Attractor for a Reaction-Diffusion System Modeling Cancer Network

Abstract

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 $\mathrm{\Omega }\subset R^n$. Some estimates and asymptotic compactness of the solutions are proved. As a result, we establish the existence of the global attractor in $L^2(\mathrm{\Omega })\times L^2(\mathrm{\Omega })$ and prove that the solution converges to stable steady states. These results can help to understand the dynamical character of cancer network and propose a new insight to study the mechanism of cancer. In the end, the numerical simulation shows that the analytical results agree with numerical simulation.