On the Global Well-Posedness for a Hyperbolic Model Arising from Chemotaxis Model with Fractional Laplacian Operator

Melkemi, Oussama; Abdo, Mohammed S.; Aiyashi, M.A.; Albalwi, M. Daher

doi:https://doi.org/10.1155/2023/1140032

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Volume 2023 | Article ID 1140032 | https://doi.org/10.1155/2023/1140032

On the Global Well-Posedness for a Hyperbolic Model Arising from Chemotaxis Model with Fractional Laplacian Operator

Oussama Melkemi,¹Mohammed S. Abdo,²M.A. Aiyashi,³and M. Daher Albalwi⁴

Academic Editor: Arzu Akbulut

Received25 Aug 2022

Revised11 Oct 2022

Accepted24 Nov 2022

Published09 May 2023

Abstract

Several cells and microorganisms, such as bacteria and somatic, have many essential features, one of which can be modeled by the chemotaxis system, which we consider to be our main interest in this article. More precisely, we studied the hyperbolic system derived from the chemotaxis model with fractional dissipation, which is a generalization for the hyperbolic system with classical dissipation. The results of this article are divided into two parts. In the first part, we used energy methods to obtain the existence of small solutions in the Besov spaces. The second one deals with the optimal decay of perturbed solutions using a refined time-weighted energy combined with the Littlewood-Paley decomposition technique. To the authors’ best knowledge, this type of system (with fractional dissipation) has not been studied in the literature.

1. Introduction

This present article aims to study the hyperbolic chemotaxis system, which is governed by the following Cauchy problem:where represents the cell density, with is the chemical concentration, and describe the cell and chemical diffusion coefficients, respectively, and the fractional operator denotes the fractional Laplacian operator. In the whole space , the operator is defined via the following Fourier transform:

Chemotaxis model (1) describes the ability of free-moving organisms to react to chemical substances or their concentration differences with specific, directed movements. On the other hand, the fractional dissipation has several applications in the molecular biology, we mention as an example the anomalous diffusion and chemical attraction to organisms in semiconductor growth, see for instance [1]. It is well known that the fractional chemotaxis system (1) is derived from the following canonical formulation of the famous “Keller-Segel” model [2, 3]:where is a constant that represents the chemosensitivity; when the coefficient , we say that the system is attractive and in the case when the coefficient , the system is repulsive. The function describes the mechanism of signal detection. In the situation, when the function with (1,2] Biler and Wu [4] investigated system (3) in the Besov spaces. In particular, they have established the local existence and uniqueness of solutions. Later, Zhai [5] studied the system in the case when ; more precisely, Zhai [5] showed that system (3) admits a mild solutions. There is a large literature on the analysis of system (3), we refer the reader to [6–10] and the references therein. For the case, when , then system (1) turns out to be the hyperbolic-parabolic chemotaxis; in this context, when the fractional Laplacian operator is substituted by the classical Laplacian operator, Zhang and Zhu [11] investigated the Cauchy problem for system (1) with small initial data. Lately, Jun et al. [12] explored the global existence of system (1) with large initial data. Recently, Hao [13] showed that system (1) admits a unique solution near to some constant equilibrium state in the critical hybrid Besov spaces . In their very recent work Nie and Yuan [14] investigated the well-posedness and the ill-posednes in the critical Besov spaces . For this hyperbolic-parabolic system, there have been a large number of results concerning the long-time dynamics to the solutions, blow-up phenomenon, and the existence of global solutions (see for instance, [15–19]). Here, we focus on the chemically diffusible model corresponding to the case of . As far as we know, the results obtained for the Hyperbolic model (1) are less compared to Hyperbolic-Parabolic model (1) . In the situation, when the fractional Laplacian is changed by the full Laplacian operator, Tao et al. [20] proved that system (1) is globally well posed; moreover, they have obtained the long-time behavior, and diffusion limit of one-dimensional large-amplitude classical solutions on finite intervals subject to the Neumann-Dirichlet boundary conditions. Thereafter, in [21], Li and Zhao explored the same issues of [20] but for the Dirichlet-Dirichlet boundary conditions. Wang et al. [22] investigated the global existence, asymptotic decay rates, and diffusion convergence rate of small solutions in the Sobolev framework. Recently, Martinez et al. [23] proved a set of results, such as, the global asymptotic stability of constant ground states and the explicit decay rate of solutions. More recently, Wu and Su [24] showed that system (1) admits global solution and also they have obtained a decay rate of solutions in the Besov spaces. For the traveling wave solution of problem (1) with and its nonlinear stability, see the series of papers [25–28].

Stimulated by the above works, especially with [24], the main objective of this article is to investigate system (1) with the fractional Laplacian operator , with . To be more precise, we have the following two outcomes:

Theorem 1. Assume that and . Let . We assume that there exists a constant such thatwhere and , for some equilibrium state . Then, system (1) admits a global unique solution satisfying for any :

Remark 1. From the identity , then for , Theorem 1 implies the global well-posedness in the scale of (homogeneous) Sobolev spaces.
Our existence theorem is based on the energy estimates. Due to the presence of the terms and , we established with the help of Bony’s decomposition (see the next section) a product estimates, thus we can get the a priori estimates of our system, which leads us to get the global small solutions in .
The second result of this article deals with the time decay rates of strong solutions for system (1), which is given as follows:

Theorem 2. Let and . Let . There exists a constants such that ifwhere and , then system (1) has a unique global solution as follows:where . Furthermore, there exists a positive constant such that

Remark 2. Theorem 2 gives the optimal decay of solutions for system (1) due to embedding

Remark 3. In the sequel and for the simplicity we assume (without loss of generality) .
The content of this article is arranged as follows. We introduced the notations and some useful definitions and results in Section 2, then we reformulated system (1) and we established some useful a priori estimates in section 3. Section 4 is devoted to the proof of the existence of solutions. Finally, we proved Theorem 2 in Section 5.

1.1. Notation

or stands for some positive constant and may represent different values in different lines, the notation means that there exist a constant such that , where is a constant depending on the initial data.

2. Preparatory

In this section, we recall some ingredients, such as the famous Littlewood–Paley operators and Bony decomposition, also some function spaces will be introduced.

2.1. Littlewood–Paley Theory

We start this section by giving the definition of the dyadic partition of unity.

Definition 1. There exists such thatFor every , we defined the operators , and as follows. Let and as follows:By the definition of localization operators above, we have some interesting properties as follows:(1)The formal Littlewood–Paley decomposition:(2)The quasiorthogonality:(i)If , then ;(ii)if , then . The paradifferential calculus plays a key role in the proof of the existence result, and its given by the following definition:(i)Bony decomposition For , we have the following equation: with The next result is the famous Bernstein inequality, for the proof see [29].

Proposition 1. Let . Assume , then for every , there exists constants such thatNow, we recall the definitions of homogeneous and nonhomogeneous Besov spaces.

Definition 2. For and . The homogeneous Besov space is defined as the set of all tempered distributions such that:Besides, if

Definition 3. For and . The nonhomogeneous Besov space is defined as the set of all tempered distributions such thatBesides, ifThe following spaces are introduced by Chemin and Lerner in [30]:

Definition 4. We assume that and , the space is the set of such thatand the space is the space such thatAs a consequence of the last definition, we have the following embedding. Let , then we have the following equation:The following Lemma is useful in our contribution, for more details see [31].

Lemma 1. (i)Let satisfying . Then(ii)Let and . ThenWe finished this section by establishing a product estimate.

Lemma 2. Let and . Then and we have the following equation:

Proof. By using the Bony decomposition (13), we split the term as follows:From the property (2) part (ii), we have the following equation:Multiplying equation (27) by and taking the norm we find out thatFor the first term we used Hölder inequality and Young inequality and obtained the following equation:Similarly, we obtain the following equation:Taking advantage of Young inequality and the Hölder inequality, it holds thatBy the Besov embedding , we get the desired result.

3. Reformulation of System and the a Priori Estimate

In this paragraph, we reformulated the original system (1), where we will assume (without loss of generality) that the equilibrium state and we set , the system becomes as follows:

We will study the a priori estimates for linearized system (32) with general source term as follows:

We note that system (33) is given by the following equation:where and

Next, we prove the A priori estimate for system (32).

3.1. A Priori Estimate

Proposition 2. We assume that is a regular solution for the system (32), then for all [0,T),

Proof. Let and we set and .
We observed that solves the following system:Taking the -scalar product of equation (37), we found out thatFor simplicity, we set . According to the identity and Bernstein’s Lemma, we get the following equation:Thus,Multiplying equation (40) by and taking the norm over , we obtained the following equation:Integrating the above estimate with respect to time, hence, we get the desired result.

4. Existence

For the proof of the existence part, we are going to use the classical Friedrichs’ regularization method combined with the energy method. First, we define the spectral cutoff as follows. Let where and denotes the characteristic function on the ball . We define the following equation:

We consider the following approximate system:where and .

The usual (Cauchy Lipschitz) theorem guarantees that system (44) admits a unique regular solution C([) ).

Denote

Letwhere and , thus we have . From Lemma 2.5, we deduce that

In a similar way,

Hence,where . We choose . Therefore, . Then, for any , we have . Our next aim is to prove that ; previously, we showed that for all . By the continuity argument, we can get , for a sufficiently small constant , which contradicts the definition of . Next, we will show that system (44) admits global solutions. For this aim, we assumed that , we have . On the other hand, we have , then we have . By the Cauchy-Lipschitz Theorem we can continue the solution beyond the time and this contradicts the definition of . Hence, . By standard arguments, we can show that converges to the solution which solves system (32) and here we omit the details.

5. Optimal Decay of Solutions

To know more about the asymptotic behavior of the solutions obtained in the previous results, we studied in this section the optimal temporal decay of perturbed solution in terms of norm, where we will estimate low frequencies and high frequencies separately. For the low frequencies, we use the good behavior of the semigroup . Regarding the high frequencies, we make use of the Fourier localization method.

Proof of Theorem 2. We assume that is a smooth solution for system (33) and , Then, employing Duhamel’s principle, we found out thatwhere and is the semigroup associated with the LHS of system (33), which is given by the following equation:wherewhere represents the identity matrix of the dimension. For convenience, we denoted the following equation:therefore,Since coincides with for all , thenBy virtue of [1], Lemma 2.4, we have the following equation:We also have the following equation:In view of Bernstein’s Lemma, we obtained the following equation:Multiplying both sides by , according to [1], Lemma 2.35, we concluded thatIn a similar way, we also get the following equation:Summing up equations (59) and (60), we deduced the following equation:For the high frequencies, we have the following equation:Similarly, we get the following equation:Consequently,According to Hölder inequality, we obtained the following equation:By using the Besov embedding for , we found thatIn a similar way, we bound . Thus, we get the following equation:According to equations (61) and (64) and Lemma 1, we infer thatWe used equations (47) and (48) with and obtained the following equation:where we have used also the embedding . According to equation (64) and Lemma 1, we found out thatIn view of equations (61) and (64), we deduced the following equation:Putting equations (68) and (70) into equation (71), we obtained the following equation:A bootstrapping argument implies that there is such that, if , then for all ,for some positive constant . Then, the Proof of Theorem 2 is now achieved.

6. Conclusions

More or less recently, several methodologies have been proposed to describe behaviors of some complex world problems emerging in several applications, especially in molecular biology. The chemotaxis model gains increasing interest from mathematicians; so far, the problem of existence and uniqueness of classical solutions to system (1) (with fractional dissipation or classical dissipation ) in multidimensions remains an open problem. In this work, we were able to give a positive answer regarding the existence of classical solutions with small initial data lying in the Besov spaces; also, we managed to get the optimal temporal decay of strong solutions. In the future direction, it will be interesting to study the current model on recent fractional derivatives, then demonstrate the effect of the fractional order through some simulations.

Data Availability

No underlying data was collected or produced in this study.

Disclosure

This work was conducted during our work at Hodeidah University.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2023 Oussama Melkemi et al. 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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