A solution of nonlinear diffusion problems by semilinear reaction-diffusion systems

Hideki Murakawa

Kybernetika (2009)

  • Volume: 45, Issue: 4, page 580-590
  • ISSN: 0023-5954

Abstract

top
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.

How to cite

top

Murakawa, Hideki. "A solution of nonlinear diffusion problems by semilinear reaction-diffusion systems." Kybernetika 45.4 (2009): 580-590. <http://eudml.org/doc/37722>.

@article{Murakawa2009,
abstract = {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.},
author = {Murakawa, Hideki},
journal = {Kybernetika},
keywords = {reaction-diffusion system approximation; degenerate parabolic problem; cross-diffusion system; cross-diffusion system; Stefan problem; porous medium equation},
language = {eng},
number = {4},
pages = {580-590},
publisher = {Institute of Information Theory and Automation AS CR},
title = {A solution of nonlinear diffusion problems by semilinear reaction-diffusion systems},
url = {http://eudml.org/doc/37722},
volume = {45},
year = {2009},
}

TY - JOUR
AU - Murakawa, Hideki
TI - A solution of nonlinear diffusion problems by semilinear reaction-diffusion systems
JO - Kybernetika
PY - 2009
PB - Institute of Information Theory and Automation AS CR
VL - 45
IS - 4
SP - 580
EP - 590
AB - 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.
LA - eng
KW - reaction-diffusion system approximation; degenerate parabolic problem; cross-diffusion system; cross-diffusion system; Stefan problem; porous medium equation
UR - http://eudml.org/doc/37722
ER -

References

top
  1. Analysis of a multidimensional parabolic population model with strong cross-diffusion, SIAM J. Math. Anal. 36 (2006), 301–322. MR2083864
  2. Analysis of a parabolic cross-diffusion population model without self-diffusion, J. Differential Equations 224 (2006), 39–59. MR2220063
  3. Spatial segregation limit of a competition-diffusion system, European J. Appl. Math. 10 (1999), 97–115. MR1687440
  4. A competition-diffusion system approximation to the classical two-phase Stefan problem, Japan J. Indust. Appl. Math. 18 (2001), 161–180. MR1842906
  5. Relative compactness in L p of solutions of some 2 m components competition-diffusion systems, Discrete Contin. Dyn. Syst. 21 (2008), 1, 233–244. MR2379463
  6. 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. MR2562164
  7. Diffusion, cross-diffusion and competitive interaction, J. Math. Biol. 53 (2006), 617–641. MR2251792
  8. Positive steady states for a prey-predator model with some nonlinear diffusion terms, J. Math. Anal. Appl. 323 (2006), 1387–1401. MR2260190
  9. Nonlinear heat conduction with absorption: space localization and extinction in finite time, SIAM J. Appl. Math. 43 (1983), 1274–1285. Zbl0536.35039MR0722941
  10. Mathematische Modelle für Transport und Sorption gelöster Stoffe in porösen Medien, Verlag Peter Lang, Frankfurt 1991. Zbl0731.35054MR1218175
  11. 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. Zbl1144.80365MR2279052
  12. Reaction-diffusion system approximation to degenerate parabolic systems, Nonlinearity 20 (2007), 2319–2332. MR2356112
  13. A regularization of a reaction-diffusion system approximation to the two-phase Stefan problem, Nonlinear Anal. 69 (2008), 3512–3524. Zbl1158.35379MR2450556
  14. A Solution of Nonlinear Degenerate Parabolic Problems by Semilinear Reaction-Diffusion Systems, Ph.D. Thesis, Graduate School of Mathematics, Kyushu University, 2008. 
  15. Discrete-time approximation to nonlinear degenerate parabolic problems using a semilinear reaction-diffusion system, Preprint. 
  16. A relation between cross-diffusion and reaction-diffusion, Preprint. 
  17. A note on the approximation of free boundaries by finite element methods, RAIRO Modél. Math. Anal. Numér. 20 (1986), 355–368. Zbl0596.65092MR0852686
  18. Diffusion and Ecological Problems: Modern Perspectives, Second edition. (Interdisciplinary Applied Mathematics 14.) Springer–Verlag, New York 2001. MR1895041
  19. Spatial segregation of interacting species, J. Theor. Biol. 79 (1979), 83–99. MR0540951
  20. Support re-splitting phenomena caused by an interaction between diffusion and absorption, Proc. Equadiff–11 2005, pp. 499–506. 

NotesEmbed ?

top

You must be logged in to post comments.