Abstract
A new type of competition-diffusion system with a small parameter is proposed. By singular limit analysis, it is shown that any solution of this system converges to the weak solution of the two-phase Stefan problem with reaction terms. This result exhibits the relation between an ecological population model and water-ice solidification problems.
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References
R.A. Adams, Sobolev Spaces. Academic Press, 1978.
D. Bothe, The instantaneous limit of a reaction-diffusion system. Preprint.
H. Brezis, Analyse Fonctionnelle. Masson, 1983.
E.N. Dancer, D. Hilhorst, M. Mimura and L.A. Peletier, Spatial segregation limit of a competition-diffusion system. European J. Appl. Math.,10 (1999), 97–115.
E. DiBenedetto, Continuity of weak solutions to a general porous medium equation. Indiana Univ. Math. J.,32 (1983), 83–118.
R. Eymard, D. Hilhorst, R. van der Hout and L.A. Peletier, A reaction-diffusion system approximation of a one-phase Stefan problem. To appear in the book in the honor of Professor Bensoussan’s sixtieth birthday, 2000.
K.P. Hadeler, Travelling fronts and free boundary value problems. Numerical Treatment of Free Boundary Problems (eds. J. Albrecht, L. Collatz and K.H. Hoffmann),8, 1981, 90–107.
D. Hilhorst, M. Mimura and R. Schätzle, Vanishing latent heat limit in a Stefan-like problem in biology. Manuscript.
D. Hilhorst, R.van der Hout and L.A. Peletier, The fast reaction limit for a reaction-diffusion system. J. Math. Anal. Appl.,199 (1996), 349–373.
D. Hilhorst, R.van der Hout and L.A. Peletier, Diffusion in the presence of fast reaction: the case of a general monotone reaction term. J. Math. Sci. Univ. Tokyo,4 (1997), 469–517.
D. Hilhorst, R.van der Hout and L.A. Peletier, Nonlinear diffusion in the presence of fast reaction. Nonlinear Anal. TMA,41 (2000), 803–823.
H. Kawarada and M. Natori, Numerical solutions of a two-phase Stefan problem. Theoretical and Applied Mechanics,23 (1975), 511–516.
M. Mimura, Y. Yamada and S. Yotsutani, A free boundary problem in ecology. Japan J. Appl. Math.,2 (1985), 151–186.
M. Mimura, Y. Yamada and S. Yotsutani, Stability analysis for free boundary problems in ecology. Hiroshima Math. J.,16 (1986), 477–498.
M. Mimura and T. Tujikawa, Aggregating pattern dynamics in a chemotaxis-model including growth. Physica A,230 (1996), 499–543.
H. Tanabe, Functional Analytic Methods for Partial Differential Equations. Dekker, 1997.
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Hilhorst, D., Iida, M., Mimura, M. et al. A competition-diffusion system approximation to the classical two-phase Stefan problem. Japan J. Indust. Appl. Math. 18, 161–180 (2001). https://doi.org/10.1007/BF03168569
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DOI: https://doi.org/10.1007/BF03168569