Journal of Applied Mathematics

Volume 2014 (2014), Article ID 724582, 5 pages

http://dx.doi.org/10.1155/2014/724582

Research Article

Asymptotic Behavior of Solutions of Free Boundary Problem with Logistic Reaction Term

Jingjing Cai

School of Mathematics and Physics, Shanghai University of Electric Power, Pingliang Road 2103, Shanghai 200090, China

Received 16 March 2014; Accepted 19 May 2014; Published 2 June 2014

Academic Editor: Junjie Wei

Copyright © 2014 Jingjing Cai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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).

References

Y. Du and B. Lou, “Spreading and vanishing in nonlinear diffusion problems with free boundaries,” http://arxiv.org/abs/1301.5373

Y. Du and Z. Lin, “Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary,” SIAM Journal on Mathematical Analysis, vol. 42, no. 1, pp. 377–405, 2010.

Y. Kaneko and Y. Yamada, “A free boundary problem for a reaction-diffusion equation appearing in ecology,” Advances in Mathematical Sciences and Applications, vol. 21, no. 2, pp. 467–492, 2011.

D. G. Aronson and H. F. Weinberger, “Multidimensional nonlinear diffusion arising in population genetics,” Advances in Mathematics, vol. 30, no. 1, pp. 33–76, 1978.

D. G. Aronson and H. F. Weinberger, “Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation,” in Partial Differential Equations and Related Topics, vol. 446 of Lecture Notes in Mathematics, pp. 5–49, Springer, Berlin, Germany, 1975.

Y. Du and Z. Guo, “The Stefan problem for the Fisher-KPP equation,” Journal of Differential Equations, vol. 253, no. 3, pp. 996–1035, 2012.

A. Friedman and B. Hu, “A Stefan problem for a photocell model,” SIAM Journal on Mathematical Analysis, vol. 30, no. 4, pp. 912–926, 1999.

H. Zhang, C. Qu, and B. Hu, “Bifurcation for a free boundary problem modeling a protocell,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 7, pp. 2779–2795, 2009.

J. Cai, B. Lou, and M. Zhou, “Asymptotic behavior of solutions of a reaction diffusion equation with free boundary conditions,” Preprint

J. Cai, “Asymptotic behavior of solutions of Fisher-KPP equation with free boundary conditions,” Nonlinear Analysis: Real World Applications, vol. 16, pp. 170–177, 2014.

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley & Sons, Chichester, UK, 2003.