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Abstract and Applied Analysis

Volume 2014 (2014), Article ID 378172, 8 pages

http://dx.doi.org/10.1155/2014/378172
Research Article

Global Asymptotic Stability in a Class of Reaction-Diffusion Equations with Time Delay

1School of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006, China

2School of Mathematics and Computer Sciences, Yichun University, Yichun 336000, China

3Key Laboratory of Mathematics and Interdisciplinary Science of Guangdong Higher Education Institutes, Guangzhou University, Guangzhou 510006, China

Received 19 February 2014; Accepted 8 April 2014; Published 29 April 2014

Academic Editor: Chuangxia Huang

Copyright © 2014 Yueding Yuan and Zhiming Guo. 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 very general class of delayed reaction-diffusion equations in which the reaction term can be nonmonotone and spatially nonlocal. By using a fluctuation method, combined with the careful analysis of the corresponding characteristic equations, we obtain some sufficient conditions for the global asymptotic stability of the trivial solution and the positive steady state to the equations subject to the Neumann boundary condition.

1. Introduction

There has been a growing interest in the dynamic behavior of spatial nonlocal and time-delayed population systems since the 1970s [1]. When the death function of such a system is linear many researchers used the theory of monotone semiflows, the comparison arguments, and the fluctuation method to study spreading speeds, traveling waves, and the global stability (see, e.g., [210]). However, researches on these problems become relatively rare for nonmonotone delayed reaction-diffusion systems in which the death function is nonlinear (see [11, 12]). The reason lies in the fact that it is difficult to establish an appropriate expression for solutions to study the solution semiflow under this case.

In this paper, we will investigate the global asymptotic stability of the positive steady state for the following time-delayed reaction-diffusion equation: where , , , denotes the Laplacian operator on , is a bounded and open domain of with a smooth boundary , is the differentiation in the direction of the outward normal to , and the kernel function is given by Here, with is the eigenvalue of the linear operator subject to the homogeneous Neumann boundary condition on , is the eigenvector corresponding to , is a complete orthonormal system in the space , for all , and is the Dirac function on [10, 13]. Throughout this paper, we assume that the functions and satisfy the following.(A1) is Lipschitz continuous with and , and for all .(A2) for all , where is Lipschitz continuous with , , and for all .(A3) There exists a positive number such that, for all , , where .

In the monotone case, where the function increases with , Xu and Zhao [12] studied the global dynamics of (1) and obtained some results on the uniqueness and global attractivity of a positive steady state by using the theory of monotone dynamical systems. In the case of , Zhao [10] proved the global attractivity of the positive constant equilibrium for (1) by using a fluctuation method of Thieme and Zhao [14], where is a positive constant. In the case where and , Yi and Zou [7] proved the global attractivity of the unique positive constant equilibrium for (1) by combining a dynamical systems argument and some subtle inequalities. In the case where and , (1) reduces to the equation derived in [3], where the numerical solutions are considered. A global convergence theorem was obtained in [11] for a special case of (1).

The aim of this paper is to establish some criteria to guarantee the global asymptotic stability of the trivial solution and the positive steady state for (1) by using a fluctuation method, combined with the careful analysis of the corresponding characteristic equations. The interesting thing is that main results obtained in this paper extend the related existing results.

The rest of this paper is organized as follows. We will present some preliminary results in Section 2. Our main results are presented and proved in Sections 3 and 4, where we obtain sufficient conditions to ensure the global asymptotic stability of the trivial solution and the positive steady state for (1) in a nonmonotone case. In Section 5, we provide four examples to illustrate the applicability of the main results.

2. Preliminaries

Firstly, we show that the kernel in (2) enjoys the following properties.

Lemma 1. For , one has (i) , (ii) , for all , where is a positive constant depending only on and ,(iii) , for all , ,(iv) , for all .

Proof. The verification of (i) is straightforward and is thus omitted. Part (ii) follows from [15, Lemma 3.2.1 and Theorem 4.4.6] since is a heat kernel of the heat equation . Part (iii) follows from , for all , . And part (iv) follows from , , and for all , where is the measure of . The proof is completed.

Let and . Then is a strongly ordered Banach space. It is well known that the differential operator generates a -semigroup on . Moreover, the standard parabolic maximum principle (see, e.g., [16, Corollary 7.2.3]) implies that the semigroup is strongly positive in the sense that .

Let and . For the sake of convenience, we will identify an element as a function from to defined by , and for each , we regard as a function on defined by . For any function , where , we define , by . Define by Then we can rewrite (1) as an abstract functional equation: Therefore, we can write (4) as an integral equation: whose solutions are called mild solutions for (1).

Since is strongly positive, we have By [17, Proposition 3 and Remark 2.4] (or [18, Corollary 8.1.3]), for each , (1) has a unique noncontinuable mild solution with , and for all . Moreover, is a classical solution of (1) for (see [18, Corollary 2.2.5]).

By the same arguments as in the proof of [12, Theorems 2.1 and 3.1], we have the following two lemmas.

Lemma 2. Let (A1)–(A3) hold. Then, for each , a unique solution of (1) globally exists on , uniformly for , and the solution semiflow , , admits a connected global attractor.

Lemma 3. Let (A1)–(A3) hold, and let be the solution of (1) with . Then the following two statements are valid. (i)If , then for any , we have uniformly for .(ii)If , then (1) admits at least one spatially homogeneous steady state , and there exists such that for any with we have uniformly for .

Note that in case (ii) above, the function satisfies , , and . Therefore, there exists at least one positive number such that , and hence, is a spatially homogeneous steady state of (1).

3. Global Attractivity

In this section, we establish the global attractivity of the positive and spatially homogeneous steady state for (1) by the fluctuation method used in [10, Theorem 3.1].

Motivated by [10, Section 3], we assume further that the functions and satisfy the following.(A4) is strictly decreasing for , and and have the property (P) that, for any satisfying , , and , we have .

Note that if is nondecreasing for , then and have the property ( ). Indeed, for any with and , we have which implies that . Combining this observation and [10, Lemma 3.1] with replaced by , where , we then have the following result.

Lemma 4. Either of the following two conditions is sufficient for the property (P) in condition (A4) to hold. (P1) is nondecreasing for .(P2) is strictly increasing for .

Now we are in a position to prove our main result in this section.

Theorem 5. Assume that (A1)–(A4) hold, and let be the solution of (1) with . Then for any with , we have uniformly for .

In order to prove Theorem 5, we will need the following lemma.

Lemma 6. Assume that (A1)–(A3) hold, and let be the solution of (1) with . Then satisfies where and the kernel function is given in (2).

Proof. Let Since , for each , there exist real numbers and , , such that Therefore, by (10), (11), and (1), we have By using the variation of constants method, we obtain Thus, by (10), (12), and (14), we further get Therefore, (8) follows immediately from (9) and (15). The proof is completed.

Proof of Theorem 5. For any given with , let be the omega limit set of the positive orbit through for the solution semiflow . By Lemma 1, we get , where is the global attractor of the solution semiflow and Note that is a maximal compact invariant set of the solution semiflow . Thus, it is sufficient to prove the global attractivity of for all with .

Let be given such that . Then it follows from Lemma 6 that where is the solution of (1) starting from the initial function . Following [19], we define a function by Then is nondecreasing in and nonincreasing in . Moreover, , , and is continuous in (see [20, Section 2]). Therefore, by (17), we have Let Then Lemmas 2 and 3 imply that On the other hand, note that . Therefore, the function is nondecreasing in . Thus, by Fatou’s lemma and (19), we further get Let Then Moreover, it follows from Lemma 1 that Therefore, by (22), we have Thus, Similarly, we have By (18), we may find such that It then follows from (27) and (28) that and hence, This, together with the strict monotonicity of for , implies that . Moreover, by (27) and (28), we also have Therefore, the property (P) implies that Thus, by (30), we obtain Since we further get This implies that

It remains to prove that uniformly for . For any , there exists a sequence such that in as . Therefore, we have uniformly for . By (37), we further get Thus, we obtain , which implies that converges to in as . The proof is completed.

4. Global Asymptotic Stability

In this section, we establish the global asymptotic stability of the trivial solution and the positive and spatially homogeneous steady state for (1) by the careful analysis of the corresponding characteristic equations. To this end, we first give the following formal definitions of stability (see, e.g., [18, Remark 2.1.3]).

Definition 7. Let be a steady state of the abstract equation (4). It is called stable if for any there exists such that the solution of (4) with satisfies , for all . It is called unstable if it is not stable. It is asymptotically stable if it is stable and there exists such that the solution of (4) with satisfies . It is globally asymptotically stable if it is stable and any solution of (4) with arbitrary satisfies .

Let be a spatially homogeneous steady state for (1) (e.g., the trivial solution and ). Define by where and . Note that is given in (2). Then we can write the linearized equation of (1) at as the following abstract functional equation where can be referred to Section 2.

For each complex number we define the -valued linear operator by where is defined by (note that we use to denote its complexification here) We will call a characteristic value of (41) if there exists solving the characteristic equation (see, e.g., [18]). Since , for any , there exist complex numbers , such that Therefore, by (2), (42), and (44), we have Thus, the characteristic value of (41) satisfies at least one of the following equations:

Lemma 8. Assume that (A1)–(A3) hold, and let be the smallest real number such that if is a characteristic value of (41), then . One has the following: (i)if , then ,(ii)if , then ,(iii)if , then .

Proof. (i) If , then, by (46) and [21, Proposition 4.6], there exists at least one characteristic value of (41) such that . Therefore, .

(ii) If , then since , we have Therefore, by (46) and [21, Proposition 4.6], all the characteristic values of (41) have negative real parts. Thus, it follows from [18, Theorem 3.1.10] that .

(iii) If , then is a characteristic value of (41). Therefore, . If , then there exists at least one characteristic value of (41) and a positive number such that and Let , where and both are real numbers. Then . By (48), we have and hence, . This implies that But, since , we have . Therefore, contradicting . This contradiction proves . The proof is completed.

Now we are ready to summarize our main results on the global stability. By Definition 7, Lemmas 3 and 8, Theorem 5, [18, Corollary 3.1.11], and the principle of linearized stability (see, e.g., [21]), we obtain the following.

Theorem 9. Assume that (A1)–(A3) hold. Then the following two statements are valid. (i)If , then the zero solution of (1) is globally asymptotically stable in .(ii)If , then the zero solution of (1) is unstable, and (1) admits at least one spatially homogeneous steady state .

Theorem 10. Assume that (A1)–(A3) hold, and . Then the following two statements for the positive and spatially homogeneous steady state of (1) are valid. (i)If , then is unstable.(ii)If and (A4) hold, then is globally asymptotically stable in .

5. Examples

In this section, we present four examples to illustrate the feasibility of our main results.

Example 1. Consider the equation resulting from letting in (1); that is, where is a positive constant.

In this case, we now formulate the following assumptions to replace (A2)–(A4): There exists a positive number such that, for all , , where . , is strictly decreasing for , and has the property ( ) that, for any satisfying , , and , we have .

By applying Theorems 9 and 10, we then obtain the following results for (52).

Theorem 11. Assume that (A1) and (A2′) hold. Then the following two statements are valid. (i)If , then the zero solution of (52) is globally asymptotically stable in .(ii)If , then the zero solution of (52) is unstable, and (52) admits at least one spatially homogeneous steady state .

Theorem 12. Assume that (A1) and (A2′) hold, and . Then the following two statements for the positive and spatially homogeneous steady state of (52) are valid. (i)If , then is unstable.(ii)If and hold, then is globally asymptotically stable in .

Remark 13. It is easy to see that (52) is discussed in [10] and some partial results of Theorems 11 and 12 have been obtained [10].

Example 2. Consider the following Nicholson’s blowfly equation resulting from letting and in (1): where and are two positive constants.

By the same arguments as in [10, Section 4], together with Theorems 11 and 12, we have the following results for (53).

Theorem 14. If , then the zero solution of (53) is globally asymptotically stable in .

If , then the zero solution of (53) is unstable, and (53) admits the unique positive constant equilibrium .

Theorem 15. If , the unique positive constant equilibrium of (53) is globally asymptotically stable in .

Remark 16. It is easy to see that the equation in [7] is a special case of of (53). Hence all main results of [7] are special cases of our Theorems 14 and 15.

Example 3. Consider the following Mackey-Glass equation resulting from letting and in (1): where is a positive constant.

By the same arguments as in [10, Section 4], together with Theorems 9 and 10, we have the following results for (54).

Theorem 17. The zero solution of (54) is always unstable, and (54) must admit the unique positive constant equilibrium , where

Theorem 18. If , the unique positive constant equilibrium is unstable, and if it is globally asymptotically stable in .

Example 4. Consider the equation resulting from letting and in (1); that is, where .

Clearly, , , and (A1)–(A3) hold, where if . Moreover, if . Therefore, (A4) is satisfied if . Thus, Theorems 9 and 10 imply the following results.

Theorem 19. The zero solution of (56) is always unstable, and (56) must admit the unique positive constant equilibrium .

Theorem 20. If , then the unique positive constant equilibrium is globally asymptotically stable in .

Remark 21. It is easy to see that the equation in [11] is a special case of of (56) and hence some partial results of Theorems 19 and 20 have been obtained [11].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was partially supported by National Natural Science Foundation of China, Research Fund for the Doctoral Program of Higher Education of China (no. 20124410110001), and Program for Changjiang Scholars and Innovative Research Team in University (IRT1226).

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