Global Solutions in the Species Competitive Chemotaxis System with Inequal Diffusion Rates

Abstract

This paper is devoted to studying the two-species competitive chemotaxis system with signal-dependent chemotactic sensitivities and inequal diffusion rates , , , , , , under homogeneous Neumann boundary conditions in a bounded and regular domain . If the nonnegative initial date and where the constants , the system possesses a unique global solution that is uniformly bounded under some suitable assumptions on the chemotaxis sensitivity functions , and linear chemical production function .

1. Introduction

In this paper, we consider the following two-species competitive chemotaxis system with signal-dependent chemotactic sensitivities and inequal diffusion rates:where is a bounded and regular domain in and is the outward unit normal vector of the boundary . and represent the populations densities, and both populations reproduce themselves and mutually compete with the other, according to the classical Lotka-Volterra kinetics [1], and the diffusion rates of the populations are 1. denotes the concentration of the chemoattractant, and the diffusion rate of the chemical substance is strictly less than 1 (i.e., ). , , , , and are positive parameters, where , are the growth coefficients and , are the competitive degradation rates of population, respectively. The chemotactic sensitivity function for , which is assumed to be positive. From a biological point of view, when , populations exhibit a tendency to move towards higher signal concentrations (chemoattraction), while conversely the choice leads to a model for chemorepulsion, where populations prefer to move away from the chemical in question [2]. Denote representing the balance between the production of the chemical substance by the populations themselves and its natural degradation (see [3] for details).

The classical chemotaxis model was first introduced by Keller and Segel using a mathematical model of two parabolic equations to describe the aggregation of Dictyostelium discoideum as well as a soil-living amoeba, in the early 1970s [4]. After the pioneering works of Keller and Segel, a large amount of chemotaxis models has been used to model the phenomena for population dynamics or gravitational collapse, among others. Winkler [5] studied the chemotaxis systemunder homogeneous Neumann boundary conditions in a smooth bounded domain . It was proved that the chemotactic collapse was absented for any nonnegative initial date and with some ; the corresponding initial-boundary value problem possessed a unique global uniformly bounded solution. Tello and Winkler [6] considered the parabolic-parabolic-elliptic systemunder homogeneous Neumann boundary conditions in a bounded domain with smooth boundary. Given the suitable positive parameters , and , , they showed that all solutions stabilized towards a uniquely determined spatially homogeneous positive steady state within a certain nonempty range of the logistic growth coefficients and . Negreanu and Tello [7] studied a two-species chemotaxis system with nondiffusive chemoattractant:under suitable boundary and initial conditions in an -dimensional open and bounded domain for . They considered the case of positive chemosensitivities and chemical production function increasing as the concentration of the species , increasing. The paper proved the global existence and uniform boundedness of solutions, and the asymptotic stability of the spatially homogeneous steady state was a consequence of the growth of and the size of for .

Reviewing the recent studies, Zhang and Li [8] considered the following fully parabolic system:under homogeneous Neumann boundary conditions in a smooth bounded domain . By extending the method in [5] (see also [9, 10]), the first step is to estimate some associated weighted functions which depend on signal density, and the second step is to obtain -bounds of solutions from -bounds using the variation-of-constants representation and a series of standard semigroup arguments (see [2, 5, 11, 12]). They proved that, if the nonnegative initial date and for some , the system possesses a unique global solution that is uniformly bounded under some appropriate conditions on the coefficients , and the chemotaxis sensitivity functions , .

Inspired by the foregoing research, the main purpose of this paper is to consider the existence of global solution for the two-species competitive chemotaxis model (1) with inequal diffusion rates. This paper is organized as follows: In Section 2, we formulate the main results of this paper by means of the theorem and establish some preliminaries which are important for our proofs. In Section 3, we firstly consider the local existence of solutions and then proceed with the extensibility criterion. Finally, under some appropriate conditions, we prove that the solutions are uniformly bounded in time using an iterative method.

2. Preliminaries and Main Results

For convenience, we denote that , , and they are quasi-monotonic decrease in . Denote .

In order to establish the global existence and uniform boundedness of solutions to (1), we need to make some restrictive conditions throughout this paper in and :

(i) Let the initial date satisfy , where and are some positive constants.

(ii) There exist positive constants and such that

(iii) There exist positive constants and such that

(iv) Assume thatwherefor defined by

(v) Finally, for some technical reasons, we also assume that

We illustrate the validity to above assumptions with the following generalized example [7].

Example 1. We take the chemosensitivity functions for positive constants fulfillingClearly, (11) holds. Take a lower bound and upper bound is to be defined later. Moreover, the initial dates and are satisfied asThen, consider the following.(1)Condition (6) is equivalent to , so choosing (2)Taking positive constants , then condition (7) holds since for (3)We notice that as well as A sufficient condition for the second inequality in (8) holding is and, for simplicity, let us take and then derive

Up to now, the above all restrictive conditions are verified, which implies that conditions (6)–(11) are sufficient to ensure the global existence of solutions.

Remark 2. In literature [7], the chemical production function , and the chemotactic sensitivity .

In this paper, the purpose is to study the global existence and uniform boundedness of solutions to (1) applying an iterative method. The main results are stated by the following theorem.

Theorem 3. Under assumptions (6)–(11), for any initial date satisfying the homogeneous Neumann boundary conditions and , , there exists a unique solution to (1):for any . Moreover the solution is uniformly bounded; that is,where <∞ is positive constant.

The proof of Theorem 3 is split into several steps. Step one: we start to consider the local existence of solutions and then proceed with the extensibility criterion. Step two: under some appropriate conditions, we prove that the solutions are uniformly bounded in time.

3. Global Existence of Solutions

We will first be devoted to dealing with local-in-time existence and uniqueness of a nonnegative solution for (1). The corresponding conclusions are written by Lemma 4.

3.1. Local Existence of Solutions

Lemma 4. Given positive initial date and the same assumptions as in Theorem 3, there exists a small enough time and a unique triple of nonnegative functions such that is a solution of (1) in . Moreover, we have .

Proof. Introduce the change of variables and given bywhere the function is defined byA direct calculation yieldssimilarly,whereSubstitute (20), (22), and (23) into (1) and denote thatfor Then system (1) becomeswith boundary conditions , , , and initial conditions , .
For sufficiently small , in the space , we implement a fixed point argument. Taking satisfying and , we define in as the unique solution to the questionfor the details, we refer the reader to [13, Remark]. Notice that the right-hand side terms of (27) are the multiplicative for and , applying the maximum principle to verify that both and are nonnegative.
Since condition (11), we easily see that the regular functions and satisfyand the right-hand side terms of (27) are quasi-decreased; looking upon the lower-solution , one may construct upper-solutions and such thatIt follows that we apply Theorem 2.1 of [7] to get a unique nonnegative solution.
We solve the parabolic equationThanks to the essential estimations of parabolic equations and the embedding theorems, we apply the Schauder fixed point theorem to obtain the local existence of solution . The smoothness of ensures the uniqueness of solution .
To prove , in view of assumption (8) and the monotone of , we have . It follows that is a lower-solution for the equationThe proof of Lemma 4 is completed.

About the above solution , we have the following extensibility criterion: The solution is extended to the interval , where has the following property:

3.2. Uniform Boundedness of Solutions

To obtain some a priori estimates, we need some technical lemmas. The following -estimate of solution is first given.

Lemma 5. For all , the solutions to (1) satisfy the following estimates:where .

Proof. Integrating the first equation of (1) over , we have Using Cauchy inequality yieldsBy the Gronwall Lemma, we derivea similar estimate holds for , which leads to Integrating the third equation of (1) over , we haveit impliesThis proves the lemma.

Lemma 6. Letting and under assumption (11), then the following estimates hold:where is defined by

Proof. It is easy to checkThen, for any , we haveClearly, if , then (40) is a consequence of (44). We in the following verify that the result holds.
In factSince , multiplying both sides by , we getand by -Yöung’s inequalityCombining (46) and (47) and substituting it into , we deriveInserting (48) into (45) yieldswe deduce from assumption (11) that for any , and the proof ends. In the same way, we achieve (41). This completes the proof of Lemma 6.

In view of and initial date , we consider defined bywhere satisfies that and for any .

Lemma 7. Letting and for any , then the solutions to (1) satisfy

Proof. In order to prove Lemma 7, we consider the terms and appearing in the proof of Lemma 6.
By the nonnegativity for and assumption (6), we obtainMultiplying (53) by and integrating over yieldBy the monotonic property of exponential function and assumption (7), we have It follows that (54) becomesMultiplying (56) by , we getFor the term , we easily achieveCombining (44), (57), and (58) leads towhich implies that (51) holds. The same arguments are used to achieve (52). The proof ends.

To establish the uniform boundedness of solutions, we apply an iterative method based on the - of solutions inspired by the Moser-Alikakos type iterations (see [10, 13]).

Lemma 8. For any , then the solutions to (1) are bounded by

Proof. Denote by . Taking arbitrary positive constant , then it holds thatHenceSubstituting (63) into (51), we haveTake . Noticing Lemma 5 and initial date , we apply the Gronwall Lemma to obtaintaking limits leads toSincecombining (66) and (67) and taking limits as , it is found that (60) holds. The same processes are used to prove (61), which ends the proof of Lemma 8.

Remark 9. In the above arguments, we used well such fact: Let be bounded and regular domain of . If , where is some positive constant and independent of , then as .

To end the arguments of uniform boundedness of solutions, it follows to prove that and by contradiction.

As in Lemmas 4–8, we define and for the unique solutions in towhere is defined in the same fashion as (32) and the truncation is defined as follows:Then it is easy to see thatas far as .

By analogy with , we definewhere satisfies that and for any . To prove , we suppose then . It follows to easily check that is a strict upper-solution to

This implies ), which contradicts (72) and proves as well as .

Thanks to , we show that is also a solution to (6). So we haveThe proof of Theorem 3 is completed.

Competing Interests

The author declares that they have no competing interests.

Acknowledgments

This work is supported by the China National Natural Science Foundation (nos. 11361055 and 11261053) and the Natural Science Research Key Project of University of Anhui Province (no. KJ2015A251).