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.Ourmain results are based on themethod 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.


Introduction
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 [1] and Keller and Segel in [2].During the past four decades, chemotaxis models have been studied extensively (see, e.g., [3][4][5][6][7][8][9][10] and the rich references therein).For instance, Othmer and Stevens in [3] 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:   = ∇ ⋅ ( () ∇) − ∇ ⋅ (∇Θ (V)) +  () ,  ∈ Ω,  > 0, V  = ΔV +  − V,  ∈ Ω,  > 0, (1) where Ω ⊂ R  is a bounded connected domain with a smooth boundary Ω,  > 0,  > 0, and  > 0 are positive constants, and  = 0, 1.The function  = (, ) denotes the cell density and V = V(, ) represents the chemotactic concentration, for example, the oxygen.The constant  is called the chemosensitive coefficient, and the sign of  corresponds to chemoattraction if  > 0, and chemorepulsion if  < 0. 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 Θ(V) is commonly referred to the chemotactic potential function.
In the absence of logistic source (i.e., () ≡ 0), 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 Ω is a ball in R 2 under the condition that ∫ Ω  0  > 8/() and ∫ Ω  0 ()|| 2  is sufficiently small.For a general domain, Nagai in [15] further showed the finite-time blowup of nonradial solutions provided that ∫  0 | − | 2  is sufficiently small and Winkler in [16] studied finite-time blowup of radially symmetric solutions to the full parabolic system with logistic sources () = −  in a ball Ω ⊂ R  with parameters  ≥ 0,  ≥ 0, and  ≥ 1.Moreover, Winkler in [8] gave the set of blowup to the system (1) by enforcing initial data with respect to the topology of   (Ω)× 1,2 (Ω) for any  ∈ (1, 2/(+2)), where Ω is a ball in R  with  ≥ 3.In summary, the solution of system (1) never blows up when  = 1, whereas there is finite-time or infinite-time blowup when  ≥ 3.Moreover, recent results in [17] confirmed that the attraction-repulsion is a plausible mechanism to regularize the classical Keller-Segel system (2) with () = 0 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: (2) System (2) with () = 0 was proposed in [20] to describe the aggregation of microglia observed in Alzheimer's disease.
Here (, ) represents the cell density and V(, ) denotes the concentration of the chemoattractant and (, ) is the concentration of the chemorepulsion.The constants satisfy  ≥ 0 and  ≥ 0, 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 crossdiffusive 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 () ≡ 0, the theorem of competing effects has been established in [17] with  = 0.Moreover, the global existence, asymptotic behavior, and steady states of classical solution were studied in [21] for one-dimensional case with  = 1.

Motivation
In this paper, we consider the following attraction-repulsion chemotaxis system including three parabolic equations: with the initial-boundary value conditions: where Ω ⊂ R  is a bounded  2+ domain for some 0 <  < 1 and ] denotes the unit outer normal vector field on Ω.The function of the cell density  = (,) is the fraction of volume occupied by cells, whereas the fast diffusion coefficient of the cell is described as ( It is easy to see that the function 1/(1 − )  is a monotonically increasing function of  which guarantees that the solution will not blow up.The functions V = V(, ) and  = (, ) denote the concentration of the chemoattraction and chemorepulsion, respectively.The constants  V > 0 and   > 0 are the diffusion coefficients of the chemoattraction and chemorepulsion, respectively. V and   denote the chemosensitive coefficients, and () describes the kinetic function.
In this paper, we will prove the existence of global classical solutions to the generalized system (3) with the initialboundary conditions and the no-flux boundary condition (4).In addition, we need the following assumptions.
The main result is the following.
Then there exists a unique global triple solution of (, V, ) to system (3) satisfying initial conditions ( 4) and Furthermore, there exist two constants  V ≥ 0 and   ≥ 0 such that Throughout this paper, we introduce some notations.Ω is a bounded open interval in R = (−∞, +∞) and  denotes a general constant which may have different values in different place.
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.

Local Existence
To prove Theorem 1, we need to establish local existence of solutions to system (3) and some priori estimates in this section.
(2) There is a global classical solution of system (3) if (, ) is bounded away from 1 for each finite time  > 0, which means  0 = +∞.
The proof of (2) in Theorem 2 is completed by applying the theorem of quasilinear parabolic equations in [26] since system ( 9) is an upper triangular system.

A Priori Estimates
Next, we recall the Gagliardo-Nirenberg inequality for functions satisfying the boundary condition for  > 0 (see [27,28]).
Proof.We take  > 0 satisfying  0 ≤ 1 −  for all  ∈ Ω and notice that |Ω| = 1 by scaling  and  in the governing equation.To complete the proof, we have to divide the proof into three steps.

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  1+/2,2+ solution for the maximal time  0 < ∞.Then according to Lemma 6 we know 0 ≤  ≤ 1.Thus, if we treat  as a source term in the second and third equations of (9), we can obtain that ‖V‖  (48) By employing the   estimate for parabolic equation in [27] when  goes to infinity, we obtain the  ∞ norm bound of V, |∇V|, and , |∇|.So we have  0 being infinite from the second conclusion in Theorem 1 which contradicts our original assumption.That is, the maximal time  0 = ∞.