Some entire solutions of the Allen-Cahn equation (Q1880532)

The paper deals with entire solutions of a bistable reaction-diffusion equation with Nagumo type nonlinearity, the so called Allen-Cahn equation. The entire solutions are the solutions defined for all \((x,t)\in (\mathbb{R}\times \mathbb{R}).\) The authors show the existence of an entire solution which behaves as two traveling front solutions coming from both sides of the \(x\)-axis and annihilating in a finite time, using the explicit expression of the traveling front and the comparison theorem. They also show the existence of an entire solution emanating from the unstable standing pulse solution and converging to the pair of diverging traveling fronts as the time tends to infinity. Then in terms of the comparison principle a rather general result is proved on the existence of an unstable set of an unstable equilibrium to apply to the present case.