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In this paper, we would consider the dynamical behaviors of the chemical model represented by Satnoianu et al. (2001). Using the Kuratowski measure of noncompactness method, we prove the existence of global attractor for the weak solution semiflow of system. Finally, several numerical experiments confirm the theoretical results.

Satnoianu et al. [

This leads to the dimensionless reactor model

Here,

Satnoianu et al. discussed that diffusion-distributed structures (FDS) are predicted in a wider domain and are more robust than the classical Turing instability patterns. FDS also represent a natural extension of flow-distributed oscillations. Nonlinear bifurcation analysis and numerical simulations in one-dimensional spatial domains show that FDS have much richer solution behavior than Turing structures. Tang and Wang [

For the infinite-dimensional dynamical systems about the chemical model You [

In this paper we will discuss the long-time behavior of the solutions to (

Define the product Hilbert spaces as follows:

By the poincaré inequality and the homogenous Dirichlet boundary condition there is a constant

Using the analytic semigroup generation theorem and Lumer-Phillips theorem, it is easy to check that the densely sectorial operator

By the fact that

A function

By the Galerkin method, analogous to the arguments in [

For any initial

In order to prove the existence of global attractor, we will investigate the absorbing property of the solution semiflow of problem (

Assume

Taking the inner-product of the first equation of (

In this section we will prove that the solution semiflow

Let

There is a constant

From (

Recall the definition of the Kuratowski measure of noncompactness for bounded sets in a Banach space

A semiflow

Let

Then

Let

There exists a bounded absorbing set

For any

For any

Since

Let

Taking the inner-product (

Inequality (

For any

Taking the inner-product (

Let

Taking the inner-product (

If

By Lemma

In this section, using the finite difference method to (

Now we set the parameters

Numerical simulation for (

Numerical simulation for (

In this paper, it can be seen from the numerical results that the global attractor

The authors declare that there is no conflict of interests.

This work is supported by National Natural Science Foundation of China (11272277, 11471129), Innovation Scientists and Technicians Troop Construction Projects of Henan Province (134100510013), Innovative Research Team in University of Henan Province (13IRTSTHN019), and Key Project of the Education Department Henan Province (15A110043).