A chemotaxis model with reproduction term in a bounded domain

This paper deals with global attractor of a quasilinear parabolic system introduced in [

The classical chemotaxis model has been extensively studied in the last few years (see [

In this paper, we are concerned with the following chemotaxis model:

there exists

there exists

To our knowledge it has never been analyzed whether the global attractor of system (

The existence of the global attractor for semilinear reaction diffusion equations in bounded and unbounded domains has been studied extensively [

For readers' convenience, the following standard result on attractor is first presented here (see [

Suppose that

The semigroup

We first show that there is a unique global solution to (

Some well-known inequalities and embedding results that will be used in the sequel are presented.

If

Let

Let

Let

Suppose that

Let

The local existence of a solution to system (

Assume that there exist

The proof of the lemma can be found in another paper [

Suppose

Choose

Next, we prove that

Similarly, for

By Lemma

Equations (

Equation (

Since

Now, we discuss the regularity of the solution to (

By semigroup techniques and Schauder estimates (Theorem IV. 5.1–5.3 in [

In this section, the global-in-time existence of a solution to system (

Suppose that

Integrating the first equation of (

Suppose that

In the process of the proof, we denote any positive constant by

If

Let

Using the same method as the above analysis, for any

By Lemma

From the above analysis, if

Equations (

Suppose that nonnegative functions

From the estimates in Lemma

Next, by the Sobolev embedding theorem, the asymptotical compactness of the semigroup

Assume that

If

Since the two embedding

This paper is supported by the National Natural Science Foundation of China (no. 11272277), Program for New Century Excellent Talents in University (NCET-10-0238), the Key Project of Chinese Ministry of Education (no. 211105) and Innovation Scientists and Technicians Troop Construction Projects of Henan Province (134100510013), Innovative Research Team in University of Henan Province (13IRTSTHN019), Foundation of Henan Educational Committee (no. 13A110737), and Hall of Henan Province Science and Technology Project (no. 12A110020).