On heat equations associated with fractional harmonic oscillators

Abstract

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

17 pages