Entire Solutions with Merging Fronts to Reaction–Diffusion Equations

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.…”

Entire solution originating from three fronts for a discrete diffusive equation

“…From the results shown in [15,5,7,10,12], there are entire solutions which behave as two traveling fronts of (1.6) on the left x-axis and right x-axis as t → −∞. We call this type of entire solution by entire solution originating from two fronts.…”

“…In the study of the entire solution originating from two fronts of (1.6), Morita and Ninomiya find some auxiliary rational functions with certain properties to help them to construct super-sub-solutions (see [12]). In [9], this method was used to prove some types of entire solution originating from two fronts of (1.1).…”

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.…”