Speed of fronts of generalized reaction-diffusion equations

Abstract

Recent work on generalized diffusion equations has given analytical and numerical evidence that, as in the standard reaction-diffusion equation, most initial conditions evolve into a traveling wave which corresponds to a minimum speed front joining a stable to an unstable state. We show that this minimal speed derives from a variational principle; from this we recover linear constraints on the speed ͑the linear marginal stability value͒ and provide upper and lower bounds on the speed. This enables us to characterize the functions for which linear marginal stability holds and also to provide a tool to calculate the speed when the marginal value does not predict its correct value. ͓S1063-651X͑98͒09906-1͔ PACS number͑s͒: 03.40. Kf, 47.10.ϩg, 02.30.Hq Different problems may be described in terms of the reaction-diffusion equation u t ϭu xx ϩ f (u), where f is a nonlinear term with at least two equilibrium points. It has been established rigorously ͓1͔ that sufficiently localized initial conditions evolve asymptotically into a traveling monotonic wave front u(xϪct) joining two equilibrium states. For simplicity we will consider reaction terms f Ͼ0 which vanish at uϭ0 and at uϭ1. There is a wide class of reaction terms f (u) for which this asymptotic speed is given by ͓2͔ the. This is the minimal value of the speed that follows from a linear analysis at the equilibrium points. The extension of this behavior to pattern-forming systems is called the ͓3͔ marginal stability hypothesis. For other reaction terms this linear or c KPP value represents a lower bound to the speed. There exist local ͓4͔ and global ͓5,6͔ variational principles that enable one to determine the speed for arbitrary reaction terms f (u), as well as methods based on approximate solutions of the differential equation itself ͓7,8͔. While all the above results have been established rigorously, other problems are described by other types of reactiondiffusion equations in which nonlinearities may appear on derivative terms. Generalized diffusion equations of the form u t ϭu xx ϩF(u x ,u) and the stability of its traveling fronts have been studied ͓9,10͔. While the stability of its traveling waves has been established, the problem of determining which of the possible traveling waves will be the asymptotic state has not been clarified in general. In recent work ͓11͔ one such type of generalized reaction diffusion equation, namely,with f ()ϭ(1Ϫ), has been considered. This equation has traveling wave fronts (xϪct) joining the stable state ϭ1 to ϭ0. Exact analytical solutions to the partial differential equation ͑PDE͒ have been found ͓11͔ for the case mϭ2, which show explicitly that in the time evolution the front of minimal speed is selected. Numerical integrations for other values of m and for reaction terms of the form f ()ϭ(1Ϫ)͓1Ϫ(1Ϫ) ␤ )] support this conclusion. For these reaction terms it is found that the system evolves into the front of minimal speed which, for these reaction terms, is the so called marginal stability value obtained from linear consi...

Progress in front propagation research

“…which is valid for any function F (n) that satisfies F (n = 0) = 0, F (n = 0) = 1 with F (n) 0 for 0 n 1 and ∂n/∂r | n=0 = 0 and ∂n/∂r | n=1 = 0 with ∂n/∂r 0 for 0 n 1 [78].…”

“…Le = 0) and transport coefficients as a function of temperature. [78] has been recently applied to equation (110) in order to obtain bounds on the propagation speed of flames [34]. For β below a critical value β c given by…”

Front propagation into unstable states: universal algebraic convergence towards uniformly translating pulled fronts

“…2 As mentioned by Murray [[36], p. 277], the equation was apparently already considered in 1906 by Luther, who obtained the same analytical form as Fisher for the wave front. 3 For some recent more mathematical advances within the physics literature, see [54,115]. 4 For a recent extension to multidimensional cases, and for an entry into the mathematical literature, see, e.g., [55].…”

“…In this section, we provide the necessary background information on fronts propagating into unstable states by reviewing a number of results on the multiplicity and stability of uniformly translating front solutions of the nonlinear diffusion equation [32,36,38,[49][50][51]54,61,63,[65][66][67][68][69][70]84,85,87,88,[90][91][92][114][115][116] . 13 We also summarize to what extent the linear stability analysis of these uniformly translating fronts allows us to solve the selection problem, i.e., to determine the basins of attraction of these solutions in the space of initial conditions and for different nonlinearities f , and to what extent it allows us to answer the related question of the convergence rate and mechanism.…”

Lack of symmetry in linear determinacy due to convective effects in reaction-diffusion-convection problems

“…An example of such convection terms arises in a simple one-dimensional model of the motion of chemotactic cells, based on a model of Keller and Segel [9]. This model is presented in Benguria, Depassier and Mendez [2], where ρ denotes the density of bacteria chemotactic to a single chemical element of concentration s, the density evolves according to ρ t = [Dρ x − ρχs x ] x + f (ρ), D is a diffusion constant and χ is the chemotactic sensitivity. For travelling front solutions, s = s(x −c t ), ρ = ρ(x −c t ), we have s t = −c s x , s x = K ρ/c, and the problem then reduces to a single differential equation for ρ, namely…”