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Attractor for a reaction-diffusion system modeling cancer network. (English) Zbl 1470.35178

Summary: 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 \(\Omega \subset \mathbb R^n\). Some estimates and asymptotic compactness of the solutions are proved. As a result, we establish the existence of the global attractor in \(L^2(\Omega) \times L^2(\Omega)\) 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.

MSC:

35K51 Initial-boundary value problems for second-order parabolic systems
35K57 Reaction-diffusion equations
35Q92 PDEs in connection with biology, chemistry and other natural sciences
92C15 Developmental biology, pattern formation
92C17 Cell movement (chemotaxis, etc.)
92D20 Protein sequences, DNA sequences
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References:

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