We study the speed of propagation of fronts for the scalar reaction-diffusion

We give an integral variational characterization for the speed of fronts of the nonlinear diffusion equation u t = u xx + f (u) with f (0) = f (1) = 0, and f > 0 in (0, 1), which permits, in principle, the calculation of the exact speed for arbitrary f .

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...

We consider the problem of the speed selection mechanism for the one dimensional nonlinear diffusion equation u t = u xx +f (u). It has been rigorously shown by Aronson and Weinberger that for a wide class of functions f , sufficiently localized initial conditions evolve in time into a monotonic front which propagates with speed c * such that 2 f ′ (0) ≤ c * < 2 sup(f (u)/u). The lower value c L = 2 f ′ (0) is that predicted by the linear marginal stability speed selection mechanism. We derive a new lower bound on the the speed of the selected front, this bound depends on f and thus enables us to assess the extent to which the linear marginal selection mechanism is valid. 03.40.Gc

We give an explicit formula for the change of speed of pushed and bistable fronts of the reaction diffusion equation when a small cutoff is applied at the unstable or metastable equilibrium point.The results are valid for arbitrary reaction terms and include the case of density dependent diffusion.