Sharp Time-Decay Estimates for Fractional Heat Semigroups Associated with Polynomial Anharmonic Oscillators

Abstract

We investigate fractional heat semigroups generated by a class of anharmonic oscillators on ${\mathbb{R}}^n$ of the form ${\mathcal{H}}_{P,Q}=Q(D)+P(x),$ where $P\in {\mathcal{P}}_{2k}$ and $Q\in {\mathcal{P}}_{2\ell }$ are real-valued polynomials with anisotropic growth. Using the Weyl--Hörmander calculus associated with the natural metric determined by $(P,Q)$, we show that the fractional powers ${\mathcal{H}}_{P,Q}^s$, $s>0$, are pseudo-differential operators with symbols in adapted classes $\hat{I}{\pounds }_{P,Q}^{2s}$. We prove fixed-time decay estimates for the fractional anharmonic heat semigroup $e^{-t{\mathcal{H}}_{P,Q}^s}$ on both Lebesgue and modulation spaces. In the Lebesgue setting, we establish sharp $L^p$--$L^q$ estimates for the full range $1\le p,q\le \infty$. For large time, the decay is exponential and governed by the smallest eigenvalue $\hat{I}{»}_0$ of ${\mathcal{H}}_{P,Q}$, namely through the factor $e^{-t\hat{I}{»}_0^s}$, while for small time the estimates reveal two distinct phase-space scales associated with the coercive growth of $P$ and $Q$, leading to anisotropic $L^p$--$L^q$ smoothing. As applications, we study nonlinear fractional heat equations associated with ${\mathcal{H}}_{P,Q}^s$. We prove local well-posedness in the supercritical Lebesgue range $p>\frac{n({\hat{I}}^2-1)}{2\ell s},$ derive a lower blow-up rate for finite-time blow-up solutions, and obtain critical small-data global existence. We further prove global well-posedness and exponential decay for small initial data in modulation spaces. These results extend the heat semigroup theory for harmonic and model anharmonic oscillators to a broad class of anisotropic polynomial Hamiltonians.

35 pages