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Existence and Stability of Traveling Wave Solutions in Reaction–Diffusion Equations

Author(s) Rajesh Kumar
Country India
Abstract Traveling fronts are the coherent structures of scalar reaction–diffusion equations: a fixed profile translating at constant speed, the mathematical image of invasion. In this expository article we develop the classical existence and stability theory for monotone fronts of the scalar equation u_t = Du_xx + f(u) in one space dimension. The traveling wave ansatz reduces the partial differential equation to a second-order ordinary differential equation whose heteroclinic orbits are the fronts. We present the existence theory in its two principal cases: the bistable case, where phase-plane shooting (Fife [6]) yields a unique speed whose sign is the sign of the integral of f, and the monostable (KPP) case, where the linear prediction c^* = 2√(Df'(0) ) of Kolmogorov, Petrovsky, and Piskunov [2] is sharp (Aronson–Weinberger [3, 4]) and governs both the minimal front speed and the asymptotic spreading speed of localized disturbances that the same number answers two questions, and that coincidence is one of the miracles of the subject. For stability we linearize about the front: the essential spectrum is read off from the constant-coefficient asymptotic operators, the point spectrum from the Evans function [8], and the translation eigenvalue λ = 0 is the principal eigenvalue by Sturm–Liouville theory. Nonlinear asymptotic stability follows from the comparison principle — via the squeezing argument of Fife and McLeod [5] for bistable fronts and weighted-norm estimates (Sattinger [7]) for KPP fronts. The Fisher–KPP equation serves throughout as the running example, including the closed-form front of Ablowitz and Zeppetella [12]. The results are classical; proofs are sketched when they illuminate and referenced otherwise.
Keywords Reaction–diffusion equations; traveling wave solutions; existence; stability; Evans function; comparison principle.
Field Mathematics
Published In Volume 8, Issue 1, January-June 2017
Published On 2017-04-12
DOI https://doi.org/10.71097/IJAIDR.v8.i1.2186
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Crossref DOI prefix of IJAIDR is 10.71097/IJAIDR

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