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We study the propagation profile of the solution $u(x,t)$ to the nonlineardiffusion problem $u_t-\Delta u=f(u)\; (x\in \mathbb R^N,\;t>0)$,$u(x,0)=u_0(x) \; (x\in\mathbb R^N)$, where $f(u)$ is of multistable type:$f(0)=f(p)=0$, $f'(0)<0$, $f'(p)<0$, where $p$ is a positive constant, and $f$may have finitely many nondegenerate zeros in the interval $(0, p)$. The classof initial functions $u_0$ includes in particular those which are nonnegativeand decay to 0 at infinity. We show that, if $u(\cdot, t)$ converges to $p$ as$t\to\infty$ in $L^\infty_{loc}(\mathbb R^N)$, then the long-time dynamicalbehavior of $u$ is determined by the one dimensional propagating terracesintroduced by Ducrot, Giletti and Matano [DGM]. For example, we will show thatin such a case, in any given direction $\nu\in\mathbb{S}^{N-1}$, $u(x\cdot \nu,t)$ converges to a pair of one dimensional propagating terraces, one moving inthe direction of $x\cdot \nu>0$, and the other is its reflection moving in theopposite direction $x\cdot\nu<0$.  Our approach relies on the introduction of the notion "radial terracesolution", by which we mean a special solution $V(|x|, t)$ of $V_t-\DeltaV=f(V)$ such that, as $t\to\infty$, $V(r,t)$ converges to the corresponding onedimensional propagating terrace of [DGM]. We show that such radial terracesolutions exist in our setting, and the general solution $u(x,t)$ can be wellapproximated by a suitablly shifted radial terrace solution $V(|x|, t)$. Thesewill enable us to obtain better convergence result for $u(x,t)$.  We stress that $u(x,t)$ is a high dimensional solution without any symmetry.Our results indicate that the one dimensional propagating terrace is a ratherfundamental concept; it provides the basic structure and ingredients for thelong-time profile of solutions in all space dimensions.

Radial terrace solutions and propagation profile of multistable reaction-diffusion equations over $\mathbb R^N$