Entire solutions with and without radial symmetry in balanced bistable reaction–diffusion equations

Let \(n\ge 2\) $n\ge 2$ be a given integer. In this paper, we assert that an n-dimensional traveling front converges to an \((n-1)\) $(n-1)$ -dimensional entire solution as the speed goes to infinity in a balanced bistable reaction–diffusion equation. As the speed of an n-dimensional axially symmetric or asymmetric traveling front goes to infinity, it converges to an \((n-1)\) $(n-1)$ -dimensional radially symmetric or asymmetric entire solution in a balanced bistable reaction–diffusion equation, respectively. We conjecture that the radially asymmetric entire solutions obtained in this paper are associated with the ancient solutions called the Angenent ovals in the mean curvature flows.