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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. Our main results are based on the method 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.

Chemotaxis is known as the active orientation of moving organisms along the chemical gradient. It is observed in many natural systems. For example,

In the absence of logistic source (i.e.,

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:

In this paper, we consider the following attraction-repulsion chemotaxis system including three parabolic equations:

In particular, the global existence of solutions was obtained by increasing the diffusion coefficient

In this paper, we will prove the existence of global classical solutions to the generalized system (

Let

Assume that functions

Let

Let

Throughout this paper, we introduce some notations.

The rest of this paper is organized as follows. In Section

To prove Theorem

Let assumptions (A1)–(A4) hold. Then

There is a positive constant

There is a global classical solution of system (

Let

where

Next we rewrite the first equation of (

The proof of (2) in Theorem

Next, we recall the Gagliardo-Nirenberg inequality for functions satisfying the boundary condition for

Let

Let the conditions in Lemma

Recalling

Let a nonnegative numerical sequence

By the definition of the sequence

Let conditions (A1)–(A4) hold. Then there exists a unique global solution

After appropriate scaling, the system is limited in a region

Suppose

We take

Combing with the assumption of Lemma

For convenience, we define

In this section, we will complete the proof of Theorem

Suppose there exists a

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors are very grateful to Professor Yong Li for providing valuable advice. Moreover, they very thankful to the anonymous referees for the careful reading and various corrections which greatly improved the exposition of the paper. This work partially supported by the NSF of China (under Grants nos. 11171350 and 11204019), the Scientific and Technological Research Project of Jilin Province’s Education Department (nos. 2013287 and 2014312), the Youth Project of Jilin Province’s Science and Technology Department (nos. 20130522099JH and 201201140), and the Twelfth Five-Year Plan Project of Jilin Province’s Educational Science (no. ZD2014078).