Summability of solutions of the heat equation with inhomogeneous thermal conductivity in two variables

TL;DR

This paper studies the summability properties of solutions to a two-variable heat equation with inhomogeneous conductivity, providing conditions for 1-summability and extending previous results to variable coefficients.

Contribution

It establishes necessary and sufficient conditions for 1-summability of solutions in two variables, generalizing prior constant coefficient results with a new proof.

Findings

Derived conditions for 1-summability in specific directions

Extended previous constant coefficient results to variable coefficients

Provided a new proof of existing summability criteria

Abstract

We investigate Gevrey order and 1-summability properties of the formal solution of a general heat equation in two variables. In particular, we give necessary and sufficient conditions for the 1-summability of the solution in a given direction. When restricted to the case of constants coefficients, these conditions coincide with those given by D.A. Lutz, M. Miyake, R. Schaefke in a 1999 article, and we thus provide a new proof of their result.