Local well-posedness of subcritical non-linear heat equations with Gaussian initial data

TL;DR

This paper proves local well-posedness for certain non-linear heat equations with Gaussian initial data in fractional dimensions below 4, extending previous results from dimension 3 to a broader subcritical range.

Contribution

It extends the well-posedness results of non-linear heat equations with Gaussian initial data from dimension 3 to all subcritical dimensions below 4.

Findings

Established local well-posedness for subcritical non-linear heat equations with Gaussian initial data in fractional dimensions.

Extended previous results from dimension 3 to the entire subcritical regime below dimension 4.

Demonstrated the applicability of Gaussian free fields as initial conditions in these equations.

Abstract

We show that any non-linear heat equation with scaling critical dimension $-1$ is locally well-posed when its initial condition is taken as the Gaussian free field in fractional dimension $d < 4$. Our results in particular extend the well-posedness results of arXiv:2111.10652, arXiv:2201.03487 from $d=3$ to the entire subcritical regime.