On heat equations associated with fractional harmonic oscillators

TL;DR

This paper investigates decay estimates for fractional heat equations linked to harmonic oscillators and establishes well-posedness results for related nonlinear equations.

Contribution

It provides new fixed-time decay estimates and well-posedness results for nonlinear fractional heat equations involving harmonic oscillators.

Findings

Established decay estimates in Lebesgue spaces for fractional heat propagators.

Proved local and global well-posedness for nonlinear fractional heat equations.

Extended understanding of fractional heat equations associated with harmonic oscillators.

Abstract

We establish some fixed-time decay estimates in Lebesgue spaces for the fractional heat propagator $e^{-tH^{\beta}}$, $t, \beta>0$, associated with the harmonic oscillator $H=-\Delta + |x|^2$. We then prove some local and global wellposedness results for nonlinear fractional heat equations.