Planar Propagating Terraces and the Asymptotic One-dimensional Symmetry of Solutions of Semilinear Parabolic Equations

Abstract

We consider the equation $u_t=\Delta u+f(u)$ on $\mathbb{R}^N$. Under suitable conditions on $f$ and the initial value $u_0=u(\cdot,0)$, we show that as $t\to\infty$ the solution $u(\cdot,t)$ approaches a planar propagating terrace, or a stacked family of planar traveling fronts. Using this result, we show the asymptotic one-dimensional symmetry of $u(\cdot,t)$ as well as its quasi convergence in $L_{loc}^\infty(\mathbb{R}^N)$.