Propagating terraces and the dynamics of front-like solutions of reaction-diffusion equations on R

We consider semilinear parabolic equations of the form u_t = u_xx + 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), u_x(x, t)): x ∈ R}, t > 0, of the solutions in question.