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- W4287693399 abstract "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 $M^{p,q}$, $0< p,qleq infty$. We prove the complete range of fixed-time estimates for the semigroup $e^{-tH^beta}$ when acting on $M^{p,q}$, for every $0< p,qleq infty$, 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 $H^beta$. This 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 - hence in $M^{infty,1}$." @default.
- W4287693399 created "2022-07-26" @default.
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- W4287693399 date "2020-08-03" @default.
- W4287693399 modified "2023-09-30" @default.
- W4287693399 title "Phase space analysis of the Hermite semigroup and applications to nonlinear global well-posedness" @default.
- W4287693399 doi "https://doi.org/10.48550/arxiv.2008.01226" @default.
- W4287693399 hasPublicationYear "2020" @default.
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