Entire solutions of reaction–advection–diffusion equations with bistable nonlinearity in cylinders

Abstract

This paper deals with entire solutions and the interaction of traveling wave fronts of bistable reaction-advection-diffusion equation with infinite cylinders. Assume that the equation admits three equilibria: two stable equilibria 0 and 1, and an unstable equilibrium θ . It is well known that there are different wave fronts connecting any two of those three equilibria. By considering a combination of any two of those different traveling wave fronts and constructing appropriate subsolutions and supersolutions, we establish three different types of entire solutions. Finally, we analyze a model for shear flows in cylinders to illustrate our main results.

Entire solutions for nonlocal dispersal equations with spatio-temporal delay: Monostable case

“…These solutions can not only describe the interaction of traveling waves but also characterize new dynamics of diffusion equations. For the study of such entire solutions, we refer to [4,[14][15][16]19,23,27] for reaction-diffusion equations without delay, [22,35,37,42] for reaction-diffusion equations with nonlocal delay, [38,39] for delayed lattice differential equations with nonlocal interaction, [21,34] for nonlocal dispersal equations without delay ((1.9) below), [28,40,44] for reaction-diffusion systems, and [43] for periodic lattice dynamical systems. However, to the best of our knowledge, the issues on entire solutions for nonlocal dispersal equations with spatio-temporal delay have not been addressed, especially for infinite delay equations.…”

Entire solutions in reaction-advection-diffusion equations with bistable nonlinearities in heterogeneous media

“…By using a similar argument to that of [19, Proofs of Theorems 1.3 and 1.4], the problem (3.9) is well posed and the maximum principle holds (see also [8,22]). Let u n (t, x, y) denote the unique classical solution of (3.9) that satisfies 0 < u n (t, x, y) < 1 for any (t, x, y) ∈ [−n, −T ) ×Σ and n > T (n ∈ N).…”

“…In our previous paper [22], an amazingly rich class of entire solutions is obtained for the bistable nonlinearity, by splicing the traveling fronts which connect 0 and α, α and 1, and 0 and 1, respectively. However, the global uniqueness and the stability of such an entire solution under the heterogeneous bistable assumption remain open.…”

“…It needs to be point out that, Li et al [20] and Wang et al [33] established the existence of entire solutions for the nonlocal reaction-diffusion model with delayed nonlinearities, and Wang et al [34] constructed new types of entire solutions for a class of monostable delayed lattice differential equations with global interaction. For high dimensional spaces, Li et al [19] considered the existence of entire solutions of a reactionadvection-diffusion equation with monostable and ignition temperature nonlinearities in infinite-cylinders by considering a pair of traveling curved fronts, and Liu et al [22] considered the similar problem for bistable nonlinearity. See also [16] for an amazingly rich class of entire solutions by using planar traveling wave fronts and [21] for periodic media.…”

Entire Solutions with Annihilating Fronts to a Nonlocal Dispersal Equation with Bistable Nonlinearity and Spatio-Temporal Delay

“…Recently, many types of front-like entire solutions have been observed for various evolution equations by mixing the traveling wave solutions and some spatially independent solutions, see [11][12][13][15][16][17]20,21,25,[27][28][29][30][31][32][33]. For examples, Hamel and Nadirashvili [12] established three-, four-and five-dimensional manifolds of entire solutions for the Fisher-KPP equation.…”