Some Entire Solutions of the Allen–cahn Equation

Abstract

This paper is dealing with entire solutions of a bistable reactiondiffusion equation with Nagumo type nonlinearity, so called the Allen-Cahn equation. Here the entire solutions are meant by the solutions defined for all (x; t) 2 R £ R. In this article we first show the existence of an entire solution which behaves as two traveling front solutions coming from both sides of x-axis and annihilating in a finite time, using the explicit expression of the traveling front and the comparison theorem. We also show the existence of an entire solution emanating from the unstable standing pulse solution and converges to the pair of diverging traveling fronts as the time tends to infinity. Then in terms of the comparison principle we prove a rather general result on the existence of an unstable set of an unstable equilibrium to apply to the present case.

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

“…We note that the above argument about p(t) was first given by Guo and Morita [20] (see also Fukao et al [16]). In the sequel of this section, we always assume that (1.1) has an increasing traveling wave solution φ with wave speed c > 0.…”

“…Furthermore, Chen et al [9] considered entire solutions of reaction-diffusion equations with bistable nonlinearities for the case c = 0. Morita and Ninomiya [28] showed some novel entire solutions which are completely different from these observed in [8,16,20,21,22,47]. However, the above mentioned results are only concerned with entire solutions of reaction-diffusion equations in the absence of time delay and nonlocality.…”

“…The stability and uniqueness of entire solutions were also considered. Yagisita's argument was substantially simplified by Fukao et al [16], and the existence of an entire solution emanating from the unstable standing pulse solution of (1.1) was also obtained. For the Fisher-KPP equation, Hamel and Nadirashvili [21] established five-dimensional, four-dimensional and three-dimensional manifolds of entire solutions, respectively, by combining two traveling wave solutions with different speeds and coming from both sides of the real axis and some spatially independent solution.…”

“…On the other hand, it has been observed that traveling wave solutions are special examples of the so-called entire solutions that are defined in the whole space and for all time t ∈ R. In particular, Chen and Guo [8], Fukao et al [16], Guo and Morita [20], Hamel and Nadirashvili [21,22], Morita and Ninomiya [28] and Yagisita [47] have shown that the study of entire solutions is essential for a full understanding of the transient dynamics and the structure of the global attractors. These studies showed the great diversity of different types of entire solutions of reaction-diffusion equations in the absence of time delay.…”

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

“…− n (t; θ) := φ(−n + ct + θ), where θ varies in R (note that the wave speed c is unique in the bistable case). For reaction-diffusion equations with continuous spatial variables, Chen and Guo [14], Chen et al [15], Crooks and Tsai [20], Fukao et al [23], Guo and Morita [25], Hamel and Nadirashvili [26,27], Morita and Ninomiya [35] and Yagisita [50] showed the existence of new types of entire solutions other than the traveling wave type by using the well-known results of planar traveling wave solutions. As reported by Hamel and Nadirashvili [27, Theorems 1.7 and 1.8], reaction-diffusion equations usually have more types of entire solutions in high dimensional spatial spaces, which even includes some other classes of solutions of traveling wave type other than planar traveling waves.…”