2017/11/02 by Du, Yihong, Matano, Hiroshi
#35J60 #35K20 #35R35 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1711.00952
We study the propagation profile of the solution u(x,t) to the nonlinear diffusion problem ut-Δu=f(u) (x∈ \mathbb RN, t>0), u(x,0)=u0(x) (x∈\mathbb RN), 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 class of initial functions u0 includes in particular those which are nonnegative and decay to 0 at infinity. We show that, if u(⋅, t) converges to p as t→∞ in L^∞loc(\mathbb RN), then the long-time dynamical behavior of u is determined by the one dimensional propagating terraces introduced by Ducrot, Giletti and Matano [DGM]. For example, we will show that in such a case, in any given direction ν∈\mathbbSN-1, u(x⋅ ν, t) converges to a pair of one dimensional propagating terraces, one moving in the direction of x⋅ ν>0, and the other is its reflection moving in the opposite direction x⋅ν<0. Our approach relies on the introduction of the notion "radial terrace solution", by which we mean a special solution V(|x|, t) of Vt-ΔV=f(V) such that, as t→∞, V(r,t) converges to the corresponding one dimensional propagating terrace of [DGM]. We show that such radial terrace solutions exist in our setting, and the general solution u(x,t) can be well approximated by a suitablly shifted radial terrace solution V(|x|, t). These will 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 rather fundamental concept; it provides the basic structure and ingredients for the long-time profile of solutions in all space dimensions.