## Phase space analysis of the Hermite semigroup and applications to nonlinear global well-posedness.(English)Zbl 1476.35297

Summary: We study the Hermite operator $$H = - \Delta + | x |^2$$ in $$\mathbb{R}^d$$ and its fractional powers $$H^\beta$$, $$\beta > 0$$ 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 $$M^{p , q}$$, $$0 < p, q \leq \infty$$. We prove the complete range of fixed-time estimates for the semigroup $$e^{- t H^\beta}$$ when acting on $$M^{p , q}$$, for every $$0 < p, q \leq \infty$$, 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 $$H^\beta$$ 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 $$e^{- c t}$$ as the solution of the corresponding linear equation, where $$c = d^\beta$$ is the bottom of the spectrum of $$H^\beta$$. 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 $$M^{\infty , 1})$$.

### MSC:

 35R11 Fractional partial differential equations 35K15 Initial value problems for second-order parabolic equations 35K58 Semilinear parabolic equations 35S05 Pseudodifferential operators as generalizations of partial differential operators 42B35 Function spaces arising in harmonic analysis 47D06 One-parameter semigroups and linear evolution equations
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