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).