Propagating Terraces and the Dynamics of Front-Like Solutions of Reaction-Diffusion Equations on ℝ

Abstract

We consider semilinear parabolic equations of the form ut = uxx + f(u), x ∈ R, t > 0, where f a C1 function. Assuming that 0 and γ > 0 are constant steady states, we investigate the large-time behavior of the front-like solutions, that is, solutions u whose initial values u(x, 0) are near γ for x ≈ −∞ and near 0 for x ≈ ∞. If the steady states 0 and γ are both stable, our main theorem shows that at large times, the graph of u(·, t) is arbitrarily close to a propagating terrace (a system of stacked traveling fonts). We prove this result without requiring monotonicity of u(·, 0) or the nondegeneracy of zeros of f . The case when one or both of the steady states 0, γ is unstable is considered as well. As a corollary to our theorems, we show that all front-like solutions are quasiconvergent: their ω-limit sets with respect to the locally uniform convergence consist of steady states. In our proofs we employ phase plane analysis, intersection comparison (or, zero number) arguments, and a geometric method involving the spatial trajectories {(u(x, t), ux(x, t)) : x ∈ R}, t > 0, of the solutions in question.