On a Multistable Type of Free Boundary Problem with a Flux at the Boundary

Ren, Haitao; Cai, Jingjing; Xu, Li

doi:https://doi.org/10.1155/2023/8944465

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Research Article | Open Access

Volume 2023 | Article ID 8944465 | https://doi.org/10.1155/2023/8944465

On a Multistable Type of Free Boundary Problem with a Flux at the Boundary

Haitao Ren,¹Jingjing Cai,¹and Li Xu¹

Academic Editor: Mohammad W. Alomari

Received08 Oct 2022

Revised22 Dec 2022

Accepted25 Mar 2023

Published12 Apr 2023

Abstract

This paper studies the free boundary problem of a multistable equation with a Robin boundary condition, which may be used to describe the spreading of the invasive species with the solution representing the density of species and the free boundary representing the boundary of the spreading region. The Robin boundary condition means that there is a flux of species at . By studying the asymptotic properties of the bounded solution, we obtain the following two situations: (i) four types of survival states: the solution is either big spreading (the solution converges to a big stationary solution defined on the half-line) or small spreading (the solution converges to a small stationary solution defined on the half-line) or small equilibrium state (the survival interval tends to a finite interval and the solution tends to a small compactly supported solution) or vanishing happens (the solution and the interval shrinks to 0 as for ); (ii) a trichotomous survival states of solutions: big spreading, big equilibrium state, and vanishing.

1. Introduction

Now, we study the problem having multistable nonlinearity where is a moving boundary, and are given constants, and the initial function belongs to , when , where

Here, is a multistable nonlinearity, is a function, and there are constants , which satisfies

From this condition, one can regard as a combination of two bistable functions. is the space of functions with second derivative in , and is the space of functions with first derivative.

There are some explanations for our problem. Usually, such problem used to describe the invading of new species or the development of chemical substances. Since the survival region of the species depends on time , we use the moving interval representing such region; the spreading speed of front satisfies the classical Stefan condition, but there is a decay rate caused by the environment at the left boundary. We are primarily focused on the spreading of the solution for direction of right side, and there is a flux on the left boundary.

In the case and , under monostable type of nonlinearity (cf. ) [1, 2] used such problem to explain the expanding of a species moving into a new place. Such species’s density is given by , and the interval is a region occupied by the species at time . They proved that, when , only spreading happens for ; when , there is a spreading/vanishing dichotomy conclusion: spreading (the solution and as ) or vanishing (the solution and tends to a positive number when ). For bistable and combustion types of nonlinearity, [2, 3] studied the asymptotic behavior of solutions (i.e., the limits of the solutions when the time goes to ) for such a free boundary problem (also, the case ). Moreover, [4] also obtained a dichotomy result with a advection term but under the condition and replaced by . Besides, [5] studied the free boundary problems in high-dimensional space. For the situation , [6, 7] studied the convergence of solutions for a free problem and obtained a trichotomy result. When the nonlinearity is Fisher-KPP, [8] studied problem (1) when and obtained a trichotomy result. For bistable nonlinearity, [2] obtained a trichotomy result for a free boundary problem when . When is composed with a monostable and a bistable type of function, [9] studied the free boudary problem with Dirichlet boundary and obtained a richer phenomenon; [10] obtained another convergence conclusion. When is multistable, some papers studied the different travelling waves of reaction-diffusion equation, such as [11–15], and they considered propagating terrace.

Here, we will study the longtime limits of solutions of free boundary problem with and . The constant is the decay rate of the species at the spreading boundary. Such a condition also widely is used in protocell growth models (cf. [16, 17]). This boundary condition can also be deduced by reaction-diffusion equations (cf. [18]). Additionally, the boundary condition in (1) means that there exists a flux of the species at the boundary . Under a multi-bistable nonlinearity, we consider the spreading of solution when the left boundary has a flux while the right boundary producing a decay rate of the solution (or, say the species). There are two critical points and (see details in Section 2) playing important roles in the long-time behavior of solutions; when , we have four types of diffusion: big spreading, small spreading, small equilibrium, and vanishing; when , we have big spreading, big equilibrium state, and vanishing.

The results in this paper are different from others; there are some different methods compared before ones, such as, we use a special stationary solution and an upper solution to prove that vanishing happens within a finite time. In the spreading situation, we first give the upper and lower bound of the limit of , then use a solution of another fixed boundary problem as a lower solution to prove that spreading happens. As for the sufficient conditions for the small and big spreading, we construct a moving lower solution.

In this paper, we use a multistable nonlinearity which often used in the study of travelling waves. Besides this, we add the influence of the boundary on the spreading of solution. Moreover, since the species is spreading on the right side, there may be an influx of species at the left boundary; so, we use the third boundary condition.

According to previous discussions [1, 2], we proved that equation (1) has a unique solution defined on , and for . Additional references [19–27] further support this claim. Furthermore, if , the solution can be extended to a larger interval with . Additionally, using ([7], Lemma 2.8), we get that exists.

We mainly consider the influences of the flux and the decay rate at the boundary on the asymptotic behavior of the solutions. We firstly obtain the following conclusions.

Theorem 1. Assume and . Then, the solution of the problem (1) is either (i)Big spreading: , , as locally uniformly in with is the solution of or(ii)Small spreading: , as locally uniformly in , where is the solution of (4) with 1 replaced by ; or(iii)Vanishing: , as or(iv)In the small equilibrium state case: and where is the smaller solution of

Theorem 2. Assume and , . Then, (i)When , big spreading happens(ii)When , vanishing happens(iii)When , big equilibrium state happens: and where is the bigger solution of (6) (cf. Section 2).

Theorem 3. Assume ; then, the solution is either vanishing or big spreading. When , only vanishing happens for any solutions of (1).

Remark 4. From the aspect of spreading for some species, , that is, when the decay rate at the boundary is large, the environment at the boundary is so bad that the species cannot spread outside, and only vanishing happens.

The structure of this paper is organized as follows. In Section 2, we provide the stationary solutions of equation (1), in Section 3, we analyze the asymptotic behavior of solutions and present several sufficient conditions for spreading and vanishing in Section 4. In Section 5, we present the complete proof of our main theorems.

2. Stationary Solutions

This section focuses on the examination of stationary solutions for the given problem (1). Specifically, we consider

Let , then equation (8) is changed into

According to the phase plane analysis (cf. [28]), the stationary solutions of (1) have the following cases (see Figure 1): (i)Constant solutions: (ii)Increasing solutions defined on the half-line : is the unique solution of (8) and satisfies where or 1 . By the phase plane analysis, always exists for all (cf. points B and D in Figure 1)(iii)Decreasing solutions defined on the half-line : is the unique solution of (8) and satisfies where or 1 , denoted by (iv)Compactly supported solutions: on the phase plane, for any , the problem has a unique solution . In addition, when , denoted its solution as (cf. points A and C in Figure 1); when , denoted as (cf. point A in Figure 1); when , denoted as (cf. point C in Figure 1). Besides, for any , we have .(v)Compactly supported travelling wave : consider the problem

When , for any smaller than the speed of the travelling wave of the equation , then (14) has a unique solution, denoted by . Besides, (8) has other solutions, such as travelling wave and travelling semiwave solutions, groundstate solution which are not used in this paper.

3. Asymptotic Behavior of Solutions

Using the same method as in ([7], Lemma 2.5) with minor modifications, we have the following estimates.

Lemma 5. Assume (3). Let be a solution of (1) defined for , where . Then, there exists a positive constant (depends on and ) such that and for , and .

Also, there exists depending on but independent of , such that for .

Proof. We first prove the bondedness of . By the property of (i.e., (3)) and the comparison principle shows that for all and .
Hence, , for , and .
We next consider the estimate of . Choose a large satisfying with .
Construct for and . Then, for .
The definitions of and derive that for , .
Moreover, The classical comparison principle implies that Note that Denoted by := and , according to the classical parabolic estimates, there exists a constant such that when and define ; we get the conclusion.

Lemma 6. Let be the solution of problem (1) for ; if as , then and .

Proof. From and the estimates in Lemma 5, we can derive that, for any , there is a time such that, when , for .
By the property as (cf. Section 2), we have as . And there is such that for .
Now, we show . Define with . We choose a small such that Also, there is such that for and all .
Construct a function defined on By calculation, is an upper solution. From the Hopf lemma and the choice of , we have Therefore, so as .

Lemma 7. Assume , and is the solution of (1). Then, and in any subset of , where is the solution of (6).

Remark 8. When and , the convergence results in the above lemma never happens. Besides, if , as , we can show that and . This is the result in ([1], Lemma 3.1): vanishing happens when .

Proof. Since , we deduce that, for any given , there is , such that for . We now define an upper solution which is the solution of where for . By the convergence result of the solution (cf. [2], Theorem 1.1), we have as , where is the solution of Since is an upper solution, so for and . Therefore, On the other hand, we can similarly prove where is the solution of We derive from the standard compactness and uniqueness argument that, as , where satisfies Change to ; furthermore utilizing the standard regularity theory of parabolic equation, we can obtain This implies that, as , From , we must have . This and the boundary condition means that . From Section 2, the solution of problem (34) with is nothing but , that is, Moreover, when , ; when , .

Lemma 9. Let be a solution of problem (1); if , then where is the solution of (4) with 1 replaced by ; is the solution of (4).

Proof. Step 1. Firstly, choose ; let be the solution of (13) (cf. Section 2); we define an upper solution which is the solution of with conditions Then, the comparison principle derives that for . By the convergence theorem of the solution, we have as locally uniformly for . Therefore, Step 2. From , for any fixed large , there is , such that and for . We now define a lower solution which is the solution of the problem Then, we deduce from the comparison principle that By the convergence result, as locally uniformly in , where is the solution of Therefore locally uniformly in .
Step 3. As , or uniformly in any compact subset of .
Hence, Therefore, (38) follows from (41) and (45).

4. Sufficient Conditions

It is commonly known that the asymptotic behavior of solutions is affected by the initial data; here, we only give simple sufficient conditions for small (big) spreading, vanishing, and equilibrium state; these conditions will be utilized in the proof of the main theorems.

Lemma 10. Let be the solution of (1) with initial data . (i)Assume . If and for , then small spreading happens(ii)Assume . Choose ; if for , then big spreading happens(iii)Assume . If in , then small (big) equilibrium state happens(iv)For any . Choose ; if for , then vanishing happens

Proof. (i)From the comparison principle and the definition of (which satisfies the boundary condition, cf. Section 2), we have for . Moreover, let be the compactly supported travelling wave in Section 2; we have, for some small , for . Then, it follows from the comparison principle that So, as . Moreover, from , we have for ; combining this and Lemma 9, we get that small spreading happens.(ii)By the comparison principle, we have So, (note that ). If , it then follows from Lemma 7 that the solution converges to , but this is impossible since . Therefore, . However, from Figure 1 in Section 2, we see that ; combining this, (47) and Lemma 9, we get that big spreading happens.(iii)By the comparison principle, for all and for . This means that equilibrium happens(iv)By the comparison principle, By Section 2 (cf. Figure 1), . So, small and big spreading cannot happen. But Lemma 7 implies that is impossible since (notice that ). So, converges to 0 within a finite time. This means that vanishing happens.

5. Proof of Main Theorems

Proof of Theorem 1. Theorem 1 follows from Lemmas 6, 7, and 9.

Proof of Theorem 2. Vanishing, big equilibrium state, and big spreading follow from Lemmas 6, 7, and 9, respectively. To complete the proof of Theorem 1, we only need to prove the sharp result. Denote by and by . Define

Proof. We divide the proof into three cases.
Case 1. . We deduce from the definition of and the classical comparison principle that vanishing happens for .
Case 2. . If vanishing happens, then there exists such that , where is given in Lemma 10 (iv). For any given sufficiently small , we have where . From Lemma 10, vanishes. This also contradicts . If big spreading happens when .
This means and as . So, there exists such that and for . Also, there exists sufficiently small such that Lemma 9 implies big spreading for , a contradiction with . Therefore, big spreading and vanishing cannot happen. Combine this and ; we have big equilibrium state happens.
Case 3. . The definition of implies that vanishing cannot happen. We now prove that big equilibrium state is impossible. Otherwise, there is some such that as . By the comparison principle, for any , Since converges to , for small , there is large such that where is the compactly supported solution (cf. Section 2). By the comparison principle, for , This contradicts case 2 that converges to the big equilibrium state. Therefore, combining this and Lemma 10, we have big spreading which happens when .

Proof of Theorem 3. (i)We first prove that, when , only vanishing happens. Actually, small and big equilibrium states are impossible when , since there is no compactly supported solutions (cf. Section 2). We next show that big spreading is also impossible, and the proof for small spreading follows a similar approach. Suppose on the contrary that spreading happens, then Furthermore, the property and ([29], Proposition A) imply that, for some , , and some , the following holds: For some , construct an upper solution: Therefore, . So, big spreading cannot happen. Suppose , it follows from Lemma 7 that equilibrium state happens, but this is impossible since there are no compactly supported solutions when . Therefore, only vanishing happens.(ii)When , vanishing follows from Lemma 6, small spreading cannot happen using the same method as in the proof of (i). From Lemma 10 (ii), big spreading happens

6. Conclusions

This paper explores the free boundary problem with multistable nonlinearity and presents two cases for the solution’s spreading. The first one is four types of diffusion: big spreading, small spreading, small equilibrium and vanishing; the second one is a trichotomy result: big spreading, big equilibrium state and vanishing. Additionally, it is a good idea to consider the free boundary problem of fractional differential equation (see also [30]). Also, we will consider the problem in time/space environment. Besides, we will find some interesting models from other background (cf. [31–33]).

Data Availability

All the data and formulas are in the manuscript.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2023 Haitao Ren et al. 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.

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