Heat equation with singular thermal conductivity

TL;DR

This paper investigates the heat equation with highly irregular, distribution-valued thermal conductivity, establishing existence, uniqueness, and consistency of solutions even with singular coefficients like the Delta function.

Contribution

It introduces a framework for solving the heat equation with distributional thermal conductivity, extending classical theory to include highly singular coefficients.

Findings

Existence of solutions in a very weak sense

Uniqueness of solutions under irregular coefficients

Consistency with classical solutions when they exist

Abstract

In this paper, we study the heat equation with an irregular spatially dependent thermal conductivity coefficient. We prove that it has a solution in an appropriate very weak sense. Moreover, the uniqueness result and consistency with the classical solution if the latter exists are shown. Indeed, we allow the coefficient to be a distribution with a toy example of a Delta-function.