Global Solutions in the Species Competitive Chemotaxis System with Inequal Diffusion Rates

This paper is devoted to studying the two-species competitive chemotaxis system with signal-dependent chemotactic sensitivities and inequal diffusion rates u1t = Δu1 − ∇ ⋅ (u1χ1(V)∇V) + μ1u1(1 − u1 − a1u2), x ∈ Ω, t > 0, u2t = Δu2 − ∇ ⋅ (u2χ2(V)∇V) + μ2u2(1 − a2u1 − u2), x ∈ Ω, t > 0, Vt = τΔV − γV + u1 + u2, x ∈ Ω, t > 0, under homogeneous Neumann boundary conditions in a bounded and regular domain Ω ⊂ R (n ≥ 1). If the nonnegative initial date (u10, u20, V0) ∈ (C1(Ω))3 and V0 ∈ (V, V) where the constants V > V ≥ 0, the system possesses a unique global solution that is uniformly bounded under some suitable assumptions on the chemotaxis sensitivity functions χ1(V), χ2(V) and linear chemical production function −γV + u1 + u2.


Introduction
In this paper, we consider the following two-species competitive chemotaxis system with signal-dependent chemotactic sensitivities and inequal diffusion rates: = 0,  ∈ Ω,  > 0,  1 (, 0) =  10 () ,  2 (, 0) =  20 () , where Ω is a bounded and regular domain in R  ( ≥ 1) and ] is the outward unit normal vector of the boundary Ω. 1 (, ) and  2 (, ) represent the populations densities, and both populations reproduce themselves and mutually compete with the other, according to the classical Lotka-Volterra kinetics [1], and the diffusion rates of the populations are 1.
V(, ) denotes the concentration of the chemoattractant, and the diffusion rate of the chemical substance is strictly less than 1 (i.e., 0 <  < 1). 1 ,  2 ,  1 ,  2 , and  are positive parameters, where  1 ,  2 are the growth coefficients and  1 ,  2 are the competitive degradation rates of population, respectively.The chemotactic sensitivity function   (V) ∈  1,∞ loc (R + ) ∩  1 (R + ) for  = 1,2, which is assumed to be positive.From a biological point of view, when   (V) > 0, populations exhibit a tendency to move towards higher signal concentrations (chemoattraction), while conversely the choice   (V) < 0 leads to a model for chemorepulsion, where populations prefer to move away from the chemical in question [2].Denote ℎ( 1 ,  2 , V) = −V +  1 +  2 representing the balance between the production of the chemical substance by the populations themselves and its natural degradation (see [3] for details).
The classical chemotaxis model was first introduced by Keller and Segel using a mathematical model of two parabolic equations to describe the aggregation of Dictyostelium discoideum as well as a soil-living amoeba, in the early 1970s [4].After the pioneering works of Keller and Segel, a large amount of chemotaxis models has been used to model the phenomena for population dynamics or gravitational collapse, among others.Winkler [5] studied the chemotaxis system under homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ R  .It was proved that the chemotactic collapse was absented for any nonnegative initial date (⋅, 0) ∈  0 (Ω) and V(⋅, 0) ∈  1, (Ω) with some  > ; the corresponding initial-boundary value problem possessed a unique global uniformly bounded solution.Tello and Winkler [6] considered the parabolic-parabolic-elliptic system under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ R  ( ≥ 1) with smooth boundary.Given the suitable positive parameters  1 ,  2 and  1 ,  2 , they showed that all solutions stabilized towards a uniquely determined spatially homogeneous positive steady state within a certain nonempty range of the logistic growth coefficients  1 and  2 .Negreanu and Tello [7] studied a two-species chemotaxis system with nondiffusive chemoattractant: under suitable boundary and initial conditions in an dimensional open and bounded domain Ω for  ≥ 1.They considered the case of positive chemosensitivities and chemical production function ℎ increasing as the concentration of the species , V increasing.The paper proved the global existence and uniform boundedness of solutions, and the asymptotic stability of the spatially homogeneous steady state was a consequence of the growth of ℎ,   and the size of   for  = 1, 2.
Reviewing the recent studies, Zhang and Li [8] considered the following fully parabolic system: under homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ R  ( ≥ 1).By extending the method in [5] (see also [9,10]), the first step is to estimate some associated weighted functions which depend on signal density, and the second step is to obtain  ∞ -bounds of solutions from   -bounds using the variation-of-constants representation and a series of standard semigroup arguments (see [2,5,11,12]).They proved that, if the nonnegative initial date ((⋅, 0), V(⋅, 0)) ∈ ( 0 (Ω)) 2 and (⋅, 0) ∈  1, (Ω) for some  > , the system possesses a unique global solution that is uniformly bounded under some appropriate conditions on the coefficients  1 ,  2 and the chemotaxis sensitivity functions  1 (),  2 ().
Inspired by the foregoing research, the main purpose of this paper is to consider the existence of global solution for the two-species competitive chemotaxis model (1) with inequal diffusion rates.This paper is organized as follows: In Section 2, we formulate the main results of this paper by means of the theorem and establish some preliminaries which are important for our proofs.In Section 3, we firstly consider the local existence of solutions and then proceed with the extensibility criterion.Finally, under some appropriate conditions, we prove that the solutions are uniformly bounded in time using an iterative method.

Preliminaries and Main Results
For convenience, we denote that In order to establish the global existence and uniform boundedness of solutions to (1), we need to make some restrictive conditions throughout this paper in ℎ( 1 ,  2 , V) and   (V) ( = 1, 2): (i) Let the initial date V 0 satisfy 0 ≤ V < V 0 < V, where V and V are some positive constants.
(ii) There exist positive constants  1 and  2 such that (iii) There exist positive constants  01 and  02 such that (iv) Assume that where for   (V) defined by (v) Finally, for some technical reasons, we also assume that We illustrate the validity to above assumptions with the following generalized example [7].
Example 1.We take the chemosensitivity functions   (V) =   /(  + V) ( = 1, 2) for positive constants   ,   fulfilling Clearly, (11) holds.Take a lower bound V = 0 and upper bound V is to be defined later.Moreover, the initial dates  10 and  20 are satisfied as Then, consider the following.
(3) We notice that ℎ(0, 0, 0) = 0 as well as A sufficient condition for the second inequality in (8) holding is and, for simplicity, let us take and then derive Up to now, the above all restrictive conditions are verified, which implies that conditions ( 6)-( 11) are sufficient to ensure the global existence of solutions.
Remark 2. In literature [7], the chemical production function ℎ( 1 ,  2 , V) = −V +  1 +  2 , and the chemotactic sensitivity In this paper, the purpose is to study the global existence and uniform boundedness of solutions to (1) applying an iterative method.The main results are stated by the following theorem.
The proof of Theorem 3 is split into several steps.
Step one: we start to consider the local existence of solutions and then proceed with the extensibility criterion.Step two: under some appropriate conditions, we prove that the solutions are uniformly bounded in time.

Global Existence of Solutions
We will first be devoted to dealing with local-in-time existence and uniqueness of a nonnegative solution for (1).The corresponding conclusions are written by Lemma 4.
We solve the parabolic equation Thanks to the essential estimations of parabolic equations and the embedding theorems, we apply the Schauder fixed point theorem to obtain the local existence of solution V(, ).The smoothness of ℎ( 1 (V)ũ 1 ,  2 (V)ũ 2 , V) ensures the uniqueness of solution V(, ).
About the above solution ( 1 ,  2 , V), we have the following extensibility criterion: The solution is extended to the interval (0,  max ), where  max has the following property: (32)

Uniform Boundedness of Solutions.
To obtain some a priori estimates, we need some technical lemmas.The following  1 -estimate of solution is first given.
Proof.Integrating the first equation of (1) over Ω, we have Using Cauchy inequality yields Integrating the third equation of ( 1) over Ω, we have This proves the lemma.Lemma 6. Letting  > 1 and under assumption (11), then the following estimates hold: where   (V) is defined by Proof.It is easy to check Then, for any  > 1, we have Clearly, if  1 ≤ 0, then (40) is a consequence of (44).We in the following verify that the result holds.
To end the arguments of uniform boundedness of solutions, it follows to prove that  * =  max and  max = +∞ by contradiction.