A global variational structure and propagation of disturbances in reaction-diffusion systems of gradient type

Discrete and Continuous Dynamical Systems - Series B

Volumn 4, Issue 4, 2004, Pages 867-892

  Muratov, C B ^a  

^a NEW JERSEY INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

Long time asymptotics; Reaction diffusion systems; Traveling waves; Wave selection

Indexed keywords

EID: 4944241607     PISSN: 15313492     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcdsb.2004.4.867     Document Type: Article

References (43)

1. M. Cross, P. C. Hohenberg, Pattern formation outside of equilibrium, Rev. Mod. Phys. 65 (1993) 851-1112.
2. A. G. Merzhanov, E. N. Rumanov, Physics of reaction waves, Rev. Mod. Phys. 71 (1999) 1173-1210.
3. R. J. Field, M. Burger (Eds.), Oscillations and Traveling Waves in Chemical Systems, Wiley Interscience, New York, 1985.
4. J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1989.
5. J. D. Buckmaster and G. S. S. Ludford, Theory of Laminar Flames, Cambridge University Press, Cambridge, 1982.
6. J.-C. Toledano, P. Toledano, The Landau theory of phase transitions: application to structural, incommensurate, magnetic, and liquid crystal systems, World Scientific, Singapore, 1987.
7. L. D. Landau, E. M. Lifshits, Course of Theoretical Physics, Vol. 8, Pergamon Press, London, 1984.
8. P. C. Hohenberg, B. I. Halperin, Theory of dynamic critical phenomena, Rev. Mod. Phys. 49 (1977) 435-479.
9. A. N. Kolmogorov, I. G. Petrovsky, N. S. Piskunov, Study of the diffusion equation with growth of the quantity of matter and its application to a biological problem, Bull. Univ. Moscou, Ser. Int., Sec. A 1 (1937) 1-25.
10. Y. I. Kanel', Stabilization of solutions of the Cauchy problem for equations encountered in combustion theory, Mat. Sbornik 59 (1962) 245-288.
11. Y. I. Kanel', On the stabilization of the solutions of the equations of combusion theory with initial data of compact support, Mat. Sbornik 65 (1964) 398-413.
12. P. C. Fife, J. B. McLeod, The approach of solutions of nonlinear diffusion equations to traveling front solutions, Arch. Rat. Mech. Anal. 65 (1977) 335-361.
13. D. G. Aronson, H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in: Proceedings of the Tulane Program in Partial Differential Equations and Related Topics, Vol. 446 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1975, pp. 5-49.
14. G. Medvedev, K. Ono, P. Holmes, Traveling wave solutions of the degenerate Kolmogorov-Petrovski-Piskunov equation, Eur. J. Appl. Math. (to be published).
15. J. Billingham, Phase plane analysis of one-dimensional reaction-diffusion waves with degenerate reaction terms, Dynam. Stabil. Syst. 15 (2000) 23-33.
16. D. G. Aronson, H. F. Weinberger, Multidimensional diffusion arising in population genetics, Adv. Math. 30 (1978) 33-58.
17. R. Gardner, Existence of multidimensional traveling wave solutions of an initial-boundary value problem, J. Differ. Equations 61 (1986) 335-379.
18. J. M. Vega, Travelling wavefronts of reaction-diffusion equations in cylindrical domains, Comm. Partial Differential Equations 18 (1993) 505-531.
19. H. Berestycki, B. Larrouturou, P.-L. Lions, Multidimensional traveling wave solutions of a flame propagation model, Arch. Rat. Mech. Anal. 111 (1990) 33-49.
20. H. Berestycki, L. Nirenberg, Traveling fronts in cylinders, Ann. Inst. H. Poincaré Anal. Non Linéaire 9 (1992) 497-572.
21. V. Volpert, A. Volpert, Multidimensional traveling waves in the monostable case, C. R. Acad. Sci Paris 328 (1999) 123-128.
22. V. Volpert, A. Volpert, Existence of multidimensional traveling waves and systems of waves, Comm. Partial Differential Equations 26 (2001) 421-459.
23. S. Heinze, A variational approach to traveling waves (preprint).
24. J.-M. Roquejoffre, Convergence to traveling waves for solutions of a class of semilinear parabolic equations, J. Differ. Equations 108 (1994) 262-295.
25. J.-M. Roquejoffre, Eventual monotonicity and convergence to traveling fronts for the solutions of parabolic equations in cylinders, Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997) 499-552.
26. J.-M. Roquejoffre, D. Terman, V. A. Volpert, Global stability of traveling fronts and convergence towards stacked families of waves in monotone parabolic systems, SIAM J. Math. Anal. 27 (1996) 1261-1269.
27. W. van Saarloos, Front propagation into unstable states: marginal stability as a dynamical mechanism for velocity selection, Phys. Rev. A 37 (1988) 211-229.
28. W. van Saarloos, Front propagation into unstable states. 2: Linear versus nonlinear, marginal stability and rate of convergence, Phys. Rev. A 39 (1989) 6367-6390.
29. J. A. Powell, A. C. Newell, C. K. R. T. Jones, Competition between generic and non-generic fronts in envelope equations, Phys. Rev. A 44 (1991) 3636-3652.
30. P. K. Brazhnik, Exact solutions for the kinematic model of autowaves in two-dimensional excitable media, Physica D 94 (1996) 205-220.
31. A. Bonnet and F. Hamel, Existence of nonplanar solutions of a simple model of premixed Bunsen flames, SIAM J. Math. Anal. 31 (1999) 80-118.
32. G. J. Lord, D. Peterhof, B. Sandstede, A. Scheel, Numerical computation of solitary waves in infinite cylindrical domains, SIAM J. Numer. Anal. 37 (2000) 1420-1454.
33. H. F. Weinberger, M. A. Lewis, B. Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol. 45 (2002) 183-218.
34. P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Springer-Verlag, Berlin, 1979.
35. A. I. Volpert, V. A. Volpert, V. A. Volpert, Traveling wave solutions of parabolic systems, AMS, Providence, 1994.
36. M. B. Taylor, Partial Differential Equations III: Nonlinear Equations, Springer-Verlag, Berlin, 1996.
37. E. Risler, Invasion of a stable equilibrium for spatially extended gradient-like systems (preprint).
38. R. D. Benguria, M. C. Depassier, Speed of fronts of the reaction-diffusion equation, Phys. Rev. Lett. 77 (1996) 1171-1173.
39. S. Heinze, M. Lucia, C. B. Muratov, M. Novaga, A variational approach to the existence of traveling wave solutions in reaction-diffusion systems of gradient type (in preparation).
40. M. Lucia, C. B. Muratov, M. Novaga, Linear vs. nonlinear selection of the propagation speed in scalar reaction-diffusion equations (preprint).
41. W. Walter, Nonlinear parabolic differential equations and inequalities, Discrete Cont. Dyn. S. 8 (2002) 451-468.
42. H. P. McKean, A correction to: "Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov" (Comm. Pure Appl. Math. 28 (1975), no. 3, 323-331). Comm. Pure Appl. Math. 29 (1976) 553-554.
43. H. P. McKean, A correction to: "Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov" (Comm. Pure Appl. Math. 28 (1975), no. 3, 323-331). Comm. Pure Appl. Math. 29 (1976) 553-554.