Abstract and Applied Analysis

Singularly perturbed periodic parabolic equations with alternating boundary layer type solutions

Adelaida B. Vasil'eva and Leonid V. Kalachev

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Abstract

We consider a class of singularly perturbed parabolic equations for which the degenerate equations obtained by setting the small parameter equal to zero are algebraic equations that have several roots. We study boundary layer type solutions that, as time increases, periodically go through two fairly long lasting stages with extremely fast transitions in between. During one of these stages the solution outside the boundary layer is close to one of the roots of the degenerate (reduced) equation, while during the other stage the solution is close to the other root. Such equations may be used as models for bio-switches where the transitions between various stationary states of biological systems are initiated by comparatively slow changes within the systems.

Article information

Source
Abstr. Appl. Anal., Volume 2006 (2006), Article ID 52856, 21 pages.

Dates
First available in Project Euclid: 30 January 2007

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1170202964

Digital Object Identifier
doi:10.1155/AAA/2006/52856

Mathematical Reviews number (MathSciNet)
MR2211658

Zentralblatt MATH identifier
1155.34031

Citation

Vasil'eva, Adelaida B.; Kalachev, Leonid V. Singularly perturbed periodic parabolic equations with alternating boundary layer type solutions. Abstr. Appl. Anal. 2006 (2006), Article ID 52856, 21 pages. doi:10.1155/AAA/2006/52856. https://projecteuclid.org/euclid.aaa/1170202964


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References

  • N. D. Alikakos, P. W. Bates, and X. Chen, Periodic traveling waves and locating oscillating patterns in multidimensional domains, Transactions of the American Mathematical Society 351 (1999), no. 7, 2777--2805.
  • J. Keener and J. Sneyd, Mathematical Physiology, Interdisciplinary Applied Mathematics, vol. 8, Springer, New York, 1998.
  • J. D. Murray, Mathematical Biology, 2nd ed., Biomathematics, vol. 19, Springer, New York, 1993.
  • N. N. Nefedov, An asymptotic method of differential inequalities for the investigation of periodic contrast structures: existence, asymptotics, and stability, Differential Equations 36 (2000), no. 2, 298--305.
  • A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspectives, 2nd ed., vol. 14, Springer, New York, 2001.
  • A. B. Vasil'eva, Periodic solutions to a parabolic problem with a small parameter multiplying the derivatives, Computational Mathematics and Mathematical Physics 43 (2003), no. 7, 932--943.
  • A. B. Vasil'eva, V. F. Butuzov, and L. V. Kalachev, The Boundary Function Method for Singular Perturbation Problems, SIAM Studies in Applied Mathematics, vol. 14, SIAM, Pennsylvania, 1995.
  • A. B. Vasil'eva, A. P. Petrov, and A. A. Plotnikov, On the theory of alternating contrast structures, Computational Mathematics and Mathematical Physics 38 (1998), no. 9, 1471--1480.