Existence of a Nontrivial Steady-State Solution to a Parabolic-Parabolic Chemotaxis System with Singular Sensitivity

Abstract

This paper establishes the existence of a nontrivial steady-state solution to a parabolic-parabolic coupled system with singular (or logarithmic) sensitivity and nonlinear source arising from chemotaxis. The proofs mainly rely on the maximum principle, the implicit function theorem, and the Hopf bifurcation theorem.

1. Introduction

Chemotaxis is the biased movement of cells toward the concentration gradient of a chemical. It plays a critical role in a wide range of biological phenomena. For example, cells migrate toward resources of food and stay away from harmful substances. The first mathematical model of chemotaxis was introduced by Patlak in [1] and Keller and Segel in [2]. There are numerous works dedicated to the analysis of chemotaxis models. For example, Othmer and Stevens in [3] modeled myxobacteria as individual random walkers and proposed a microscopic model based on a velocity jump process. By taking the parabolic limit of the microscopic model, they obtain the macroscopic chemotaxis model, which is the well-known Keller-Segel systemwhere is a bounded connected domain with a smooth boundary . The function denotes the cell density and represents the chemical concentration, for example, oxygen. The constant is called the chemotactic coefficient, and the sign of corresponds to chemoattraction if and chemorepulsion if . Parameters and are the diffusion coefficients of the cells and the chemical, respectively. The function represents the kinetic function describing production and degradation of cells, and is commonly referred to as the chemotactic potential function. Function describes the production and degradation of the chemical.

The existence of global solutions, blow-up, and traveling wave solutions to the chemotactic system (1) were extensively studied during the past four decades (see, e.g., [4–12] and references therein). The authors studied the roles of growth, death and random in promoting population persistence through band popation in [4], and the work was related to the significance of cell motility and chemotaxis in microbial ecology.

X.F. Wang addressed the trivial and nontrivial steady states with small and of the quasi-linear system (1) in [13]. In the absence of population dynamics (i.e., ), there have been extensive studies. The main feature of solutions to the Keller-Segel model is the possibility of blow-up in finite time in [9, 14, 15]. Moreover, recent results in [6, 10, 11, 16, 17] proved the global existence of solutions under some conditions. Moreover, the global existence, asymptotic behavior, and steady states of classical solutions were studied in [18] for the one-dimensional case.

The purpose of this paper is to study the existence of the nontrivial steady-state solution to a parabolic-parabolic coupled system arising from chemotaxis with singular sensitivity. We consider the following systemwith the initial-boundary value conditionswhere and “”. The function denotes the density of the cells and denotes the concentration of the chemical. The parameters , and are all positive constants, with being the diffusion coefficient of cells and the chemotactic sensitivity coefficient as above. The function is assumed to be smooth.

The steady-state problem corresponding to system (2)-(3) iswhere . The linear chemotactic potential was considered in [19], and X.F. Wang’s model in [13] is similar to ours.

In this paper, we establish the existence of nontrivial steady-state solutions to the parabolic-parabolic coupled chemotactic system (2)-(3) with singular sensitivity. Here we assume that the function satisfiesObviously, and are the trivial solutions to system (4).

Our main results in this paper are presented in the following theorem.

Theorem 1. Consider system (4) with condition (5). Then the following alternative holds: (1)If , then system (4) has a pair of nonnegative steady-state solutions and (2)If , then there exists at least a pair of nontrivial steady-state solutions .

The rest of this paper is organized as follows. In Section 2, we give some preliminary lemmas. In Section 3, we complete the proof of Theorem 1.

2. Lemmas

To prove Theorem 1, we need to establish boundary estimates of solutions to system (4). Since the state variables represent densities, we only consider nonnegative solutions, that is, and on . Firstly, we state one result concerning the estimate of a solution .

Lemma 2. Assume the solution is a nonnegative steady state. Then we obtain the following: (i), and .(ii).

Proof. (i)By the second equation of (4), we have and, integrating once, we obtain . Hence, and .(ii)Integrating the first equation of (4) yields

Lemma 3. Let solve system (2) with the initial-boundary value condition of (3). Then we have the lower-bound estimateand is defined in (11) below.

Proof. From the second equation in (4), we obtain Integrating the above equation once, we have Integrating by part and applying the condition , we rearrange the above equationFrom the boundary value condition of in (3), we have . Define the functionFrom the definition of the function , we easily getApplying the implicit function theorem, we have the estimate (7).

Lemma 4. There are two positive constants and such that a nonnegative steady state of system (4) satisfies

Proof. By the steady-state system (4), we know , and for every , we haveApplying the condition , we obtain the following equationSimilarly, we also getOn the other hand, integrating once the first equation of (4), we getwhich is equal toThen, applying the boundary condition and rewriting (18), we have the following estimateNext, we review the condition , which implies that . Combined with the condition of in Lemma 3, inequality (19) can be rewritten asIntegrating (20) once, we haveHere and . Hence, we obtain which implies that Define the function . Then we have . Combining the above inequalities, we obtain the following boundBy Gronwall’s inequality, we getSubstituting the definition of into inequality (25), we obtainCombining this result with in Lemma 2, we obtain estimate (13).

Next, we suppose that for any . Given , we consider the equationThen, we prove the following lemma.

Lemma 5. For any small , we pick a sufficiently small such that . Next, we chooseand setFinally, we denote by a nonnegative solution to system (4) when . Then we have .

Proof. If not, there exists some such that . According to Lemma 2 and the definition of , we haveFrom (13), we can easily getHowever, .
In fact, if not we have which contradicts the fact that .
Next, we consider which is a contradiction. In other words, we have .

Lemma 6. (i)For each , system (27) has a unique solution .(ii) is continuous.(iii)If , then .

Proof. For any given , it is obvious that and are a pair of sup-sub-solutions. According to the standard comparison theorem, we easily obtain the estimate of the solution to system (27). If and are solutions to system (27), then we haveThe difference between the first and second equation of (34) iswith the notation .
By the maximum principle, we obtain . Moreover, is uniquely defined and .
Next we give the proof of (ii) and (iii) in Lemma 6, respectively. (i)Assume that there exists a sequence , which converges to . Then we only point out that there exists a subsequence of such that on . By regularity theory, we know that is the solution to system (34) and , which contradicts the uniqueness of .(ii)We can directly prove (iii) from the maximum principle.

3. Proof of Theorem 1

We dedicate this section to the proof of Theorem 1.

Assume there exists such that satisfies the following systemIt is easy to see that is the nonnegative solution to system (36) by the definition of and .

Define to be the unique solution to the systemwhere is a constant satisfying . For any given , we define to be the unique solution to the following systemThen we rewrite (36) and obtain the following operator equationWe can reformulate (39) as the following operator equationNext, we introduce some notation:It is easy to see that the operator is compact and linear.

Define the operator as the mapSimilar to the proof of Lemma 5(i), we can show that the operator is continuous and bounded from to . We also haveHence, is compact operator from to with the norm . Moreover, the operator is also compact from to and .

From the above lemmas, we can show the proof of Theorem 1.

Proof of Theorem 1. Assume that is the nonnegative solution to system (2). Then we have However, from the proof of Lemma 3, we have which is a contradiction. Thus, we obtain .
Assume that there is a constant such that .
Next, we choose a variable as the parameter of bifurcation. Furthermore, is a single-value characteristic root. Thus we know that system produces a bifurcation in a small neighbourhood of the point according to the classical bifurcation theorem. Then there is a bifurcating solution , represented aswhere and is sufficiently small. Furthermore, and . Therefore, when is sufficient small, the bifurcation solution is in the set ofwhere is the characteristic eigenvector of characteristic eigenvalue and is a positive constant. There is a nontrivial connected set satisfying the following alternative conclusion: (i) connects the two points and , where is another characteristic eigenvalue of .(ii) is unbounded and . We firstly prove . Otherwise, there exists a point such that andChoose a sufficient large constant . Then by (36), we can obtain However, which means that is equal to zero at some point. We can directly have by the maximum principle. That is, is a bifurcation point of (40).
Denote . Because is a compact operator, there exists a convergent subsequence of . For the convenience, we still write it as which satisfies . Next, we divide by both sides of (40) and let go to infinity. Then we obtain where . By the definition of we know that . However, has a unique characteristic eigenvalue with positive characteristic function. Thus, we have which contradicts the assumption. So conclusion (i) is impossible. In other words, is unbounded and .
According to (i) in Theorem 1, we know that if , then , which is equivalent to the condition . However, from Lemma 5, we have that is bounded for any . By Lemma 4, we have that is bounded. Furthermore, because the set is unbounded and connected, we can obtain that , which means that when , system (4) has nontrivial steady-state solution .
Finally, we show that system (2)-(3) has a positive solution when . In fact, we know that because . Next we prove that . If is zero at some point, then the initial value problem has a solution , which contradicts the boundary condition .

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interest.

Acknowledgments

This work is partially supported by the National Natural Science Foundation of China (No. 41701054), the China Postdoctoral Science Foundation (No. 2018M631890), the Scientific and Technological Research Project of Jilin Province’s Education Department (No. 2016285), Teaching Reform of Higher Education Research Project of Jilin Province (No. SJYB17-01). Finally, the author is grateful for the support by the China Scholarship Council (No. 201707535005).