Time periodic traveling curved fronts of bistable reaction–diffusion equations in $${\mathbb {R}}^3$$ R 3

Abstract

This paper is concerned with the existence and stability of time periodic traveling curved fronts for reaction–diffusion equations with bistable nonlinearity in \({\mathbb {R}}^3\). We first study the existence and other qualitative properties of time periodic traveling fronts of polyhedral shape. Furthermore, for any given \(g\in C^{\infty }(S^1)\) with \(\min \nolimits _{0\le \theta \le 2\pi }g(\theta )=0\) that gives a convex bounded domain with smooth boundary of positive curvature everywhere, which is included in a sequence of convex polygons, we show that there exists a three-dimensional time periodic traveling front by taking the limit of the solutions corresponding to the convex polyhedrons as the number of the lateral surfaces goes to infinity.