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

Abstract

We study the Hermite operator in and its fractional powers , in phase space. Namely, we represent functions f via the so-called short-time Fourier, alias Fourier-Wigner or Bargmann transform (g being a fixed window function), and we measure their regularity and decay by means of mixed Lebesgue norms in phase space of , that is in terms of membership to modulation spaces , . We prove the complete range of fixed-time estimates for the semigroup when acting on , for every , 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 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 as the solution of the corresponding linear equation, where is the bottom of the spectrum of . 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 ).