A solution of the constant coefficient heat equation on R with exceptional asymptotic behavior: An explicit construction

摘要

In this paper, we construct a solution of the one-dimensional heat equation on R which has the following properties: Given any 0 < mu < 1, there exists a sequence t(k) -> infinity such that parallel to u(t(k))parallel to(L)infinity approximate to t(k)(-mu/2) -> infinity. In other words, this solution exhibits all the possible decay rates. In addition, for essentially all values of mu is an element of (0, 1), we characterize the set a omega(mu)(u) of all limit points in C-0(R) as t -> infinity of t(mu/2)u(t, x root t). More precisely, if mu is rational, then omega(mu)(u) is all of C-0(R). Furthermore, for almost all mu is an element of (0, 1), omega(mu) (u(0)) is equal to a fixed nonempty set N not subset of C-0(R). (c) 2005 Elsevier SAS. All rights reserved.