Global Existence to an Attraction-Repulsion Chemotaxis Model with Fast Diffusion and Nonlinear Source
This paper deals with the global existence of solutions to a strongly coupled parabolic-parabolic system of chemotaxis arising from the theory of reinforced random walks. More specifically, we investigate the attraction-repulsion chemotaxis model with fast diffusive term and nonlinear source subject to the Neumann boundary conditions. Such fast diffusion guarantees the global existence of solutions for any given initial value in a bounded domain. Our main results are based on the method of energy estimates, where the key estimates are obtained by a technique originating from Moser’s iterations. Moreover, we notice that the cell density goes to the maximum value when the diffusion coefficient of the cell density tends to infinity.
Chemotaxis is known as the active orientation of moving organisms along the chemical gradient. It is observed in many natural systems. For example, myxobacteria produce so-called slime trails on which their cohorts can move more readily. The mathematical models of chemotaxis were introduced by Patlak in  and Keller and Segel in . During the past four decades, chemotaxis models have been studied extensively (see, e.g., [3–10] and the rich references therein). For instance, Othmer and Stevens in  modeled myxobacteria as individual random walkers and proposed the microscopic model based on the velocity jump process. By taking the parabolic limit of microscopic model, we can obtain the macroscopic chemotaxis model which is the well-known Keller-Segel system:where is a bounded connected domain with a smooth boundary , , , and are positive constants, and . The function denotes the cell density and represents the chemotactic concentration, for example, the oxygen. The constant is called the chemosensitive coefficient, and the sign of corresponds to chemoattraction if , and chemorepulsion if . The function and the constant are the diffusion coefficient of the cell motility and the chemical, respectively. The function represents the kinetic function describing production and degradation of chemicals, and is commonly referred to the chemotactic potential function.
In the absence of logistic source (i.e., ), there have been extensive studies to system (1). The main feature of solution to the Keller-Segel model is the possibility of blowup in finite time in [3, 8, 11, 12]. For instance, the first result on finite-time blowup for a radially symmetric solutions was shown in [13, 14] when is a ball in under the condition that and is sufficiently small. For a general domain, Nagai in  further showed the finite-time blowup of nonradial solutions provided that is sufficiently small and if or if with . Winkler in  studied finite-time blowup of radially symmetric solutions to the full parabolic system with logistic sources in a ball with parameters , , and . Moreover, Winkler in  gave the set of blowup to the system (1) by enforcing initial data with respect to the topology of for any , where is a ball in with . In summary, the solution of system (1) never blows up when , whereas there is finite-time or infinite-time blowup when . Moreover, recent results in  confirmed that the attraction-repulsion is a plausible mechanism to regularize the classical Keller-Segel system (2) with whose solutions may blow up in higher dimensions. Authors in [5, 9, 10, 18, 19] proved the global existence of solutions.
However, in many biology progresses, the cells usually interact with not only the attractive combination but also repulsive signalling. Therefore, it is necessary to study the attraction-repulsion chemotaxis model:System (2) with was proposed in  to describe the aggregation of microglia observed in Alzheimer’s disease. Here represents the cell density and denotes the concentration of the chemoattractant and is the concentration of the chemorepulsion. The constants satisfy and , where and measure the strength of chemotactic signal of attraction and repulsion, respectively. The constants , , , and are positive, and they denote the production and degradation rates of the two chemicals, respectively. The first cross-diffusive term in the first equation of (2) means that the orientation movement of the cell is directed to the chemorepulsion, whereas the second cross-diffusive term implies that cells move down the chemoattraction. The second and third equations in (2) elaborate that the two chemicals of chemoattraction and chemorepulsion are released by cells and go through decay. For the case of , the theorem of competing effects has been established in  with . Moreover, the global existence, asymptotic behavior, and steady states of classical solution were studied in  for one-dimensional case with .
In this paper, we consider the following attraction-repulsion chemotaxis system including three parabolic equations:with the initial-boundary value conditions:where is a bounded domain for some and denotes the unit outer normal vector field on . The function of the cell density is the fraction of volume occupied by cells, whereas the fast diffusion coefficient of the cell is described asIt is easy to see that the function is a monotonically increasing function of which guarantees that the solution will not blow up. The functions and denote the concentration of the chemoattraction and chemorepulsion, respectively. The constants and are the diffusion coefficients of the chemoattraction and chemorepulsion, respectively. and denote the chemosensitive coefficients, and describes the kinetic function.
In particular, the global existence of solutions was obtained by increasing the diffusion coefficient : for example, , in , and in . Liu and Tao in  established global boundedness of classical solutions to the parabolic attraction-repulsion chemotaxis system (2) when . For the case of including logistic source, global existence and boundedness of classical solution were studied in [23, 24]. For the case , the existence of families of traveling impulses and fronts was analyzed in .
In this paper, we will prove the existence of global classical solutions to the generalized system (3) with the initial-boundary conditions and the no-flux boundary condition (4). In addition, we need the following assumptions.(A1)Let . Suppose that functions are of class, and for and for .(A2)Assume that functions are of class and that , , and for all , where .(A3)Let be of class, where .(A4), , and for and for .The main result is the following.
Throughout this paper, we introduce some notations. is a bounded open interval in and denotes a general constant which may have different values in different place. is the usual Lebesgue space with the norm for and When , we denote for convenience. is the th-order Sobolev space with the norm For notational convenience, we write and as and , respectively. Moreover, we denote
The rest of this paper is organized as follows. In Section 2, we establish the local existence and give some preliminary lemmas. Some necessary priori estimates will be established in Section 3. We will complete the proof of Theorem 1 in Section 4.
3. Local Existence
Theorem 2 (local existence). Let assumptions (A1)–(A4) hold. Then(1)There is a positive constant depending on initial value such that system (3) has a unique maximal solution in the space with .(2)There is a global classical solution of system (3) if is bounded away from 1 for each finite time , which means .
Proof. Let . System (3) can be written aswhereSystem (9) satisfies the initial valueFor some and given initial conditions , it is clear that the eigenvalues of matrix are positive at . Thus, system (9) is normally parabolic and there exists local solution by Theorem 7.3 in ; that is, there is a such that the unique solution .
Next we rewrite the first equation of (3) as Recalling that assumption (A2) ensures that for all and for all , this means that for all and , where and . Hence, the source term is zero when at some . By applying the maximum principle to , therefore, we obtain that whenever and . Similarly, we infer (or ) from the second equation of (3) (or the third of (3)) whenever . The proof of (1) in Theorem 2 is finished.
The proof of (2) in Theorem 2 is completed by applying the theorem of quasilinear parabolic equations in  since system (9) is an upper triangular system.
4. A Priori Estimates
Lemma 3. Let be a bounded open domain satisfying the uniform cone property in with :Then there exists a positive constant , which depends on , , and , such that for all : where and .
Lemma 4. Let the conditions in Lemma 3 hold: and. Then there exists a constant independent of such that
Proof. Recalling for and employing Lemma 3 with and , we obtain ; thus,where we have used the inequality for .
Lemma 5. Let a nonnegative numerical sequence satisfying with and for . Suppose and . Then one has
Proof. By the definition of the sequence , we obtain that for
Lemma 6. Let conditions (A1)–(A4) hold. Then there exists a unique global solution to system (3) such that , , and are in Furthermore, there exists two positive constants and satisfying
Proof. After appropriate scaling, the system is limited in a region satisfying . Next, we introduce the auxiliary scalar equation:with the initial-boundary condition:where , , and are given functions. To complete the proof of Lemma 6, we will establish the following lemma.
Lemma 7. Suppose for all . LetLet be a classical solution to system (20)-(21) satisfying for . Then, for any , there exists a constant such that for all , where the constant depends only on , , , and .
Proof. We take satisfying for all and notice that by scaling and in the governing equation. To complete the proof, we have to divide the proof into three steps.
Step 1. We will firstly prove that for any . Multiplying both sides of the first equation of (3) by and integrating the resultant equation, we yieldwhere the last inequality follows from (A3); then (23) is rewritten asNoticing , we obtain the fact thatSubstituting (24) into (25), we havewhere we have applied Young’s inequality with .
Combing with the assumption of Lemma 6, and , we obtain the following inequality:where .
For convenience, we define . Therefore, for some constant , we have from which we yield that for some positive constant which is independent of . Thus, for any , we conclude that which means that for any and . We notice that depends only on and and .
Step 2. Choose a satisfying and . Then we obtain from (25) thatRecalling -Young’s inequality , , we havewhere and Combining (31) with (32), we yieldwhere . We notice that Hence, we choose a sufficiently small such thatFurthermore, we have the estimatewhose proof is similar to Lemma 4. Thus, we omit it. Squaring (36), we obtain that Thus, That is,Substituting the above inequality into (39), we havefor some constants .
Step 3. We notice that , and define We also see that is bounded and depends only on , , , and . It is easy to observe that is nondecreasing in with and is a nondecreasing function with respect to . Combining the above analysis, we yield That is,Integrating (43) in , we haveAssuming by the definition of , we getwhereFrom (44)–(46), we obtain thatBy a similar argument as in , we immediately yield that . Thus, the proof of Lemma 7 is completed.
5. Proof of Theorem 1
In this section, we will complete the proof of Theorem 1 by the local existence and some priori estimates as given below.
Proof. Suppose there exists a solution for the maximal time . Then according to Lemma 6 we know . Thus, if we treat as a source term in the second and third equations of (9), we can obtain thatBy employing the estimate for parabolic equation in  when goes to infinity, we obtain the norm bound of , , and , . So we have being infinite from the second conclusion in Theorem 1 which contradicts our original assumption. That is, the maximal time .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors are very grateful to Professor Yong Li for providing valuable advice. Moreover, they very thankful to the anonymous referees for the careful reading and various corrections which greatly improved the exposition of the paper. This work partially supported by the NSF of China (under Grants nos. 11171350 and 11204019), the Scientific and Technological Research Project of Jilin Province’s Education Department (nos. 2013287 and 2014312), the Youth Project of Jilin Province’s Science and Technology Department (nos. 20130522099JH and 201201140), and the Twelfth Five-Year Plan Project of Jilin Province’s Educational Science (no. ZD2014078).
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