Time-Periodic Planar Fronts Around an Obstacle

This paper is concerned with a time-periodic reaction-diffusion equation in exterior domains Omega = R-N \ K, where K is a compact set in R-N and is called an obstacle. We first prove the existence of the entire solution u(t, x) emanating from a timeperiodic planar front phi(t, x(1) - ct) as t -> -infinity. Then, under the assumption that the propagation of u(t, x) is complete, we prove that u(t, x) converges to the same time-periodic planar front phi(t, x(1)- ct) as t ->+infinity uniformly in Omega. Finally, we show some examples of geometrical shapes of K such that the propagation of u(t, x) is complete or incomplete.