On heat equations associated with fractional harmonic oscillators

Abstract

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