Linear vs. nonlinear selection for the propagation speed of the solutions of scalar reaction-diffusion equations invading an unstable equilibrium

Communications on Pure and Applied Mathematics

Volumn 57, Issue 5, 2004, Pages 616-636

  Lucia, Marcello ^a   Muratov, Cyrill B ^b   Novaga, Matteo ^c  

^a RUTGERS UNIVERSITY   (United States)

^b NEW JERSEY INSTITUTE OF TECHNOLOGY   (United States)

^c UNIVERSITY OF PISA   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 2442629430     PISSN: 00103640     EISSN: None     Source Type: Journal    
DOI: 10.1002/cpa.20014     Document Type: Article

References (30)

1. Aronson, D. G.; Weinberger, H. F. Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. Partial differential equations and related topics (Program, Tulane Univ., New Orleans, La., 1974), 5-49. Lecture Notes in Mathematics, 446. Springer, Berlin, 1975.
2. Aronson, D. G.; Weinberger, H. F. Multidimensional nonlinear diffusion arising in population genetics. Adv. in Math. 30 (1978), no. 1, 33-76.
3. Ben-Jacob, E.; Brand, H.; Dee, G.; Kramer, L.; Langer, J. S. Pattern propagation in nonlinear dissipative systems. Phys. D 14 (1985), 348-364.
4. Benguria, R. D.; Depassier, M. C. Validity of the linear speed selection mechanism for fronts of the nonlinear diffusion equation. Phys. Rev. Lett. 73 (1994), 2272-2274.
5. Benguria, R. D.; Depassier, M. C. Speed of fronts of the reaction-diffusion equation. Phys. Rev. Lett. 77 (1996), 1171-1173.
6. Berestycki, H.; Lions, P.-L. Nonlinear scalar field equations. I. Existence of a ground state. Arch. Rational Mech. Anal. 82 (1983), no. 4, 313-345.
7. Berestycki, H.; Nirenberg, L. Travelling fronts in cylinders. Ann. Inst. H. Poincaré Anal. Non Linéaire 9 (1992), no. 5, 497-572.
8. Billingham, J. Phase plane analysis of one-dimensional reaction diffusion waves with degenerate reaction terms. Dyn. Stab. Syst. 15 (2000), no. 1, 23-33.
9. Cheney, W. Analysis for applied mathematics. Graduate Texts in Mathematics, 208. Springer, New York, 2001.
10. Cross, M. C.; Hohenberg, P. C. Pattern formation outside of equilibrium. Rev. Mod. Phys. 65 (1993), no. 3, 851-1112.
11. Dal Maso, G. An introduction to Γ-convergence. Progress in Nonlinear Differential Equations and Their Applications, 8. Birkhäuser, Boston, 1993.
12. Field, R. J.; Burger, M., eds. Oscillations and traveling waves in chemical systems. Wiley Interscience, New York, 1985.
13. Fife, P. C. Mathematical aspects of reacting and diffusing systems. Lecture Notes in Biomathematics, 28. Springer, Berlin-New York, 1979.
14. Fisher, R. A. The advance of advantageous genes. Ann. Eugenics 7 (1937), 355-369.
15. Gilbarg, D.; Trudinger, N. S. Elliptic partial differential equations of second order. Springer, Berlin, 1983.
16. Hadeler, K. P.; Rothe, F. Travelling fronts in nonlinear diffusion equations. J. Math. Biol. 2 (1975), 251-263.
17. Heinze, S. A variational approach to traveling waves. Submitted to Z. Anal. Anwendungen.
18. Kolmogorov, A. N.; Petrovskii, I. G.; Piskunov, N. S. 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.
19. Lucia, M.; Muratov, C. B.; Novaga, M. Existence of traveling wave solutions in Ginzburg-Landau-type problems in infinite cylinders. In preparation.
20. Medvedev, G. S.; Ono, K.; Holmes, P. J. Travelling wave solutions of the degenerate Kolmogorov-Petrovski-Piskunov equation. European J. Appl. Math. 14 (2003), no. 3, 343-367.
21. Merzhanov, A. G.; Rumanov, E. N. Physics of reaction waves. Rev. Mod. Phys. 71 (1999), 1173-1210.
22. Muratov, C. B. A global variational structure and propagation of disturbances in reaction-diffusion systems of gradient type. Discrete Contin. Dyn. Syst. Ser. B, to appear.
23. Murray, J. D. Mathematical biology. Biomathematics, 19. Springer, Berlin, 1989.
24. Powell, J. A., Newell, A. C.; Jones, C. K. R. T. Competition between generic and non-generic fronts in envelope equations. Phys. Rev. A 44 (1991), 3636-3652.
25. Rothe, F. Convergence to pushed fronts. Rocky Mountain J. Math. 11 (1981), no. 4, 617-633.
26. Roquejoffre, J.-M. Eventual monotonicity and convergence to traveling fronts for the solutions of parabolic equations in cylinders. Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997), no. 4, 499-552.
27. van Saarloos, W. Front propagation into unstable states: marginal stability as a dynamical mechanism for velocity selection. Phys. Rev. A (3) 37 (1988), no. 1, 211-229.
28. van Saarloos, W. Front propagation into unstable states. II. Linear versus nonlinear marginal stability and rate of convergence. Phys. Rev. A (3) 39 (1989), no. 12, 6367-6390.
29. van Saarloos, W. Front propagation into unstable states. Phys. Rep. 386 (2003), 29-222.
30. Volpert, A. I.; Volpert, V. A.; Volpert, V. A. Traveling wave solutions of parabolic systems. Translations of Mathematical Monographs, 140. American Mathematical Society, Providence, R.I. 1994.