Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay

Abstract

This paper is concerned with the existence, uniqueness and globally asymptotic stability of traveling wave fronts in the quasi-monotone reaction advection diffusion equations with nonlocal delay. Under bistable assumption, we construct various pairs of super-and subsolutions and employ the comparison principle and the squeezing technique to prove that the equation has a unique nondecreasing traveling wave front (up to translation), which is monotonically increasing and globally asymptotically stable with phase shift. The influence of advection on the propagation speed is also considered. Comparing with the previous results, our results recovers and/or improves a number of existing ones. In particular, these results can be applied to a reaction advection diffusion equation with nonlocal delayed effect and a diffusion population model with distributed maturation delay, some new results are obtained.

Entire Solutions in Lattice Delayed Differential Equations with Nonlocal Interaction: Bistable Cases

“…Under the assumptions (H1)-(H3), it follows from Ma and Wu [32] and Wang et al [43] that (1.2) admits a strictly increasing traveling wave front ϕ (x + ct) satisfying ϕ (−∞) = 0 and ϕ (+∞) = K with wave speed c ∈ R. When c = 0, we know from [44] that for any (θ 1 , θ 2 ) ∈ R 2 , there exists a unique entire solution U (x, t) := U (x, t; θ 1 , θ 2 ) with 0 < U (x, t) < K for all (x, t) ∈ R 2 such that…”

“…Since the influence of maturation period and the random walk of individuals in space, time delay and global interaction have to be taken into account (Britton [7], Gourley et al [24], So et al [40], Wang et al [42,43]). Recently, Weng et al [47] derived a lattice delayed differential equation with global interaction for a single species with two age classes distributed over a patchy environment consisting of all integer nodes of a one-dimensional lattice.…”

Entire solutions in bistable reaction-diffusion equations with nonlocal delayed nonlinearity

“…Under the bistable assumption, they proved the existence, uniqueness and global asymptotic stability of traveling wave solutions of (1.6). Recently, Wang et al [42] studied the reaction advection diffusion equation with nonlocal delay and a bistable nonlinear term of the form (1.7) ∂u ∂t = d∆u + B ∂u ∂x + g u (x, t) ,…”

“…In [42], we showed that (1.11) has an increasing traveling wave solution under the following conditions:…”

“…[42]. That is, take the smaller between 1 and the supersolution in Chen [7] and Smith and Zhao [38] (or the larger between 0 and the subsolution) as a new supersolution (a new subsolution).…”