Abstract and Applied Analysis

Regularity of Global Attractor for the Reaction-Diffusion Equation

Hong Luo

Full-text: Open access

Abstract

By using an iteration procedure, regularity estimates for the linear semigroups, and a classical existence theorem of global attractor, we prove that the reaction-diffusion equation possesses a global attractor in Sobolev space H k for all k > 0 , which attracts any bounded subset of H k ( ) in the H k -norm.

Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 917190, 16 pages.

Dates
First available in Project Euclid: 14 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1355495777

Digital Object Identifier
doi:10.1155/2012/917190

Mathematical Reviews number (MathSciNet)
MR2947748

Zentralblatt MATH identifier
1246.35106

Citation

Luo, Hong. Regularity of Global Attractor for the Reaction-Diffusion Equation. Abstr. Appl. Anal. 2012 (2012), Article ID 917190, 16 pages. doi:10.1155/2012/917190. https://projecteuclid.org/euclid.aaa/1355495777


Export citation

References

  • A. M. Turing, “The chemical basis of morphogenesis,” Philosophical Transactions of the Royal Society B, vol. 237, pp. 37–72, 1952.
  • L. G. Harrison, Kinetic Theory of Living Pattern, Cambridge University Press, 1993.
  • H. Meinhardt, Models of Biological Pattern Formation, Academic Press, 1982.
  • J. D. Murray, Mathematical Biology, vol. 19 of Biomathematics, Springer, Berlin, Germany, 1989.
  • P. De Kepper, V. Castets, E. Dulos, and J. Boissonade, “Turing-type chemical patterns in the chlorite-iodide-malonic acid reaction,” Physica D, vol. 49, no. 1-2, pp. 161–169, 1991.
  • K. R. Schneider, “A note on the existence of periodic travelling wave solutions with large periods in generalized reaction-diffusion systems,” Journal of Applied Mathematics and Physics, vol. 34, no. 2, pp. 236–240, 1983.
  • S. J. Ruuth, “Implicit-explicit methods for reaction-diffusion problems in pattern formation,” Journal of Mathematical Biology, vol. 34, no. 2, pp. 148–176, 1995.
  • Y. Liu, Z. Li, and Q. Ye, “The existence, uniqueness and stability of positive periodic solution for periodic reaction-diffusion system,” Acta Mathematicae Applicatae Sinica, vol. 17, no. 1, pp. 1–13, 2001.
  • T. Miura and P. K. Maini, “Speed of pattern appearance in reaction-diffusion models: implications in the pattern formation of limb bud mesenchyme cells,” Bulletin of Mathematical Biology, vol. 66, no. 4, pp. 627–649, 2004.
  • M. Marion, “Approximate inertial manifolds for reaction-diffusion equations in high space dimension,” Journal of Dynamics and Differential Equations, vol. 1, no. 3, pp. 245–267, 1989.
  • J. Mallet-Paret and G. R. Sell, “Inertial manifolds for reaction diffusion equations in higher space dimensions,” Journal of the American Mathematical Society, vol. 1, no. 4, pp. 805–866, 1988.
  • J. K. Hale, Asymptotic Behavior of Dissipative Systems, vol. 25 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, USA, 1988.
  • S. Lu, H. Wu, and C. Zhong, “Attractors for nonautonomous 2D Navier-Stokes equations with normal external forces,” Discrete and Continuous Dynamical Systems. Series A, vol. 13, no. 3, pp. 701–719, 2005.
  • Q. Ma, S. Wang, and C. Zhong, “Necessary and sufficient conditions for the existence of global attractors for semigroups and applications,” Indiana University Mathematics Journal, vol. 51, no. 6, pp. 1541–1559, 2002.
  • C. Zhong, C. Sun, and M. Niu, “On the existence of global attractor for a class of infinite dimensional dissipative nonlinear dynamical systems,” Chinese Annals of Mathematics. Series B, vol. 26, no. 3, pp. 393–400, 2005.
  • R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, vol. 68 of Applied Mathematical Sciences, Springer, New York, NY, USA, 2nd edition, 1997.
  • C. K. Zhong, M.-H. Yang, and C.-Y. Sun, “The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations,” Journal of Differential Equations, vol. 223, no. 2, pp. 367–399, 2006.
  • M. Marion, “Attractors for reaction-diffusion equations: existence and estimate of their dimension,” Applicable Analysis, vol. 25, no. 1-2, pp. 101–147, 1987.
  • M. Marion, “Finite-dimensional attractors associated with partly dissipative reaction-diffusion systems,” SIAM Journal on Mathematical Analysis, vol. 20, no. 4, pp. 816–844, 1989.
  • T. Ma and S. H. Wang, Bifurcation Theory and Applications, vol. 53 of World Scientific Series on Nonlinear Science, Series A, World Scientific, Singapore, 2005.
  • T. Ma and S. H. Wang, Phase Transition Dynamics in Nonlinear Sciences, Springer, NewYork, NY, USA, 2012.
  • T. Ma and S. H. Wang, Stability and Bifurcation of Nonlinear Evolution Equations, Academic Press, China, 2006.
  • B. Nicolaenko, B. Scheurer, and R. Temam, “Some global dynamical properties of a class of pattern formation equations,” Communications in Partial Differential Equations, vol. 14, no. 2, pp. 245–297, 1989.
  • L. Song, Y. Zhang, and T. Ma, “Global attractor of the Cahn-Hilliard equation in ${H}^{k}$ spaces,” Journal of Mathematical Analysis and Applications, vol. 355, no. 1, pp. 53–62, 2009.
  • A. Pazy, Semigroups of Linear Pperators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1983.