Attractor for a Reaction-Diffusion System Modeling Cancer Network

and Applied Analysis 3 Next, we prove ψ is a contractive mapping from S into itself forT small enough andR sufficiently large. By Lemma 4, then 󵄩󵄩󵄩󵄩ψ1 (u, V) 󵄩󵄩󵄩󵄩L2 ≤ 󵄩󵄩󵄩󵄩 e −tA u 0 (x) 󵄩󵄩󵄩󵄩L2 + δ 1 ∫ t 0 󵄩󵄩󵄩󵄩 e −(t−s)A u 󵄩󵄩󵄩󵄩L2 ds


Introduction
In this paper, we discuss the asymptoti cbehavior of solutions for reaction-diffusition equations which studied by Aguda et al. [1] and Shen et al. [2].The system describes cancer network regulated by microRNA (miRNA).MicroRNAs are an abundant class of small noncoding RNA that function to regulate the activity and stability of specific mRNA targets through posttranscriptional regulatory mechanism and play a role of repressing translation of mRNA or degrading mRNAs.Recent studies show that miRNAs play a central role in many biological (cellular) processes, including developmental timing, cell proliferation, apoptosis, metabolism, cell differentiation, somitogenesis, and tumour-genesis.In addition, there is diffusion when molecules interact (see, e.g., [2][3][4][5][6]), so we should consider the diffusion process and its dynamical behavior.
In order to understand further the miR-17-92 involved in the network with Myc and E2F, we would investigate the cancer network [1,2] with diffusion term and consider the attractor system of cancer network with diffusion as follows: and here Ω is an open bounded subset of   with the boundary of class  3 .(, V) are two real smooth nonlinear functions for , V ∈ [0, +∞) satisfying the following conditions: (i) there exists  > 0 such that (, V) ≤  1  2 +  2 V as  ≥ 0, V ≥ 0; (ii) there exists  1 ,  2 > 0 such that |   | <  1 , |  V | <  2 for all , V ∈ [0, +∞).
To our knowledge, the long time behavior of solution for reaction-diffusion system has been studied by several authors (see [7][8][9][10][11][12][13][14][15]).But for different nonlinear reaction function, there are some different extra difficulties.We will study the existence of global attractor for the system (1) in  2 (Ω) ×  2 (Ω).The key point to our method relies on the regularity and estimates on solutions which show that the solutions are uniformly bounded in  2 (Ω) ×  2 (Ω).
We construct a local solution of system (1) by the semigroup method and fixed-point theorem and then discuss its regularity by priori estimate method.We study the asymptotic behavior of solution and show the existence of global uniformly bounded solution to the system in a bounded domain Ω ⊂   .Some estimates and asymptotic compactness of the solutions are proved.As a result, we establish the existence of global attractor in  2 (Ω) ×  2 (Ω) and prove that the solution converges to stable steady states.In the end, we apply these results to the cancer network model and give the numerical test.The numerical simulation shows that the analytical results agree with numerical simulation.This paper is more motivated from the mathematical point of view than from the biological one, but it will help to get more insights in the understanding of the behavior of the problem.
Proposition 1. Suppose that  is a metric space and () ≥0 is a semigroup of continuous operators in .If () ≥0 has a bounded absorbing set and is asymptotically compact, then () ≥0 possesses a global attractor which is a compact invariant set and attracts every bounded set in .

Preliminary
Some well-known inequalities and embedding results that will be used in the sequel are presented.
Note that, for any given smooth function V,  = 0 is the subsolution of the following problem: and for any given smooth function , V = 0 is the subsolution of the following problem: By the comparison principle, for any  ∈ [0,  max ) and  ∈ Ω, (, ) ≥ 0; (, ) ≥ 0. Now, we discuss the regularity of the solution to (1).From the above analysis, (, ), V(, ) is bounded in which implies that ‖(, )‖   is bounded for any  ∈ [, ].

Global Solution and Some A Priori Estimates
In this section, the global-in-time existence of a solution to system (1) is proved.The following a priori estimates will play a crucial role in the proof of our result.
Proof.In the process of the proof, we denote any positive constant by  which may change from line to line and let  be a small enough constant.
Taking the inner product on both sides of the second equation with −ΔV, for any  2 > 0, Taking the inner product of the first equation of ( 1) with −Δ in  2 (Ω), then By Lemma 6, there exist  = 1/2,  0 = 0, and  1 = 2 such that (31) Since ‖‖  2 is bounded, for any  2 > 0, there is a constant   2 such that With the same analysis, it easy to know that there exist  3 > 0,   3 such that From the above analysis and ( 29) and (30), we choose  2 ,  3 small enough; then there exists positive constant  2 = 1/2 > 0,   2 , 3 > 0 such that By Gronwall's lemma, In this section, we denote any positive constant by  whose value may change from line to line.Equations (30), (35) and the choice of  (in Theorem 7) depending on ‖ 0 ‖  2 (Ω) + ‖V 0 ‖  2 (Ω) , then ((, ), (, )) ∈ ; it is clear by a standard argument that the solution (, ) to (1) can be extended up to some  max ≤ ∞.With the same method as in the proof of Lemma 8, for any finite  max ,      ( max ) which implies that  max = +∞.The global existence of the solution to (1) is obtained as the following theorem.

Global Attractor
The existence of a global attractor to system (1) is given in this section.
From the estimates in Lemma 8, there exists fixed constant  > 0 and Denote the set where  is the constant in (38).The results of Theorem 9 imply that the existence of a dynamical system {()} ≥0 which maps  = {(, V) : (, V) ∈  2 ×  2 ;  ≥ 0, V ≥ 0} into itself and satisfying ((), V()) = ()( 0 , V 0 ).Since  is bounded, by Lemma 8, there exists () depending only on  and |Ω| such that which implies that  is a bounded absorbing set of the semigroup {()} ≥0 .Next, by the Sobolev embedding theorem, the asymptotical compactness of the semigroup {()} ≥0 is shown and then the existence of a global attractor to system (1) is given.
Theorem 10.Assume that (, V) satisfying hypothesis.Then the problem (1) has a global attractor which is a compact invariant set and attracts every bounded set in .
From the estimates in Theorem 9, it is easy to know that the solution of system (1) is exponential decay in space  2 (Ω) ×  2 (Ω) if the forcing term  is zero.So the global attractor reduces to the single point (0,0).

Conclusions and Discussion
In this paper, by the semigroup method and fixed-point theorem, we construct a local solution of system (1) and then discuss its regularity by a priori estimate method.We also study the asymptotic behavior of solution and show the existence of global uniformly bounded solution to the system in a bounded domain Ω ⊂   .Some estimates and asymptotic compactness of the solutions are proved.To prove the compactness of the semigroup, we used the fact that Sobolev embedding is compact in bounded domains.As a result, we establish the existence of the global attractor in  2 (Ω) ×  2 (Ω) and prove that the solution converges to stable steady states if the forcing term  is zero.
The hypothesis of the nonlinear function can be more general.For the case of unbounded domains or partly dissipative system, the dynamical system (1) is not compact.Then, using the similar idea in [11], we should decompose the semigroup into two parts such that one part asymptotically tends to zero and the other part is compact.But the lack of compactness of Sobolev embedding introduces some extra difficulties.In general, the space domain should be bounded in a biological process.