Travelling waves in nonlinear diffusion-convection-reaction (original) (raw)
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Traveling wavefront solutions to nonlinear reaction-diffusion-convection equations
Journal of Physics A: Mathematical and Theoretical, 2017
Physically motivated modified Fisher equations are studied in which nonlinear convection and nonlinear diffusion is allowed for besides the usual growth and spread of a population. It is pointed out that in a large variety of cases separable functions in the form of exponentially decaying sharp wavefronts solve the differential equation exactly provided a co-moving point source or sink is active at the wavefront. The velocity dispersion and front steepness may differ from those of some previously studied exact smooth traveling wave solutions. For an extension of the reactiondiffusion-convection equation, featuring a memory effect in the form of a maturity delay for growth and spread, also smooth exact wavefront solutions are obtained. The stability of the solutions is verified analytically and numerically.
Journal of Physics A: Mathematical and General, 2005
This paper concerns processes described by a nonlinear partial differential equation that is an extension of the Fisher and KPP equations including densitydependent diffusion and nonlinear convection. The set of wave speeds for which the equation admits a wavefront connecting its stable and unstable equilibrium states is characterized. There is a minimal wave speed. For this wave speed there is a unique wavefront which can be found explicitly. It displays a sharp propagation front. For all greater wave speeds there is a unique wavefront which does not possess this property. For such waves, the asymptotic behaviour as the equilibrium states are approached is determined.
Reactive-Diffusive-Advective Traveling Waves in a Family of Degenerate Nonlinear Equations
The Scientific World Journal, 2016
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 heteroclini...
On the analysis of traveling waves to a nonlinear flux limited reaction–diffusion equation
Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 2013
In this paper we study the existence and qualitative properties of traveling waves associated with a nonlinear flux limited partial differential equation coupled to a Fisher-Kolmogorov-Petrovskii-Piskunov type reaction term. We prove the existence and uniqueness of finite speed moving fronts of C 2 classical regularity, but also the existence of discontinuous entropy traveling wave solutions.
Traveling wave solutions for a reaction diffusion equation with double degenerate nonlinearities
Discrete and Continuous Dynamical Systems, 2009
This paper studies the traveling wave solutions for a reaction diffusion equation with double degenerate nonlinearities. The existence, uniqueness, asymptotics as well as the stability of the wave solutions are investigated. The traveling wave solutions, existed for a continuance of wave speeds, do not approach the equilibria exponentially with speed larger than the critical one. While with the critical speed, the wave solutions approach to one equilibrium exponentially fast and to the other equilibrium algebraically. This is in sharp contrast with the asymptotic behaviors of the wave solutions of the classical KPP and m − th order Fisher equations. A delicate construction of super-and sub-solution shows that the wave solution with critical speed is globally asymptotically stable. A simpler alternative existence proof by LaSalle's Wazewski principle is also provided in the last section.
arXiv: Analysis of PDEs, 2020
We study a family of reaction-diffusion equations that present a doubly nonlinear character given by a combination of the ppp-Laplacian and the porous medium operators. We consider the so-called slow diffusion regime, corresponding to a degenerate behaviour at the level 0, \normalcolor in which nonnegative solutions with compactly supported initial data have a compact support for any later time. For some results we will also require pge2p\ge2pge2 to avoid the possibility of a singular behaviour away from 0. Problems in this family have a unique (up to translations) travelling wave with a finite front. When the initial datum is bounded, radially symmetric and compactly supported, we will prove that solutions converging to 1 (which exist, as we show, for all the reaction terms under consideration for wide classes of initial data) do so by approaching a translation of this unique traveling wave in the radial direction, but with a logarithmic correction in the position of the front when the d...
The Characterization of Reaction-Convection-Diffusion Processes by Travelling Waves
Journal of Differential Equations, 1996
It has long been known that the heat equation displays infinite speed of propagation. This is to say that if the initial data are nonnegative and have nonempty compact support, the solution of an initial-value problem is positive everywhere after any infinitesimal time. However, since the nineteen-fifties it has also been known that certain nonlinear diffusion equations of degenerate parabolic type do not display this phenomenon. For these equations, the (generalized) solution of an initialvalue problem with compactly-supported initial data will have bounded support with respect to the spatial variable at all times. In this paper the necessary and sufficient criterion for finite speed of propagation for the general nonlinear reactionconvection-diffusion equation u t =(a(u)) xx +(b(u)) x +c is determined. The assumptions on the coefficients a, b and c are such that the classification unifies and generalizes previously-known results. The technique employed is comparision of an arbitrary solution of the equation with suitably-constructed travelling-wave solutions and subsolutions. Basically the central conclusion is that the equation exhibits finite speed of propagation if and only if it admits a travelling-wave solution with bounded support. Concurrently, the search for a travelling-wave solution with bounded support can be reduced to the study of a singular nonlinear integral equation whose solution must satisfy a certain constraint.
Emergence of Waves in a Nonlinear Convection-Reaction-Diffusion Equation
Advanced Nonlinear Studies, 2004
In this work we prove that for some class of initial data the solution of the Cauchy problem ut = (um)xx + a(um)x + u(1 - um-1), x ∈ ℝ; t > 0 u(0; x) = u0(x), u0(x) ≥ 0 approaches the travelling solution, spreading either to the right or to the left, or two travelling waves moving in opposite directions.