Abstract
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.
Highlights
The strong effect produced by the addition of the nonlinear convective term kuux on the solutions behavior of the classical Fisher-KPP equation ut = uxx + u(1 − u) is well documented in the literature
In the above equation the term ku is called the advection “speed.” It has been proved that the previous equation has monotonic decreasing traveling wave solutions (TWS) u(x, t) = φ(x − ct) ≡ φ(ξ), where c is the speed of the wave, satisfying the boundary conditions limξ→−∞φ(ξ) = 1, limξ→+∞φ(ξ) = 0 with 0 < φ(ξ) < 1, ∀ξ ∈ (−∞, +∞), if and only if c ≥ c(k), where
Our analysis is based on the assumption that to look for TWS in a functional space is equivalent to search the set of parameters for which a two-dimensional system of ordinary differential equations (ODE) possesses heteroclinic trajectories
Summary
Our analysis is based on the assumption that to look for TWS in a functional space is equivalent to search the set of parameters (in which the speed c is included) for which a two-dimensional system of ODE possesses heteroclinic trajectories This system comes from the restatement of the original problem into the appropriate traveling wave variable. According to the conditions of interest, the problem of showing the existence of the TWS satisfying such conditions in the full nonlinear PDE transforms into a dynamical systems problem This is searching for the existence of the parameter values for which there exist heteroclinic trajectories of (18) connecting P1 with P0 or with Pc. The analysis is conducted by stages.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
