×

New travelling wave solutions for the Fisher-KPP equation with general exponents. (English) Zbl 1082.35512

Summary: The Fisher-KPP equation with general nonlinear diffusion and arbitrary kinetic orders in the reaction terms is considered. The existence of oscillatory travelling wave solutions is proved for this model. Conditions for the existence of such solutions are provided.

MSC:

35K65 Degenerate parabolic equations
35K55 Nonlinear parabolic equations
35K15 Initial value problems for second-order parabolic equations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Fisher, R. A., The wave of advance of advantageous genes, Ann. Eugenics, 7, 355-369 (1937) · JFM 63.1111.04
[2] Kolmogoroff, A.; Petrovsky, I.; Piscounoff, N., Étude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un probleme biologique, Bull. Univ. Moscou, Ser. Internat. Sec. Math., 1, 1-25 (1937) · Zbl 0018.32106
[3] Aronson, D. G.; Weinberger, H. F., Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, (Partial Differential Equations and Related Topics. Partial Differential Equations and Related Topics, Lecture Notes in Mathematics, vol. 446 (1975), Springer-Verlag: Springer-Verlag Berlin), 5-49 · Zbl 0325.35050
[4] B.H. Gilding, R. Kersner, Travelling waves in nonlinear diffusion-convection-reaction (Memorandum No. 1585), Faculty of Mathematical Sciences, University of Twente, 2001; B.H. Gilding, R. Kersner, Travelling waves in nonlinear diffusion-convection-reaction (Memorandum No. 1585), Faculty of Mathematical Sciences, University of Twente, 2001
[5] Biro, Z., Attractors in a density-dependent diffusion-reaction model, Nonlinear Anal., 29, 485-499 (1997) · Zbl 0881.35060
[6] de Pablo, A.; Vázquez, J. L., Travelling waves and finite propagation in a reaction-diffusion equation, J. Differential Equations, 93, 19-61 (1991) · Zbl 0784.35045
[7] Malaguti, L.; Marcelli, C., Sharp profiles in degenerate and doubly degenerate Fisher-KPP equations, J. Differential Equations, 195, 471-496 (2003) · Zbl 1042.34056
[8] Sánchez-Garduño, F.; Maini, P. K., Travelling wave phenomena in some degenerate reaction-diffusion equations, J. Differential Equations, 117, 281-319 (1995) · Zbl 0821.35085
[9] Voit, E. O., Computational Analysis of Biochemical Systems: A Practical Guide for Biochemists and Molecular Biologists (2000), Cambridge University Press: Cambridge University Press Cambridge, UK
[10] Torres, N. V.; Voit, E. O., Pathway Analysis and Optimization in Metabolic Engineering (2002), Cambridge University Press: Cambridge University Press Cambridge, UK
[11] Murray, J. D., Mathematical Biology (1993), Springer-Verlag: Springer-Verlag New York · Zbl 0779.92001
[12] de Pablo, A.; Sánchez, A., Global travelling waves in reaction-convection-diffusion equations, J. Differential Equations, 165, 377-413 (2000) · Zbl 0961.34011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.