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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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