Spatial trajectories and convergence to traveling fronts for bistable reaction-diffusion equations

Abstract

We consider the semilinear parabolic equation ut = uxx + f(u), x∈ ℝ, t > 0, where f is a bistable nonlinearity. It is well known that for a large class of initial data, the corresponding solutions converge to traveling fronts. We give a new proof of this classical result as well as some generalizations. Our proof uses a geometric method, which makes use of spatial trajectories {(u(x, t), ux(x, t)): x∈ ℝ} of solutions of (1).