2026/07/20 by Julio Delgado, Vishvesh Kumar, Shyam Swarup Mondal
#math.FA
We investigate fractional heat semigroups generated by a class of anharmonic oscillators on \mathbb Rn of the form \mathcal HP,Q=Q(D)+P(x), where P∈\mathcal P2k and Q∈\mathcal P2ℓ 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 HP,Qs, s>0, are pseudo-differential operators with symbols in adapted classes ΣP,Q2s. We prove fixed-time decay estimates for the fractional anharmonic heat semigroup e^-t\mathcal HP,Qs on both Lebesgue and modulation spaces. In the Lebesgue setting, we establish sharp Lp--Lq estimates for the full range 1≤ p,q≤∞. For large time, the decay is exponential and governed by the smallest eigenvalue λ0 of \mathcal HP,Q, namely through the factor e-tλ0s, while for small time the estimates reveal two distinct phase-space scales associated with the coercive growth of P and Q, leading to anisotropic Lp--Lq smoothing. As applications, we study nonlinear fractional heat equations associated with \mathcal HP,Qs. We prove local well-posedness in the supercritical Lebesgue range p>(n(β-1))/(2ℓ 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.