Abstract
Based on a recent work on traveling waves in spatially nonlocal reaction–diffusion equations, we investigate the existence of traveling fronts in reaction–diffusion equations with a memory term. We will explain how such memory terms can arise from reduction of reaction–diffusion systems if the diffusion constants of the other species can be neglected. In particular, we show that two-scale homogenization of spatially periodic systems can induce spatially homogeneous systems with temporal memory. The existence of fronts is proved using comparison principles as well as a reformulation trick involving an auxiliary speed that allows us to transform memory terms into spatially nonlocal terms. Deriving explicit bounds and monotonicity properties of the wave speed of the arising traveling front, we are able to establish the existence of true traveling fronts for the original problem with memory. Our results are supplemented by numerical simulations.
Highlights
The study of traveling fronts in scalar reaction–diffusion equations with a bistable nonlinearity is a classical topic and there is a rich literature concerning the existence, uniqueness, and asymptotic stability of such fronts, see e.g. [1, 4, 10, 13] and the references therein
[10] studies the case Lu = Duxx, assumes that G(u, ·) : R → R and S : R → R are monotonously increasing and that J is a convolution with a smooth, nonnegative kernel J ≥ 0, i.e. (J f )(x) = (J ∗ f )(x) = R J (x−y) f (y) dy
The nonlinearity F is of bistable type such that (1.1) has exactly three homogeneous steady states u ≡ uα with α ∈ {−, m, +}, where u± are stable while the middle state um is unstable
Summary
The study of traveling fronts in scalar reaction–diffusion equations with a bistable nonlinearity is a classical topic and there is a rich literature concerning the existence, uniqueness, and asymptotic stability of such fronts, see e.g. [1, 4, 10, 13] and the references therein. Traveling fronts in parabolic equations with discrete time delay are treated in [28], but not including the bistable case considered here. After providing suitable a priori estimates for the front speeds (see Theorem 2.5), our main result Theorem 2.8 shows that the memory equation (1.2) has always a traveling front if Fγ : u → F(u) + γ u is a bistable nonlinearity. A second motivation for deriving memory equations is the study of traveling pulses and fronts in situations where the coefficients in the system are rapidly oscillating, modeling a periodic heterogeneous medium on a microscopic scale. 2 we develop our existence result for traveling fronts for the memory equation (1.2), see Theorem 2.8 which relies on the comparison principle in the spirit of [1, 4, 10] and on new a priori bounds for the front speed c in Theorem 2.5.
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