Abstract
Extending (Drábek and Takáč 2017), we investigate the Lyapunov stability of planar waves for the reaction-diffusion equation on , , with a -Hlder continuous (), but not necessarily smooth reaction term. We first consider an initial value problem for the equation and then construct sub- and supersolutions to the problem by a subtle modification of the planar wave. Our main result states that a bounded classical solution to the problem stays near the planar wave for all time whenever an initial data is close enough to the planar wave.
1. Introduction
We consider a reaction-diffusion equation on where , , , , , and . In particular, we are concerned with the stability of the planar wave of the form which is the most widely studied type of travelling wave to (1) with a continuously differentiable reaction term (e.g., see [1–3] and the references therein). However, in the present paper, we assume the reaction term is not necessarily Lipschitz continuous, but only -Hlder continuous () and one-sided Lipschitz continuous (see (H3) in Section 1.2). A typical example for a non-Lipschitzian reaction term is for some constants , , and , which has the singular derivatives at and . Equation (1) with this type of is more realistic and has been used extensively as biological models, in particular, Fisher’s model for population genetics. If in the reaction function (3), the product represents the classical logistic growth of the population. The assumption requiring very large birth or death rate of the population leads to the formula with in (3), which gives the restriction on the differentiability of (see the classical work of population genetics [4, 5], or [6, 7] for the derivation of ).
The restriction on the reaction function makes us unable to linearize equation (1) about the planar wave (2) and to use the spectral analysis which is a standard method to study of stability of travelling waves. Instead, we construct sub- and supersolutions to (1) by an appropriate modification of the planar wave , and then we show that both the planar wave and a unique solution to a Cauchy problem (1) with an initial data near are trapped by the sub- and supersolutions whose difference is sufficiently small. This method has also been used in [3] to prove the stability of planar waves (2) in the Allen-Cahn equation on , , with the continuously differentiable reaction term (i.e., in (3)). They studied that the planar wave is asymptotically stable under any initial perturbations that decay at space infinity or almost periodic perturbations. In their works, the continuous differentiability of the reaction term is necessary to construct sub- and supersolutions and to obtain the convergence rate by using the idea of mean curvature flow on (see [8] for the idea of mean curvature). The main purpose in our project is to study the stability of the planar wave in , , without the differentiability of the reaction term.
Our analysis is totally motivated by the results of Drábek and Takáč [6], showing that the long-time asymptotic behavior of solutions to an initial value problem of a one-dimensional reaction-diffusion equation with a non-Lipschitzian reaction term defined by (3). By constructing sub- and supersolutions without the differentiability of , they established the convergence of a solution to a travelling wave solution , that is, for some spatial shift , when the initial data is close enough to as goes to . However, they restricted the model to the simple case of one space variable by assuming a habitat of a population is a one-dimensional space, for example, a long thin strip along a straight shoreline. In order to make the model to be more realistic, we extend their method to a multidimensional space and prove the planar wave is stable under small initial perturbations in with .
The purpose of this introduction is to provide information of the profile of the planar wave (2) and the precise assumptions on the reaction term and to state our main result.
1.1. The Profile of Planar Waves
In order to study of stability of the planar wave (2), we first introduce the -moving coordinate with speed by setting so that the planar wave can be considered as a stationary solution. In the coordinates, equation (1) reads and the planar wave is then a stationary solution that satisfies the profile equation
The existence and monotonicity of such profiles have been studied in [1, DT2]. According to Section 2 of [6], the assumption (H1) stated in Section 1.2 guarantees -profiles for some speed . If satisfies (8), then its translate also satisfies (8) for any constant . Throughout this paper, we impose the condition for some constant that appears in (H1), so that the -profile satisfying (8) is unique for some unique speed . Moreover, under the assumption (H1), the profile is nondecreasing on and there is an open interval , , such that and on . The asymptotic behavior of the profile is determined solely by the behavior of as and . It is well-known that and in the classical case, i.e., . However, in our case of being non-Lipschitzian at the points and , one has . More precisely, a non-Lipschitzian satisfies that there exist positive constants and such that (see a typical example (3) of ). It is obvious that the limits of (10) are and , respectively, in the case of (see [6, 9] for further details).
In summary, the planar wave satisfies and there is an open interval , , such that
1.2. Hypotheses on the Reaction Term
Throughout this paper, following [6], we assume the reaction term satisfies the following:
(H1). is continuous such that for some . Moreover, for any , for all , and for any ,
(H2). is -Hlder continuous ().
(H3). is one-sided Lipschitz continuous, that is, there exists a positive number such that
(H4). There exists a positive constant satisfying that for any there are positive constants , and such that
As mentioned in the previous subsection, the first assumption (H1) is needed to show the existence and monotonicity of the profile . The last assumption (H4), referred to as the secant conditions, holds trivially if is a classical bistable type and is small enough. We can also assume (H4) even for a non-Lipschitzian reaction term . The reader can see Figure 1 in [6] for an example satisfying (H4).
1.3. Main Result
We now state our main theorem. Theorem 1 says that if the equation (7) starts with an initial data near the planar wave then the solution to (7) stays near it for all time.
Theorem 1 (Lyapunov stability). Let the assumptions (H1)–(H4) hold and . Suppose that a function satisfies that for all , where is sufficiently small. Then, there is a unique bounded classical solution to (7) with an initial data such that for all and all time , for some constant .
Remark 2. Theorem 1 is the extension of the one-dimensional stability result (Proposition 4.1) of [6] to a multidimensional space. Our assumption for the profile gives in their result.
The paper is organized as follows. In Section 2, we first consider the Cauchy problem with an initial data , . Under the assumption (H2), we establish regularity estimates so that a bounded mild solution to (20) is well-defined, and it becomes a bounded classical solution to (20). In Section 3, we define weak sub- and supersolutions to (20) and discuss the weak comparison principle for them under the assumption (H3). This section will also show the uniqueness of the solution to (20). Finally, in Section 4, using the assumption (H4), we modify the planar wave to construct sub- and supersolutions to (20) and prove Theorem 1 by showing both the solution to (20) and the planar wave stay between sub- and supersolutions when the initial data is close enough to .
1.4. Discussion and Open Problems
Theorem 1 does work if is sufficiently small that . We use the assumption (H4) to prove suitable modifications of the planar wave and are sub- and supersolutions of (20) for some functions and . By setting and in (H4), we can apply the inequalities (16) and (17) to the proof of Theorem 1.
The main idea of our work follows [6], but in the present paper, we do not impose one of their assumptions, saying (H5). The assumption (H5) in [6] means that the initial perturbation is small enough only at space infinity, so the assumption (18) for an initial data implies their assumption (H5). Indeed, they proved the travelling wave is asymptotically stable by showing the solution of (20) converges to the travelling wave as goes to infinity when the initial data satisfies (H5) (see (5)). So the initial perturbation does not need to be small in their work. However, unfortunately, the assumption (H5) is not enough to prove the convergence, even the stability, for our multidimensional case. It seems that we need more assumptions on the initial perturbation or the reaction term to prove the asymptotic stability of the planar wave on with , which is a interesting open problem.
Moreover, even in , the convergence rate for a non-Lipschitzian reaction term has not been proven yet. The asymptotic stability with the convergence rate of the planar wave (2) on , , for a continuously differentiable has been studied in [3]. They also modified the planar wave to construct sup- and supersolutions by using more delicate phase functions thanks to the differentiability of . The study of convergence rate on for a non-Lipschitzian would be another very interesting direction to carry out.
2. A Bounded Classical Solution
In this section, we consider the Cauchy problem on , where , , an initial data is Lebesgue-measurable, and for all . Especially, under the assumption (H2), we prove the existence of a bounded classical solution to (21) in the sense that a mild solution , defined in (25), of (21) satisfies for any given and the weak star limit in as . Here, for any and , the norm of Hlder space is given by where
Applying Duhamel’s principle to (21) yields an integral equation of : for any . Here, for any , the heat kernel is defined by
It is well known that if the initial value problem (21) has a solution, this solution is given by (25), referred to as the mild solution of (21). However, it is not trivial that every mild solution is a classical solution. So the first step is to establish the regularity estimates (22) for any bounded mild solution and then we will consider well-posedness of a bounded mild solution (25). The estimates of the heat potentials in Hlder norm have been established in Section 4 of [10], which lead to the following lemma.
Lemma 3 (Regularity estimates). Assume that satisfies the condition (H2). If is a bounded mild solution of (21), then for any given and we have the estimates for some constant , dependent upon and .
Proof. We first fix any and with . A direct calculation gives that for all , and for all with and all , The second inequality of (30) is proved in Chapter 4 of [10]. We now fix with and replace the initial time by . By (29) and the Hlder estimates in [10], we have for all , Since , to obtain the estimate (28), it is enough to show that We notice that the inequalities (29) and (30) yield that for all with , Since is -Hlder continuous, the estimates (32) give and similarly Finally, we have ☐
Noting that is the source term of the inhomogeneous initial value problem (21), we now apply the regularity estimates (28) and Tikhonov fixed point theorem to show the well-posedness of a bounded mild solution (25).
Lemma 4. Assume that satisfies the condition (H2). If an initial data satisfies , then the initial value problem (21) has a bounded mild solution such that for all .
Proof. We first consider the existence of a bounded mild solution on , where We define the locally convex vector space and the closed convex subset of By recalling a mild solution (25), let us consider an operator defined by where We first prove . For any given , since , , and for all , it follows Thus, for any which implies maps into itself. We now notice that for any , , for some constant satisfying for any , , which means that the operator is continuous from into itself. Moreover, the image is relatively compact because is uniformly bounded for all and is equicontinuous by the estimates (30). Hence, by Tikhonov’s fixed point theorem, there is such that . Recalling (39) and (40) gives that for all , which is a bounded mild solution to (21). Therefore, we obtain the existence of a bounded mild solution by replacing the initial time by for any and repeating the above procedure in each time interval .☐
3. A Bounded Weak Solution and a Weak Comparison Principle
In the previous section, we have proved that a mild solution to (21) enjoys regularity estimates for a classical solution. We now prove the uniqueness of the classical solution by showing that (21) possesses at most one weak solution. As a starting point, we define a weak sub- and supersolution of (21), and we then establish a comparison principle for them. The weak comparison principle concludes the uniqueness of a weak solution and plays an important role in the proof of stability in the next section.
Definition 5 (Weak sub- and supersolution). Let . By a weak supersolution of (21), we mean a function satisfying the following three conditions: (i)For all nonnegative test function , (ii) is Lipschitz-continuous in every set whenever , that is, , for all (iii)For all nonnegative test function , By a weak subsolution of (21), we mean a function satisfying the above three conditions (i), (ii), and (iii) replaced the reverse inequality in (45) and (46). We say that a function is a weak solution of (21) if is a weak sub- and supersolution to (21).
We now establish a weak comparison principle for a weak sub- and supersolution under the crucial hypothesis (H3) which is one-sided Lipschitz continuity of .
Theorem 6 (Weak comparison principle). Let be continuous with the one-sided Lipschitz condition (H3). Assume that are weak sub- and supersolutions to (21), respectively, such that a.e. in . Then, we have a.e. in .
Proof. Subtracting (46) from the reverse inequality of (46) for a subsolution yields for any nonnegative function . Let , and we then prove that a.e. in . For any given nonnegative , since is also a nonnegative test function, replacing by in (47) gives From a simple calculation, we evaluate that and where the last inequality is from the one-sided Lipschitz condition (H3). Hence, we find By multiplying each term of (50) by and rearranging, we obtain By setting , it follows that Since and , we find for all nonnegative . As a.e. in , a.e. in and thus satisfies the initial value problem By setting , The weak maximum principle for the heat equation concludes that a.e. in , which implies a.e. in . Therefore, ☐
Since any bounded weak solution is a bounded sub- and supersolution simultaneously, as a consequence of the weak comparison principle together with Lemmas 3 and 4, we obtain the following corollary.
Corollary 7. Let be continuous satisfying (H2) and (H3). Then the initial value problem (21) has a unique bounded classical solution.
4. Lyapunov Stability of the Planar Wave
In this section, we give a proof of Theorem 1. We prove the planar wave is stable in the sense that the unique solution to the initial value problem (21) stays near for all time when the initial data at is close enough to .
We first construct, by an appropriate modification of the planer wave , weak sub- and supersolutions and , respectively, to (21) satisfying for all . The assumption (H4) plays a significant role in the construction of the sub- and supersolution. We notice that the constant functions and are sub- and supersolutions, respectively, to (21) because the initial data satisfies for all .
Lemma 8. Suppose that satisfies the assumptions (H1)–(H4) and the initial data of (21) satisfies for all , where is sufficiently small. Then, there exists a constant such that the functions defined by where are sub- and supersolutions, respectively, to (21) satisfying (60) (see Figure 2).
Proof. To begin, we recall the profile in (12) and let be sufficiently small that
where is a constant such that . Then, by the assumption (61) for an initial data , one can say that for all ,
which implies that
We notice that is decreasing to and is increasing to as , where the positive constant will be determined later. Moreover, by the monotonicity of the profile and the inequality (66), the functions and satisfy (60).
By setting in the assumption (H4), we prove is a supersolution to (21). In order to prove is a supersolution to (21), we decompose its domain as
where
with the positive constant in the assumption (H4). That is, the decomposition is depending only on the -coordinate and .
For the region , since is decreasing and by (64),
and the profile equation gives
Since and for all , applying the inequality (16) to the last two terms on the right-hand side of (73) and recalling and yield
which implies is a supersolution to (21) on .
For the region , by the definition of and the fact that is a supersolution to (21), it is enough to prove is a supersolution only when . Since for all , applying (17) to the last two terms on the right-hand side of (73) gives
For all , applying the assumption (H3) to the last two terms on the right-hand side of (73) yields
Since the profile is a strictly increasing -function for , there exists some such that for all . If we choose satisfying , we obtain
which implies is a supersolution to (21). The proof of the subsolution follows similarly by decomposing as
where
and using (17), (16), and (H3) for , , and , respectively.☐
Remark 9. Similarly as in [6], one can also construct sub- and supersolutions satisfying (60) even if the initial data is close enough to only at space infinity in (possibly large initial perturbations). In that case, might be strictly positive, while in our case. However, the initial perturbation in [6] is not enough to obtain stability results of the planar wave in . Thus, we prove the stability under small initial perturbations (61) and so simply set .
(a)
(b)
We now prove the main theorem.
Proof of Theorem 1. Let be sufficiently small that By Lemma 8, the sub- and supersolutions and to (21) defined by (62)–(63) satisfy Moreover, since the initial data satisfies , by the weak comparison principle in the previous section, the unique bounded classical solution to (21) also satisfies Therefore, recalling (63) and is a -profile satisfying (12), the inequalities (78) and (79) yield that for all , for some constant .
Data Availability
The results of our manuscript are not based on data but only mathematical analysis.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the research grant of the Kongju National University in 2019. The research of the first author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (no. 2019R1F1A1063018).