Abstract and Applied Analysis

Attractor for a Reaction-Diffusion System Modeling Cancer Network

Xueyong Chen, Jianwei Shen, and Hongxian Zhou

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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 ΩRn. Some estimates and asymptotic compactness of the solutions are proved. As a result, we establish the existence of the global attractor in L2(Ω)×L2(Ω) 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

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

First available in Project Euclid: 6 October 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


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

Export citation


  • 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