Global Existence of a Chemotactic Movement with Singular Sensitivity by Two Stimuli

Ma, Heping

doi:https://doi.org/10.1155/2021/9323338

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Analysis, Control of Singularity, and Hybrid in Fractional-Order Systems

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Research Article | Open Access

Volume 2021 | Article ID 9323338 | https://doi.org/10.1155/2021/9323338

Global Existence of a Chemotactic Movement with Singular Sensitivity by Two Stimuli

Heping Ma¹

Academic Editor: Shiping Wen

Received29 Apr 2021

Revised11 Jul 2021

Accepted28 Jul 2021

Published05 Aug 2021

Abstract

In this study, we deal with the chemotaxis system with singular sensitivity by two stimuli under homogeneous Neumann boundary conditions in a bounded domain with smooth boundary. Under appropriate regularity assumptions on the initial data, we show that the system possesses global classical solution. Our results generalize and improve previously known ones.

1. Introduction

Chemotaxis is a well-known biological phenomenon describing the collective motion of cells or the evolution of density of bacteria driven by chemicals, including embryo development, skin wound healing, cancer invasion, and metastasis. The pioneering works of the chemotaxis model was introduced by Keller and Segel in [1], describing the aggregation of cellular slime mold toward a higher concentration of a chemical signal, which readswhere is a bounded domain with smooth boundary. The mathematical analysis of (1) and the variants thereof mainly concentrate on the boundedness and blow-up of the solutions (refer to [2–6] and the references therein), that is, it is well-known that for all suitably regular initial data , an associated Neumann initial boundary value problem, posed in a smooth -dimensional domain , Osaki and Yagi [4] proved that system (1) always possesses a global bounded classical solution in one-dimensional bounded domain, Nagai et al. [3] showed that system (1) also admits a global bounded classical solution in two-dimensional bounded domain if is small, as , Winkler [5] proved that for each , , one can find , such that if the initial data fulfill and for some , then the solution is global in time and bounded; this is complemented by corresponding findings on the occurrence of finite-time blow-up of some solutions emanating from smooth but appropriately large initial data [2, 6]. In the past few decades, system (1) has attracted extensive attentions.

Keller and Segel [7] introduced a phenomenological model of the wave-like solution behavior without any type of cell kinetics, a prototypical version of which is given bywhere represents the density of bacteria and denotes the concentration of the nutrient. The second equation models consumption of the signal. In the first equation, the chemotactic sensitivity is determined according to the Weber–Fechner law, which says that the chemotactic sensitivity is proportional to the reciprocal of signal density. Winkler [8] proved that if initial data satisfy appropriate regularity assumptions, system (2) possesses at least one global generalized solution in two-dimensional bounded domains. Moreover, he took into account asymptotic behavior of solutions to system (2) and proved that in and in as provided , , where are the positive constants. When is replaced by , , and , , , and any sufficiently regular initial data, Lankeit and Viglialoro [9] showed that system (2) has a global classical solution. Moreover, if additionally is sufficiently small, then also their boundedness is achieved. When system (2) has a logistic source , Lankeit and Lankeit [10] showed that system (2) possesses a global generalized solution for any , , and if . As , and is replaced by , , satisfying as . Zhao and Zheng [11] proved that system (2) possesses a unique positive global classical solution provided with or with . When is replaced by , and turned into , Zhao et al. [12] obtained the global existence of classical solutions with . Moreover, for any global classical solution to the case of , it is shown that converges to 0 in the -norm as with decay rate established whenever with . When is replaced by and , Liu [13] showed that for any sufficiently smooth initial data, system (2) admits a global classical solution when either and or and . When is replaced by , Lankeit [14] proved that if , system (2) admits a global classical solution or global locally bounded weak solution.

This study deals with a chemotaxis system with singular sensitivity by two stimuli, which is given bywhere is a bounded domain with smooth boundary , denotes the derivative with respect to the outer normal of , and , , and represent the density of the cell population, the concentration of the chemoattractant substances, and the concentration of the chemorepellent substances, respectively. We assume that , , where and satisfieswith , and are the constants. Furthermore, we assume that the initial data satisfy

It is different from model (3); the following is the attraction-repulsion Keller–Segel model where the signal is produced and not consumed by the cells:which was proposed in [15] to describe how the combination of chemical might interact to produce aggregates of cells. In two space dimensions, when , , and , Jin et al. [16] proved that if , then system (6) possesses a unique global uniformly in-time bounded classical solution, and if , , the same result is also obtained; to the contrary, if , the solutions blow up in finite or infinite time. Ulteriorly, Xu et al. [17] showed that if , the global classical solution of system (6) converged to the unique constant state as , where . When , , Li et al. [18] proved that if is sufficiently small, either of the following cases holds: (i) , , and ; (ii) , , and , the corresponding solution of (6) blows up in finite time. Hu et al. [19] showed that if and hold with small enough, the solution of (6) blows up in finite time. In high dimensions, when , , Jin [20] proved that if , system (6) possesses global classical solution in dimensions and weak solution in three dimensions with large initial data. When , , , , and for all with some and , Lin et al. [21] showed that if , for any nonnegative initial data, system (6) admits a unique classical solution which was global and bounded; if , the radially symmetric solutions may blow up in finite time. When , for and with and for all , Li et al. [22] proved that the corresponding initial boundary value problem possesses a unique global bounded classical solution for . In particular, in the case and , the solution is globally bounded if . When the system has a logistic source, the relevant results can be found in [23–25].

To the best of our knowledge, Dong et al. [26] first put forward the following chemotactic model with general rotational sensitivity caused by two stimuli:where is a bounded domain with smooth boundary . Under mild assumptions on , , system (7) admits at least one global generalized solution.

Throughout above analysis, compared with system (6), the theory of system (3) is so fragmentary. To the best of our knowledge, the global classical solution of model (3) in has never been touched. No matter biological relevance or mathematical meaning, we find it is worth addressing the basic solvability theory of the model (3). Inspired by the arguments in previous studies [8, 13, 14, 26, 27], we mainly investigate the global classical solution in a chemotactic movement with singular sensitivity by two stimuli. Theorem 1 partially generalizes and improves previously known ones.

In this study, we use symbols and as some generic positive constants which may vary in the context. For simplicity, is written as , the integral is written as , and is written as .

The rest of this study is organized as follows. In Section 2, we summarize some useful lemmata in order to prove the main result. In Section 3, we give some fundamental estimates for the solution to system (3) and proof of Theorem 1.

2. Preliminaries and Main Result

In this section, we give the main theorem and the local existence of the classical solution to (3) and also summarize some useful lemmata in order to prove the main result. Noting the singular chemotaxis term, we let

Then, we can rewrite (3) as

At first, we give the main result of global existence of the classical solution to (3).

Theorem 1. Let be a bounded domain with smooth boundary. Assume that , , satisfy (4) and

Then, for any choice of the initial data fulfilling (5), there exists a triple which solves (3) classically. Moreover, we have and in .

Remark 1. Theorem 1 shows that system (3) admits a global classical solution nothing to do with and .

Remark 2. Theorem 1 partially generalizes and improves the results in ([14], Theorem 1.1) and ([13], Theorem 1.1).

Remark 3. If we replaced the terms and in system (3) by and , respectively, Theorem 1 still holds provided that both and are nonnegative differentiable functions satisfying and .

In the sequel, we will consider system (9) to obtain the local boundedness of and then come back to system (3) to prove the main theorem. Under the framework of fixed point theorem, we will prove the local existence of classical solution to system (3) in the following lemma. The proof is quite standard, and a more detailed display of a similar reasoning in a related circumstance can be found in [14].

Lemma 1. Let be a bounded domain with smooth boundary. Assume that , , satisfy (4) and . Then, for any initial data fulfilling (5), there exist and a triple solving (3) classically in , where denotes the maximal existence time. Moreover, the solution satisfiesif , then

Lemma 2. The solution of (9) satisfieswhere .

Proof. Integrating the second equation of (9) with respect to , we haveDue to the positivity of , we obtain (13) immediately. Likewise, we get (14). The proof is complete.

Lemma 3. (Gagliardo–Nirenberg interpolation inequality [28]). Let . There exists a positive constant , such that for all ,is valid with .

Lemma 4. (See [29]). Let and be such thatwhere has the property thatwith some and . Then,

3. Proof of Theorem 1

In this section, we establish some priori estimates for solutions to system (9); we first establish a bound for and in the one-dimensional case, which differs from that in the multidimensional settings.

Lemma 5. Let and suppose that , (4) holds with , , and , . Then, there exists , such that the solution of (9) satisfies

Proof. Multiplying the first equation in (9) by , integrating by parts, using (4) and Young’s inequality, we havefor all , where , and we have used the facts that due to ; since , without loss of generality, we let ; by Lemma 1, we know that there exists a constant , such that , and with some rearrangements, we haveMultiplying the first equation in (9) by , similar to (23), we havewhere by Lemma 2, we know that there exists a constant , such that .Let , we haveCombining with (23) and (26), we obtain thatBy the Gagliardo–Nirenberg inequality with (16), there exist , such thatwithdue to . Similar to (28), there exists , such thatChoose appropriate , such that and . Let , ,By (26), we haveAs from Lemma 3 and Lemma 4, we readily obtain (28). (21) follows by integrating (32) in time. The proof is complete.

Lemma 6. Let and suppose that , (4) holds with , and , . Then, there exist and , such that

Proof. By the Gagliardo–Nirenberg inequality, there exist and , such thatfor all with some andas well asIn accordance with Lemma 5, there exist some , such thatSubstituting above four inequalities into (34) and (35), we havefor all with some , where we used the fact that and , and the proof similar to Lemma 2.6 in [13]. The proof is complete.
According to the proof of [14], we have from Lemma 6 thatwith some . We claim that for any and for all , there exist some , such that and in . This together with the definition of entails and in . With this in hand, we next consider system (3).

Lemma 7. Let , and suppose that , (4) holds with , , and

Then, for all , there exists , such that

Proof. Multiplying the first equation in (3) by , integrating by parts and Young’s inequality, we havefor all , here we used the fact that and in . We obtain (42) if we letThe proof is complete.
Applying the same arguments as in the proof of Lemma 2.9 and Lemma 2.10 in [13], we have the following Lemma.

Lemma 8. Assume that be a bounded domain with smooth boundary. Then, for all , there exists , such that

Lemma 9. Let . Assume , (3) holds with , , and

Let and be the same as in Lemmas 7 and 8. Then, for all , one can find some fulfillingas well as

Proof. We first consider the dimensional . Combining with Lemmas 7 and 8, there exists some , such thatfor all . By the Hölder inequality, we havefor all , where and . Moreover, by the Gagliardo–Nirenberg inequality, there exist some constants , such thatwiththe reason analogous to (2.42) and (2.43) in ([13], Lemma 2.11). Therefore, combining with (20) and (52), we getfor all , where and . Along with (55), (56) follows from ([13], Lemma 2.9), (50), and (51), and we deducefor all , with some constants determined by . Similar to (2.45) in ([13], Lemma 2.11), we know thatTherefore, using the Young inequality, we derivefor all , with some constants determined by . Once more, employing Young’s inequality, there exist some constants determined by , such thatas well asfor all . Combining with (57)–(63), we obtainfor all . Analogous to (64) and (65), we havefor all , with some constants determined by . By the Gagliardo–Nirenberg inequality, there exists a constant , such thatTogether with (49) and (64)–(68), we havefor all with some constants . The proof is complete.
Finally, we prove the main theorem.

Proof. of Theorem 1. From Lemma 9 and the Moser–Alikakos iterative technique in Lemma A.1 of [30], we have for all ,for all with some constant determined by . Thus, the global existence is a consequence of (70) and the extensibility criterion provided by Lemma 1.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interest.

Acknowledgments

The study was supported by the NSF of China (11801154).

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Copyright © 2021 Heping Ma. 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.

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