On heat equations associated with fractional harmonic oscillators

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 $H=-\mathrm{\Delta }+\mathrm{\mid }x{\mathrm{\mid }}^2$. We then prove some local and global wellposedness results for nonlinear fractional heat equations.