Travelling Fronts and Entire Solutions¶of the Fisher-KPP Equation in ℝN

Abstract

This paper is devoted to time-global solutions of the Fisher-KPP equation in R N :It is well known that this equation admits a finite-dimensional manifold of planar travelling-fronts solutions. By considering the mixing of any density of travelling fronts, we prove the existence of an infinite-dimensional manifold of solutions. In particular, there are infinite-dimensional manifolds of (nonplanar) travelling fronts and radial solutions. Furthermore, up to an additional assumption, a given solution u can be represented in terms of such a mixing of travelling fronts.

Stability of travelling waves in a model for conical flames in two space dimensions

“…Existence of smooth solutions of problem (1.1-1.2) with bistable nonlinearity f was obtained by Fife [16] for angles α < π/2 close to π/2. Conical-shaped and more general curved fronts also exist for the Fisher-KPP equation, with concave nonlinearity f (see [11], [21]). Other stability results were also obtained by Michelson [31] for Bunsen fronts solving the KuramotoSivashinsky equation, in some asymptotic regimes.…”

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

“…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.…”

“…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.…”

“…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. In [22], Hamel and Nadirashvili further considered the existence of entire solutions of the Fisher-KPP equation in high-dimensional spaces and obtained an amazingly rich class of entire solutions. Chen and Guo [8] and Guo and Morita [20] developed a unified approach based on the comparison principle to find entire solutions for both the bistable and monostable cases.…”

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.…”