# Complex powers for cone differential operators and the heat equation on manifolds with conical singularities

## TL;DR

This paper develops a framework for analyzing complex powers of cone differential operators on manifolds with conical singularities, applying it to the heat equation and revealing how solutions behave near singular points.

## Contribution

It introduces new embeddings for the domains of complex powers of cone differential operators and studies their application to the heat equation on singular manifolds.

## Key findings

- Established continuous embeddings for domains of complex powers of cone operators.
- Derived asymptotic expansions for heat equation solutions near conical points.
- Showed the preservation of asymptotic decomposition under heat evolution.

## Abstract

We obtain left and right continuous embeddings for the domains of the complex powers of sectorial $\mathbb{B}$-elliptic cone differential operators. We apply this result to the heat equation on manifolds with conical singularities and provide asymptotic expansions of the unique solution close to the conical points. We further show that the decomposition of the solution in terms of asymptotics spaces, i.e. finite dimensional spaces that describe the domains of the integer powers of the Laplacian and determined by the local geometry around the singularity, is preserved under the evolution.