Neumann heat content asymptotics with singular initial temperature and singular specific heat

TL;DR

This paper investigates the asymptotic behavior of heat content on manifolds with boundary, focusing on cases with singular initial temperature and specific heat, and derives the first three terms of the asymptotic series.

Contribution

It provides explicit asymptotic expansions for heat content with singular data under Robin boundary conditions, including detailed analysis on intervals and relations between boundary conditions.

Findings

First three terms of the asymptotic series are determined.

Recursion relations among coefficients are established.

Connections between Dirichlet and Robin boundary conditions are clarified.

Abstract

We study the asymptotic behavior of the heat content on a compact Riemannian manifold with boundary and with singular specific heat and singular initial temperature distributions imposing Robin boundary conditions. Assuming the existence of a complete asymptotic series we determine the first three terms in that series. In addition to the general setting, the interval is studied in detail as are recursion relations among the coefficients and the relationship between the Dirichlet and Robin settings.