by using (3.4) we have The estimate (4.17) follows from Hölder’s inequality and (4.21), by writing . The proof is complete.
The work is partially supported by PRC Grants NSFC 11071209, NSFC 10801115.
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