Abstract
We study a free boundary problem for a reaction diffusion equation modeling the spreading of a biological or chemical species. In this model, the free boundary represents the spreading front of the species. We discuss the asymptotic behavior of bounded solutions and obtain a trichotomy result: spreading (the free boundary tends to and the solution converges to a stationary solution defined on ), transition (the free boundary stays in a bounded interval and the solution converges to a stationary solution with positive compact support), and vanishing (the free boundary converges to 0 and the solution tends to 0 within a finite time).
1. Introduction
Consider the following free boundary problem: where is a moving boundary to be determined together with and is a given constant. The initial function belongs to for some , where
Recently, problem (1) with was studied by [1–3] and so forth. They used this model to describe the spreading of a new or invasive species; they used the free boundary which represents the expanding front of the species whose density is represented by . They obtained a spreading-vanishing dichotomy result; namely, the species either spreads to the whole environment and stabilizes at the positive state 1 (i.e., ) or vanishes (i.e., ) as time goes to infinity. Such a result shows that problem (1) with has advantages comparing with the Cauchy problems (the Cauchy problems have hair-trigger effect: any positive solution which converges to a positive constant; cf. [4, 5]). In the last two years, [6] also studied the corresponding problem of (1) with in high dimension spaces.
In this paper, we mainly study problem (1) with ; such a boundary condition represents that there is a spreading resistant force at the front for some species. Intuitively, the presence of makes the solution more difficult to spread than the case where . Indeed, only if . This boundary condition is widely used in many biological models. For example, it is often used in protocell models (cf. [7, 8]).
We give the following theorem whose proof is similar to that of [1, 2]. It suffices to repeat their arguments with obvious modification.
Theorem 1. For any given , there is a such that free boundary problem (1) has a solution where , and the solution can be extended to some interval with as long as .
Moreover, as in the proof of [9, Lemma 2.8], one can show that exist.
The main purpose of this paper is to study the asymptotic behavior of bounded solutions of (1) and obtain trichotomy result. We will prove that, for a solution of (1), one has either(i)spreading: and where is the unique positive solution of or(ii)vanishing: and or(iii)transition: and where is the solution of
Remark 2. Comparing with the results in [1–3], the phenomenon (iii) is a new one, since it does not happen in case .
Remark 3. (ii) shows that vanishing happens in a finite time and the free boundary converges to the point ; those phenomena are also new and do not happen in case .
2. Asymptotic Behavior of Solutions
In this section, we study the asymptotic behavior of solutions and obtain trichotomy result when ; namely, the solution of (1) is either vanishing (Theorem 6) or transition (Theorem 7) or spreading (Theorem 10). Then, we prove that only vanishing happens if (Theorem 11) for the completeness of the paper.
We first prepare the following comparison theorems which can be proved similarly as in [2, Lemma 3.5].
Lemma 4. Suppose that , , and with and If and in and if is a solution of (1), then
Remark 5. The pair is usually called an upper solution of problem (1) and one can define a lower solution by revising all the inequalities.
Theorem 6. Let be a solution of (1) on . If , then and
Proof. By [2, 10], one can prove that there exists a constant such that . In order to prove that converges to , we need to construct the function
over the region
where
Clearly in . By the definitions of and , we have
Moreover,
Therefore, in by the comparison principle Lemma 4. Note that ; then there exists such that for . Therefore, for and . For such and , we have
it follows that
We now prove that . By , there is some such that
Set and
where is small such that
Consider the problem
It is obvious that for all . Construct a function
over , where . Then is an upper solution of (22) over and so
Therefore, . Thus, as .
On the other hand, (18) implies that there exists some such that for all and . Clearly for . By the comparison principle, we have , and so cannot be .
Theorem 7. Assume that . Let be a solution of (1). If , then where is a unique positive solution of where with given by .
Remark 8. This is a new phenomenon. It never happens when . Moreover, by the phase plane method, one can prove that and as . This conclusion gives an explanation of Lemma 3.1 in [2]; that is, vanishing happens if .
Remark 9. It is easily seen that (26) has no positive solution when .
Proof of Theorem 7. For any , there exists such that for . Let be a function defined on and satisfies
By the comparison principle we have in , where is the solution of
It is well known that (i) as if ; or(ii) as if , where is a positive function. More precisely, when , it follows from [11, Corollary 3.4] that is the unique positive solution of
Hence,
Similarly,
where is a positive solution of
We conclude from (31) and (32) that
or when ,
where is the unique positive solution of
We now show that is impossible when . Suppose that this does not hold; there exists such that . Then using the approach of proving in Theorem 7, we can show that for some ; this contradicts the assumption . Hence, , locally uniformly in ; we next prove that .
Make a change of the variable to reduce to the fixed interval and use estimates as well as Sobolev embedding theorems on the reduced equation with Dirichlet boundary conditions to conclude that
for some . It follows that as . Hence, we conclude that is not a finite interval unless .
Theorem 10. Let be a solution of (1). If , then where is the unique positive solution of
Proof. Choose a bounded continuous function for and for . Let be the unique solution of
Then the comparison principle theorem shows that for , . Using [11, Lemma 3.4], we see that
On the other hand, since , for any large , there is such that and for all . Let be the solution of the following problem:
where is a nonnegative continuous function satisfying for . The comparison principle implies
By [11], one can obtain
where is the positive solution of
It is well known that . Combining this with (43) and (44), we have
By (41) and (46), we have
Theorem 11. Suppose that and is a solution of (1) defined on some maximal existence interval ; then , converges to as , and .
Proof. The proof of this theorem is similar to [10]; it suffices to repeat their arguments with obvious modification.
3. Example
In this section, we give some sufficient conditions for vanishing, spreading, and transition.
Example 1. Suppose that . Let and ; then the following properties hold: (i)vanishing happens when ;(ii)spreading happens if for ;(iii)transition happens if for .
Proof. (i) By [1], we see that for . Since , there is such that , by the comparison principle that , so and . It then follows from Theorem 6 that vanishing happens.
(ii) Let be a solution of (1) with initial data ; by the phase plane analysis, there is such that . It then follows from the comparison principle that , so Theorem 10 implies that and spreading happens.
(iii) It follows from the comparison principle Lemma 4 that and for all .
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This research was supported by Shanghai Natural Science Foundation (no. 13ZR1454900) and Shanghai University Young Teachers Training Scheme (no. ZZsdl13021).