Reactive-Diffusive-Advective Traveling Waves in a Family of Degenerate Nonlinear Equations.
This paper deals with the analysis of existence of traveling wave solutions (TWS) for a diffusion-degenerate (atD(0) = 0) and advection-degenerate (ath'(0) = 0) reaction-diffusion-advection (RDA) equation. Diffusion is a strictly increasing function and the reaction term generalizes the kinetic part of the Fisher-KPP equation. We consider different forms of the convection termh(u): (1)h'(u) is constantk, (2)h'(u) =kuwithk> 0, and (3) it is a quite general form which guarantees the degeneracy in the advective term. In Case 1, we prove that the task can be reduced to that for the corresponding equation, wherek= 0, and then previous results reported from the authors can be extended. For the other two cases, we use both analytical and numerical tools. The analysis we carried out is based on the restatement of searching TWS for the full RDA equation into a two-dimensional dynamical problem. This consists of searching for the conditions on the parameter values for which there exist heteroclinic trajectories of the ordinary differential equations (ODE) system in the traveling wave coordinates. Throughout the paper we obtain the dynamics by using tools coming from qualitative theory of ODE.
Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México (UNAM), Circuito Exterior, Ciudad Universitaria, 04510 Ciudad de México, Mexico.
Faustino
F
Sánchez-Garduño
Institute of Computational Biology, Helmholtz Zentrum München, German Research Center for Environmental Health, Ingolstädter Landstraße 1, 85764 Neuherberg, Germany; Technical University of Munich, Zentrum Mathematik, M12, Boltzmannstraße 3, 85747 Garching, Germany.
Judith
0000-0003-0969-075X
J
Pérez-Velázquez
J Math Biol. 1988;26(3):263-98
3411255
J Math Biol. 1994;33(2):139-61
7868990