Transition fronts for inhomogeneous monostable reaction–diffusion equations via linearization at zero

Abstract

We prove existence of transition fronts for a large class of reaction-diffusion equations in one dimension, with inhomogeneous monostable reactions. We construct these as perturbations of corresponding front-like solutions to the linearization of the PDE at u = 0. While a close relationship of the solutions to the two PDEs has been well known and exploited for KPP reactions (and our method is an extension of such ideas from [15]), to the best of our knowledge this is the first time such an approach has been used in the construction and study of fronts for non-KPP monostable reactions.

Stability of transition waves and positive entire solutions of Fisher-KPP equations with time and space dependence

“…A huge amount of research has been carried out toward the transition waves or generalized traveling waves of various time and/or space dependent monostable equations. See, for example, [7,8,9,10,11,20,22,25,27,28,32,33,34,36,39,41,42,43,44,45,46,51,52,53,58,60,61,62,63], and references therein for space and/or time dependent Fisher-KPP type equations with random dispersal, and see, for example, [15,16,17,35,48,54,55,56,57], and references therein for space and/or time dependent Fisher-KPP type equations with nonlocal dispersal.…”

“…Next, we show the stability of uniformly continuous transition waves connecting the unique strictly positive entire solution and the trivial solution zero and satisfying certain decay property at the end close to the trivial solution zero (if it exists). The existence of transition waves has been studied in [34,39,45,46,61] for random dispersal Fisher-KPP equations with time and space periodic dependence, in [41,42,43,51,52,53,58,63] for random dispersal Fisher-KPP equations with quite general time and/or space dependence, and in [17,48,56] for nonlocal dispersal Fisher-KPP equations with time and/or space periodic dependence. The stability result established in this paper implies that the transition waves obtained in many of the above mentioned papers are asymptotically stable for well-fitted perturbation.…”

On the mean speed of bistable transition fronts in unbounded domains

“…Lastly, even in the homogeneous space R N , non-standard transition fronts which are not invariant in any moving frame were also constructed in [24] under assumptions (1.2)-(1.5). More generally speaking, there is now a large literature devoted to transition fronts for bistable reactions in homogeneous or heterogeneous settings [6,13,18,22,48,53,60], as well as for other types of homogeneous or space/time dependent reactions in dimension 1 [16,29,30,34,35,37,40,42,43,51,52,57,58] and in higher dimensions [1,9,38,39,49,50,59,61].…”

Multidimensional transition fronts for Fisher–KPP reactions

“…In one dimension d = 1 the set Γ t in (iii) is just a collection of at most n points. The special case n = 1 of transition fronts with a single interface is of particular interest and has recently been studied extensively for various types of reactions (see, e.g., [3,5,9,11,12,14,[16][17][18][19][20][21][22][23][24][25][26][27][28][29]). These are entire solutions 0 < u < 1 satisfying lim x→−∞ u(t, x + x t ) = 1 and lim x→∞ u(t, x + x t ) = 0 (1.9) uniformly in t ∈ R, where x t ∶= max{x ∈ R ∶ u(t, x) = 1 2 } (or with 0 and 1 exchanged in (1.9)).…”