Phase space analysis of the Hermite semigroup and applications to nonlinear global well-posedness

We study the Hermite operator H = −Δ + |x|^2 in Rd and its fractional powers H^β, β > 0 in phase space. Namely, we represent functions f via the so-called short-time Fourier, alias Fourier-Wigner or Bargmann transform V_g f (g being a fixed window function), and we measure their regularity and decay by means of mixed Lebesgue norms in phase space of V_g f, that is in terms of membership to modulation spaces M^{p,q}, 0 < p,q ≤ ∞. We prove the complete range of fixed-time estimates for the semigroup e−{tH^β} when acting on M^{p,q}, for every 0 < p,q ≤ ∞, exhibiting the optimal global-in-time decay as well as phase-space smoothing. As an application, we establish global well-posedness for the nonlinear heat equation for H^β with power-type nonlinearity (focusing or defocusing), with small initial data in modulation spaces or in Wiener amalgam spaces. We show that such a global solution exhibits the same optimal decay e^{−ct} as the solution of the corresponding linear equation, where c = d^β is the bottom of the spectrum of H^β. Global existence is in sharp contrast to what happens for the nonlinear focusing heat equation without potential, where blow-up in finite time always occurs for (even small) constant initial data (constant functions belong to M^{∞,1}).