On the Cauchy problem for the heat equation on Riemannian manifolds with conical singularities

TL;DR

This paper investigates the existence and regularity of solutions to the heat equation on Riemannian manifolds with conical singularities, introducing new weighted function spaces and extending previous results.

Contribution

It develops a framework of weighted H"older and Sobolev spaces with discrete asymptotics for solving the heat equation on singular manifolds, generalizing prior work.

Findings

Established existence and maximal regularity of solutions.

Extended previous results to manifolds with conical singularities.

Introduced new weighted function spaces with discrete asymptotics.

Abstract

We study the existence and regularity of solutions to the Cauchy problem for the inhomogeneous heat equation on compact Riemannian manifolds with conical singularities. We introduce weighted H\"older and Sobolev spaces with discrete asymptotics and we prove existence and maximal regularity of solutions to the Cauchy problem for the inhomogeneous heat equation, when the free term lies in a weighted parabolic H\"older or Sobolev space with discrete asymptotics. This generalizes a result previously obtained by Coriasco, Schrohe, and Seiler (Thm. 7.2 in Math. Z. 244 (2003), 235--269) by different means.