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 (at D(0)=0) and advection-degenerate (at h′(0)=0) reaction-diffusion-advection (RDA) equation. Diffusion is a strictly increasing function and the reaction term generalizes the kinetic part o...
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| Format: | Article |
| Language: | English |
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Wiley
2016-01-01
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| Series: | The Scientific World Journal |
| Online Access: | http://dx.doi.org/10.1155/2016/5620839 |
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| author | Faustino Sánchez-Garduño Judith Pérez-Velázquez |
| author_facet | Faustino Sánchez-Garduño Judith Pérez-Velázquez |
| author_sort | Faustino Sánchez-Garduño |
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| description | This paper deals with the analysis of existence of traveling wave solutions (TWS) for a diffusion-degenerate (at D(0)=0) and advection-degenerate (at h′(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 term h(u): (1) h′(u) is constant k, (2) h′(u)=ku with k>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, where k=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. |
| format | Article |
| id | doaj-art-9f5e081bccb04029ba736fbf8e1cb41c |
| institution | OA Journals |
| issn | 2356-6140 1537-744X |
| language | English |
| publishDate | 2016-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | The Scientific World Journal |
| spelling | doaj-art-9f5e081bccb04029ba736fbf8e1cb41c2025-08-20T02:20:30ZengWileyThe Scientific World Journal2356-61401537-744X2016-01-01201610.1155/2016/56208395620839Reactive-Diffusive-Advective Traveling Waves in a Family of Degenerate Nonlinear EquationsFaustino Sánchez-Garduño0Judith Pérez-Velázquez1Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México (UNAM), Circuito Exterior, Ciudad Universitaria, 04510 Ciudad de México, MexicoInstitute of Computational Biology, Helmholtz Zentrum München, German Research Center for Environmental Health, Ingolstädter Landstraße 1, 85764 Neuherberg, GermanyThis paper deals with the analysis of existence of traveling wave solutions (TWS) for a diffusion-degenerate (at D(0)=0) and advection-degenerate (at h′(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 term h(u): (1) h′(u) is constant k, (2) h′(u)=ku with k>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, where k=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.http://dx.doi.org/10.1155/2016/5620839 |
| spellingShingle | Faustino Sánchez-Garduño Judith Pérez-Velázquez Reactive-Diffusive-Advective Traveling Waves in a Family of Degenerate Nonlinear Equations The Scientific World Journal |
| title | Reactive-Diffusive-Advective Traveling Waves in a Family of Degenerate Nonlinear Equations |
| title_full | Reactive-Diffusive-Advective Traveling Waves in a Family of Degenerate Nonlinear Equations |
| title_fullStr | Reactive-Diffusive-Advective Traveling Waves in a Family of Degenerate Nonlinear Equations |
| title_full_unstemmed | Reactive-Diffusive-Advective Traveling Waves in a Family of Degenerate Nonlinear Equations |
| title_short | Reactive-Diffusive-Advective Traveling Waves in a Family of Degenerate Nonlinear Equations |
| title_sort | reactive diffusive advective traveling waves in a family of degenerate nonlinear equations |
| url | http://dx.doi.org/10.1155/2016/5620839 |
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