A reaction-diffusion cancer network regulated by microRNA is considered in this paper. We study the asymptotic behavior of solution and show the existence of global uniformly bounded solution to the system in a bounded domain

In this paper, we discuss the asymptoti cbehavior of solutions for reaction-diffusition equations which studied by Aguda et al. [

In order to understand further the miR-17-92 involved in the network with Myc and E2F, we would investigate the cancer network [

there exists

there exists

To our knowledge, the long time behavior of solution for reaction-diffusion system has been studied by several authors (see [

We construct a local solution of system (

For readers’ convenience, the following standard result on attractor is first presented here (see, e.g., [

Suppose that

The semigroup

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

If

Let

Let

Suppose

In this paper, we denote the standard Sobolev spaces by

The local existence of a solution to system (

Suppose

Choose

Next, we prove

Equation (

Note that, for any given smooth function

Now, we discuss the regularity of the solution to (

If

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

Suppose that

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

Taking the inner product of the first equation of (

Taking the inner product of the second equation of (

From the analysis in Step

Taking the inner product on both sides of the second equation with

From the above analysis and (

In this section, we denote any positive constant by

Suppose that nonnegative functions

The existence of a global attractor to system (

From the estimates in Lemma

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

Assume that

If

Since the two embedding

From the estimates in Theorem

In this paper, by the semigroup method and fixed-point theorem, we construct a local solution of system (

The hypothesis of the nonlinear function can be more general. For the case of unbounded domains or partly dissipative system, the dynamical system (

The arguments in the previous sections can be applied to more general reaction diffusion systems. As we discussed in the introduction, in order to understand further the miR-17-92 involving in the network with Myc and E2F, scientists plan to model this network with mathematical model. By using the mathematical model, the researchers can detect the key points regulating main properties of biological system and find the methods to solve the different diseases. In order to explain the cancer mechanism induced by miR-17-94, Aguda et al. [

From the estimates in Theorem

Next, we will give the numerical test to the gene network model. The gene network model is simulated numerically in two spatial dimensions. Our numerical simulations employ the zero-flux boundary conditions. We set time step and space step as 0.02 and 1 and select coefficients of diffusion

The numerical simulation shows that the behavior of the solution to system (

The spotted patterns prevail over the two-dimensional space. And [(a)–(d)] are, respectively, at time 0, 10, 500, 1000. Parameter values and initial perturbation, respectively, are

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

This work is supported by National Natural Science Foundation of China (11272277 and 11301455), Foundation of Henan Educational Committee (13A110737 and 13A110756), Program for New Century Excellent Talents in University (NCET-10-0238), the Key Project of Chinese Ministry of Education (211105), Innovation Scientists and Technicians Troop Construction Projects of Henan Province (134100510013), and Innovative Research Team in University of Henan Province (13IRTSTHN019).