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reaction-diffusion system approximation; degenerate parabolic problem; cross-diffusion system
This paper deals with nonlinear diffusion problems involving degenerate parabolic problems, such as the Stefan problem and the porous medium equation, and cross-diffusion systems in population ecology. The degeneracy of the diffusion and the effect of cross-diffusion, that is, nonlinearities of the diffusion, complicate its analysis. In order to avoid the nonlinearities, we propose a reaction-diffusion system with solutions that approximate those of the nonlinear diffusion problems. The reaction-diffusion system includes only a simple reaction and linear diffusion. Resolving semilinear problems is typically easier than dealing with nonlinear diffusion problems. Therefore, our ideas are expected to reveal new and more effective approaches to the study of nonlinear problems.
[1] L. Chen and A. Jüngel: Analysis of a multidimensional parabolic population model with strong cross-diffusion. SIAM J. Math. Anal. 36 (2006), 301–322. MR 2083864
[2] L. Chen and A. Jüngel: Analysis of a parabolic cross-diffusion population model without self-diffusion. J. Differential Equations 224 (2006), 39–59. MR 2220063
[3] 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. MR 1687440
[4] D. Hilhorst, M. Iida, M. Mimura, and H. Ninomiya: A competition-diffusion system approximation to the classical two-phase Stefan problem. Japan J. Indust. Appl. Math. 18 (2001), 161–180. MR 1842906
[5] D. Hilhorst, M. Iida, M. Mimura, and H. Ninomiya: Relative compactness in $L^p$ of solutions of some $2m$ components competition-diffusion systems. Discrete Contin. Dyn. Syst. 21 (2008), 1, 233–244. MR 2379463
[6] D. Hilhorst, M. Mimura, and H. Ninomiya: Fast reaction limit of competition-diffusion systems. In: Handbook of Differential Equations: Evolutionary Equations Vol. 5 (C. M. Dafermos and M. Pokorny, eds.). Elsevier/North Holland, Amsterdam 2009, pp. 105–168. MR 2562164
[7] M. Iida, M. Mimura, and H. Ninomiya: Diffusion, cross-diffusion and competitive interaction. J. Math. Biol. 53 (2006), 617–641. MR 2251792
[8] T. Kadota and K. Kuto: Positive steady states for a prey-predator model with some nonlinear diffusion terms. J. Math. Anal. Appl. 323 (2006), 1387–1401. MR 2260190
[9] R. Kersner: Nonlinear heat conduction with absorption: space localization and extinction in finite time. SIAM J. Appl. Math. 43 (1983), 1274–1285. MR 0722941 | Zbl 0536.35039
[10] P. Knabner: Mathematische Modelle für Transport und Sorption gelöster Stoffe in porösen Medien. Verlag Peter Lang, Frankfurt 1991. MR 1218175 | Zbl 0731.35054
[11] H. Murakawa: On reaction-diffusion system approximations to the classical Stefan problems. In: Proc. Czech–Japanese Seminar in Applied Mathematics 2005 (M. Beneš, M. Kimura and T. Nakaki, eds.), COE Lecture Note Vol. 3, Faculty of Mathematics, Kyushu University 2006, pp. 117–125. MR 2279052 | Zbl 1144.80365
[12] H. Murakawa: Reaction-diffusion system approximation to degenerate parabolic systems. Nonlinearity 20 (2007), 2319–2332. MR 2356112
[13] H. Murakawa: A regularization of a reaction-diffusion system approximation to the two-phase Stefan problem. Nonlinear Anal. 69 (2008), 3512–3524. MR 2450556 | Zbl 1158.35379
[14] H. Murakawa: A Solution of Nonlinear Degenerate Parabolic Problems by Semilinear Reaction-Diffusion Systems. Ph.D. Thesis, Graduate School of Mathematics, Kyushu University, 2008.
[15] H. Murakawa: Discrete-time approximation to nonlinear degenerate parabolic problems using a semilinear reaction-diffusion system. Preprint.
[16] H. Murakawa: A relation between cross-diffusion and reaction-diffusion. Preprint.
[17] R. H. Nochetto: A note on the approximation of free boundaries by finite element methods. RAIRO Modél. Math. Anal. Numér. 20 (1986), 355–368. MR 0852686 | Zbl 0596.65092
[18] A. Okubo and S. A. Levin: Diffusion and Ecological Problems: Modern Perspectives. Second edition. (Interdisciplinary Applied Mathematics 14.) Springer–Verlag, New York 2001. MR 1895041
[19] N. Shigesada, K. Kawasaki, and E. Teramoto: Spatial segregation of interacting species. J. Theor. Biol. 79 (1979), 83–99. MR 0540951
[20] K. Tomoeda: Support re-splitting phenomena caused by an interaction between diffusion and absorption. Proc. Equadiff–11 2005, pp. 499–506.
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