Some results on a heat conduction problem by Myshkis

摘要

An infinite homogeneous d-dimensional medium initially is at zero temperature. A heat impulse is applied at the origin, raising the temperature there to a value greater than a constant value u0>0. The temperature at the origin then decays, and when it reaches u0, another equal-sized heat impulse is applied at a normalized time τ1=1. Subsequent equal-sized heat impulses are applied at the origin at the normalized times τn, n=2,3,…, when the temperature there has decayed to u0. This sequence of normalized waiting times τn can be defined recursively by a difference equation and its asymptotic behavior was known recently. This heat conduction problem was first studied in [J. Difference Equations Appl. 3 (1997) 89–91].A natural subsequent question is what happens if the problem is set in a finite region, like in a laboratory, with the temperature at the boundary being kept zero forever. In this paper we obtain the asymptotic behavior of the heating times for the one-dimensional case.