## Abstract and Applied Analysis

### 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 ${L}^{2}(\mathrm{\Omega })×{L}^{2}(\mathrm{\Omega })$ 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.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 420386, 8 pages.

Dates
First available in Project Euclid: 6 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412605857

Digital Object Identifier
doi:10.1155/2014/420386

Mathematical Reviews number (MathSciNet)
MR3198189

Zentralblatt MATH identifier
07022360

#### Citation

Chen, Xueyong; Shen, Jianwei; Zhou, Hongxian. Attractor for a Reaction-Diffusion System Modeling Cancer Network. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 420386, 8 pages. doi:10.1155/2014/420386. https://projecteuclid.org/euclid.aaa/1412605857

#### References

• B. D. Aguda, Y. Kim, M. G. Piper-Hunter, A. Friedman, and C. B. Marsh, “MicroRNA regulation of a cancer network: consequences of the feedback loops involving miR-17-92, E2F, and Myc,” Proceedings of the National Academy of Sciences of the United States of America, vol. 105, no. 50, pp. 19678–19683, 2008.
• J. Shen, L. Chen, and K. Aihara, “Self-induced stochastic resonance in a cancer network of microRNA regulation,” in Lecture Notes in Operations Research, vol. 13, pp. 251–257, 2010.
• V. Ambros, “The functions of animal microRNAs,” Nature, vol. 431, no. 7006, pp. 350–355, 2004.
• J. D. Murray, Mathematical Biology. I: An Introduction, vol. 17 of Interdisciplinary Applied Mathematics, Springer, 3rd edition, 2002.
• S. Kondo and T. Miura, “Reaction-diffusion model as a framework for understanding biological pattern formation,” Science, vol. 329, no. 5999, pp. 1616–1620, 2010.
• A. L. Gartel and E. S. Kandel, “miRNAs: little known mediators of oncogenesis,” Seminars in Cancer Biology, vol. 18, no. 2, pp. 103–110, 2008.
• X. Chen and W. Liu, “Global attractor for a density-dependent sensitivity chemotaxis model,” Acta Mathematica Scientia B, vol. 32, no. 4, pp. 1365–1375, 2012.
• R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, vol. 68, Springer, New York, NY, USA, 2nd edition, 1997.
• A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, NY, USA, 1969.
• A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44, Springer, New York, NY, USA, 1983.
• A. Rodriguez-Bernal and B. Wang, “Attractors for partly dissipative reaction diffusion systems in ${R}^{n}$,” Journal of Mathematical Analysis and Applications, vol. 252, no. 2, pp. 790–803, 2000.
• K. Osaki, T. Tsujikawa, A. Yagi, and M. Mimura, “Exponential attractor for a chemotaxis-growth system of equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 51, pp. 119–144, 2002.
• A. V. Babin and M. I. Vishik, “Attractors of partial differential evolution equations in an unbounded domain,” Proceedings of the Royal Society of Edinburgh A: Mathematics, vol. 116, no. 3-4, pp. 221–243, 1990.
• X. Chen and J. Shen, “Global attractor for a chemotaxis model with reaction term,” Journal of Applied Mathematics, vol. 2013, Article ID 536381, 8 pages, 2013.
• L. E. Payne and B. Straughan, “Decay for a Keller-Segel chemotaxis model,” Studies in Applied Mathematics, vol. 123, no. 4, pp. 337–360, 2009. \endinput