Reaction-diffusion equation with a bistable nonlocal nonlinearity is considered in the case where the reaction term is not quasi-monotone. For this equation, the existence of travelling waves is proved by the Leray-Schauder method based on the topological degree for elliptic operators in unbounded domains and a priori estimates of solutions in properly chosen weighted spaces.

This paper is concerned with the existence, non-existence and qualitative properties of cylindrically symmetric travelling fronts for time-periodic reaction–diffusion equations with bistable nonlinearity in ℝm with m ≥ 2. It should be mentioned that the existence and stability of two-dimensional time-periodic V-shaped travelling fronts and three-dimensional time-periodic pyramidal travelling fronts have been studied previously. In this paper we consider two cases: the first is that the wave speed of a one-dimensional travelling front is positive and the second is that the one-dimensional wave speed is zero. For both cases we establish the existence, non-existence and qualitative properties of cylindrically symmetric travelling fronts. In particular, for the first case we furthermore show the asymptotic behaviours of level sets of the cylindrically symmetric travelling fronts.

This paper is concerned with entire solutions of a class of bistable delayed latticedifferential equations with nonlocal interaction. Here an entire solution is meant by asolution defined for all (n,t) ∈ ℤ × ℝ. Assuming that the equation has anincreasing traveling wave front with nonzero wave speed and using a comparison argument,we obtain a two-dimensional manifold of entire solutions. In particular,it is shown that the traveling wave fronts are on the boundary of the manifold.Furthermore, uniqueness and stability of such entire solutions are studied.