The Cauchy problem for the vibrating plate equation in modulation spaces

J. Pseudo-Differ. Oper. Appl.

2021

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In this note we consider the nonlinear heat equation associated to the fractional Hermite operator $$H^\beta =(-\Delta +|x|^2)^\beta $$ H β = ( - Δ + | x | 2 ) β , $$0<\beta \le 1$$ 0 < β ≤ 1 . We show the local solvability of the related Cauchy problem in the framework of modulation spaces. The result is obtained by combining tools from microlocal and time-frequency analysis. As a byproduct, we compute the Gabor matrix of pseudodifferential operators with symbols in the Hörmander class $$S^m_{0,0}$$ S 0 , 0 m , $$m\in \mathbb {R}$$ m ∈ R .

Section: Introduction and Resultsmentioning

confidence: 99%

“…The tools employed follow the pattern of similar Cauchy problems studied for other equations such as the Schrödinger, wave and Klein-Gordon equations [1,2,8,12].…”

Section: Introduction and Resultsmentioning

confidence: 99%

On the local well-posedness of the nonlinear heat equation associated to the fractional Hermite operator in modulation spaces

J. Pseudo-Differ. Oper. Appl.

2021

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“…[22,26], has found important applications in Signal Processing and related problems in Numerical Analysis, see for example [8,42] and references therein. More recently, time-frequency methods have been applied to the study of the partial differential equations, in particular constant coefficient wave, Klein-Gordon, parabolic and Schrödinger equations [2,3,12,21,31,32,33,36,45,46], let us also refer to the survey [41] and the monograph [47]. The analysis of variable coefficient Schrödinger type equations was carried out in [4,9,10,13,14,15,16,19,43], see also [17,18] in the analytic category.…”

Section: Introductionmentioning

confidence: 99%

Gabor Analysis for Schrödinger Equations and Propagation of Singularities

Recent Trends in Operator Theory and Partial Differential Equations

Nicola

Rodino

2017

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Abstract. We consider the Schrödinger equationwhere the Hamiltonian a(z), z = (x, η), is assumed real-valued and smooth, with bounded derivatives |∂ α a(z)| ≤ C α , for every |α| ≥ 2, z ∈ R 2d . For such equation results are known concerning well-posedness of the Cauchy problem for initial data in L 2 (R d ) and local representation of the propagator e itH by means of Fourier integral operators.In the present paper we give a global expression for e itH in terms of Gabor analysis and we deduce boundedness in modulation spaces. Moreover, by using time-frequency techniques, we obtain a result of propagation of microsingularities for e itH .

“…Gabor frames turned out to be appropriate tools for many problems in time-frequency analysis, with relevant applications to signal processing and related issues in Numerical Analysis, see for example [7,43], and references there. More recently, attention has been addressed to the analysis of partial differential equations, especially the Schrödinger, wave and Klein-Gordon equations with constant coefficients [2,3,9,16,30,31,32,34,49,50,51,52]; see also the recent survey [40]. The main results and techniques are now also available in the monograph [53].…”

Section: Introductionmentioning

confidence: 99%

Gabor representations of evolution operators

Trans. Amer. Math. Soc.

Nicola

Rodino

2015

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Abstract. We perform a time-frequency analysis of Fourier multipliers and, more generally, pseudodifferential operators with symbols of Gevrey, analytic and ultraanalytic regularity. As an application we show that Gabor frames, which provide optimally sparse decompositions for Schrödinger-type propagators, reveal to be an even more efficient tool for representing solutions to a wide class of evolution operators with constant coefficients, including weakly hyperbolic and parabolic-type operators. Besides the class of operators, the main novelty of the paper is the proof of super-exponential (as opposite to super-polynomial) off-diagonal decay for the Gabor matrix representation.