Global dynamics of the solution for a bistable reaction diffusion equation with nonlocal effect

Abstract

<p style='text-indent:20px;'>This paper is devoted to studying the Cauchy problem corresponding to the nonlocal bistable reaction diffusion equation. It is the first attempt to use the method of comparison principle to study the well-posedness for the nonlocal bistable reaction-diffusion equation. We show that the problem has a unique solution for any non-negative bounded initial value by using Gronwall's inequality. Moreover, the boundedness of the solution is obtained by means of the auxiliary problem. Finally, in the case that the initial data with compactly supported, we analyze the asymptotic behavior of the solution.</p>

Pyramidal traveling fronts in the Belousov-Zhabotinskii reaction-diffusion systems in R^3

“…In fact, studies of traveling wave solutions on reaction-diffusion equations u t (x, t) = ∆u(x, t) + f (u(x, t)), x ∈ R N , t > 0, which can be originated from the pioneer work of Fisher [6], have attracted a lot of attention [4,10,11,23,24,42], which mainly focus on 1-D traveling wave solutions and planar traveling wave solutions in R N (N ≥ 2). The gradually mature theory of 1-D traveling wave solutions promotes the research on multidimensional traveling wave solutions, which are proper to describe the traveling wave phenomena in multidimensional space, see [2,3,13,14,15,28,29,34,35,44] for the scalar equation and [5,16,17,18,19,27,36,43,45,46] for the reaction diffusion system.…”