Rad HAZU, Matematičke znanosti, Vol. 21 (2017), 99-116.

SHADOW LIMIT FOR PARABOLIC-ODE SYSTEMS THROUGH A CUT-OFF ARGUMENT

Anna Marciniak-Czochra and Andro Mikelić

Univ Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, 43 blvd. du 11 novembre 1918, F-69622 Villeurbanne cedex, France
e-mail: andro.mikelic@univ-lyon1.fr

Abstract.   We study a shadow limit (the infinite diffusion coefficient-limit) of a system of ODEs coupled with a diagonal system of semilinear heat equations in a bounded domain with homogeneous Neumann boundary conditions. The recent convergence proof by the energy approach from [19], developed for the case of a single PDE, is revisited and generalized to the case of the coupled system. Furthermore, we give a new convergence proof relying on the introduction of a well-prepared related cut-off system and on a construction of the barrier functions and comparison test functions, new in the literature. It leads to the L^∞-estimates proportional to the inverse of the diffusion coefficient.

2010 Mathematics Subject Classification.   35B20, 34E13, 35B25, 35B41, 35K57.

Key words and phrases.   Shadow limit, reaction-diffusion equations, model reduction.

DOI: https://doi.org/10.21857/ydkx2c3rp9

References:

1. A. Bobrowski, Singular perturbations involving fast diffusion, J. Math. Anal. Appl. 427 (2015), 1004-1026.
   MathSciNet     CrossRef

2. J. Carr, Applications of Centre Manifold Theory, Springer-Verlag, New York, 1981.
   MathSciNet

3. L. Y. Chen, N. Goldenfeld and Y. Oono, Renormalization group theory for global asymptotic analysis, Phys. Rev. Lett. 73 (1994), 1311-1315.
   MathSciNet     CrossRef

4. H. Chiba, C¹ approximation of vector fields based on the renormalization group method, SIAM J. Appl. Dyn. Syst. 7 (2008), 895-932.
   MathSciNet     CrossRef

5. P. Gray and S. K. Scott, Autocatalytic reactions in the isothermal continuous stirred tank reactor: isolas and other forms of multistability, Chem. Eng. Sci. 38 (1983), 29-43.
   CrossRef

6. R. E. Lee DeVille, A. Harkin, M. Holzer, K. Josić and T. Kaper, Analysis of a renormalization group method and normal form theory for perturbed ordinary differential equations, Phys. D 237 (2008), 1029-1052.
   MathSciNet     CrossRef

7. A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik 12 (1972), 30-39.
   CrossRef

8. J. K. Hale and K. Sakamoto, Shadow systems and attractors in reaction-diffusion equations, Appl. Anal. 32 (1989), 287-303.
   MathSciNet     CrossRef

9. D. Henry, Geometric Theory of semilinear Parabolic Equations, Lecture Notes in Math. 840, Springer-Verlag, 1981.
   MathSciNet

10. S. Hock, Y. Ng, J. Hasenauer, D. Wittmann, D. Lutter, D. Trümbach, W. Wurst, N. Prakash and F. J. Theis, Sharpening of expression domains induced by transcription and microRNA regulation within a spatio-temporal model of mid-hindbrain boundary formation, BMC Syst Biol 7 (2013), 48.
    CrossRef

11. J. P. Keener, Activators and inhibitors in pattern formation, Studies in Applied Mathematics 59 (1978), 1-23.
    MathSciNet     CrossRef

12. V. Klika, R. E. Baker, D. Headon and E. A. Gaffney, The influence of receptor-mediated interactions on reaction-diffusion mechanisms of cellular self-organization, Bull. Math. Biol. 74 (2012), 935-957.
    MathSciNet     CrossRef

13. O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, Translations of Mathematical Monographs Reprint Vol. 23, American Mathematical Society, Providence, RI, 1968.
    MathSciNet

14. I. Lengyel and I. R. Epstein, A chemical approach to designing Turing patterns in reaction-diffusion systems, Proc. Natl. Acad. Sci. USA 89 (1992), 3977-3979.
    CrossRef

15. A. Marciniak-Czochra, S. Härting, G. Karch and K. Suzuki, Dynamical spike solutions in a nonlocal model of pattern formation, arXiv:1307.6236 (2013).

16. A. Marciniak-Czochra, G. Karch and K. Suzuki, Unstable patterns in reaction-diffusion model of early carcinogenesis, J. Math. Pures Appl. 99 (2013), 509-543.
    MathSciNet     CrossRef

17. A. Marciniak-Czochra and M. Kimmel, Modelling of early lung cancer progression: influence of growth factor production and cooperation between partially transformed cells, Math. Models Methods Appl. Sci. 17 (2007), suppl., 1693-1719.
    MathSciNet     CrossRef

18. A. Marciniak-Czochra and M. Kimmel, Reaction-diffusion model of early carcinogenesis: the effects of influx of mutated cells, Math. Model. Nat. Phenom. 3 (2008), 90-114.
    MathSciNet     CrossRef

19. A. Marciniak-Czochra and A. Mikelić, Shadow limits via the renormalization group method and the center manifold method, Vietnam J. Math. 45 (2017), 103-125.
    MathSciNet     CrossRef

20. J. D. Murray, Mathematical biology II. Spatial models and biomedical applications. 3rd edition, Interdisciplinary Applied Mathematics 18, Springer-Verlag, New York, 2003.
    MathSciNet

21. F. Rothe, Global Solutions of Reaction-Diffusion Systems, Lecture notes in mathematics 1072, Springer-Verlag, Berlin, 1984.
    MathSciNet     CrossRef