A solution of the constant coefficient heat equation on \(\mathbb R\) with exceptional asymptotic behavior: an explicit construction (Q2489996)

The goal of this paper is to construct a classical solution of the one-dimensional heat equation on \(\mathbb{R}\) which has an unexpected rich long-time asymptotic structure. In the present study the authors construct a solution which has the following properties: given any \(0<\nu<1\), there exists a sequence \(t_k\to\infty\) as \(k\to\infty\) such that \(\|U(t_k) \|_{L^\infty}\approx t_k^{-\nu/2}\). Here \(U(t,x)\) is a solution. In other words, this solution exhibits all the possibly decay rates (except perhaps \(t^{-1/2})\). Moreover, for essentially all values of \(\mu\in(0,1)\), the authors characterize the set of all possible limit points in \(C_0(\mathbb{R})\) as \(t\to\infty\) of \(t^{\nu/2}U(t,x\sqrt t)\).