Weighted energy estimates for p-evolution equations in SG classes

Ascanelli, Alessia; Cappiello, Marco

doi:10.1007/s00028-015-0274-6

Cited by 13 publications

(13 citation statements)

References 24 publications

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“…In the present paper we want to adapt the techniques used in [9] to study the problem (1.1) in arbitrary dimension n ≥ 1 and assuming a weaker condition on the behavior at infinity of the imaginary parts of the coefficients of the lower order terms, namely…”

Section: Introductionmentioning

confidence: 99%

Schrödinger-type equations in Gelfand-Shilov spaces

Ascanelli

Cappiello

2019

Journal de Mathématiques Pures et Appliquées

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We study the initial value problem for Schrödinger-type equations with initial data presenting a certain Gevrey regularity and an exponential behavior at infinity. We assume the lower order terms of the Schrödinger operator depending on (t, x) ∈ [0, T ] × R n and complex valued. Under a suitable decay condition as |x| → ∞ on the imaginary part of the first order term and an algebraic growth assumption on the real part, we derive global energy estimates in suitable Sobolev spaces of infinite order and prove a well posedness result in Gelfand-Shilov type spaces. We also discuss by examples the sharpness of the result.RÉSUMÉ. Nous étudions le problème de Cauchy pour des équations de Schrödinger avec des données initiales présentent une régularité de type Gevrey et un comportement exponentiel a l'infini. Nous supposons les coefficients des terms d'ordre inferieur dépendant de (t, x) ∈ [0, T ]×R n et à valeurs complexes. Sous une condition de décroissance convenable pour |x| → ∞ pour la partie imaginaire du terme du premier ordre et une hypotèse de croissance algébrique sur la partie réelle, nous obtenons des estimations de l'énergie globales dans des espaces de Sobolev d'ordre infini et prouvons que le problème de Cauchy est bien posé dans des espaces de type Gelfand-Shilov. Nous discutons également par des exemples de l'optimalité du résultat.

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Section: Introductionmentioning

confidence: 99%

Schrödinger-type equations in Gelfand-Shilov spaces

Ascanelli

Cappiello

2019

Journal de Mathématiques Pures et Appliquées

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“…This phenomenon is quite common in the theory of hyperbolic partial differential equations with SG type coefficients, see [2,4,5]. We remark that in the symmetric case = 1 the Cauchy problem (5.1) turns out to be well-posed also in H r, (R n ).…”

Section: Fundamental Solution To Hyperbolic Systems In Sg Classesmentioning

confidence: 64%

“…Concerning applications to SG hyperbolic problems and propagation of singularities, see, e.g., Ascanelli and Cappiello [2][3][4], Cappiello [8], Coriasco et al [13], Coriasco and Maniccia [14]. Concerning applications to anisotropic evolution equations of Schrödinger type see, e.g., Ascanelli and Cappiello [5].…”

Section: Introductionmentioning

confidence: 99%

Fourier integral operators algebra and fundamental solutions to hyperbolic systems with polynomially bounded coefficients on $$\mathbb {R}^n$$ R n

Ascanelli¹,

Coriasco²

2015

J. Pseudo-Differ. Oper. Appl.

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We study the composition of an arbitrary number of Fourier integral operators A j , j = 1, . . . , M, M ≥ 2, defined through symbols belonging to the so-called SG classes. We give conditions ensuring that the composition A 1 • · · · • A M of such operators still belongs to the same class. Through this, we are then able to show well-posedness in weighted Sobolev spaces for first order hyperbolic systems of partial differential equations in SG classes, by constructing the associated fundamental solutions. These results expand the existing theory for the study of the properties "at infinity" of the solutions to hyperbolic Cauchy problems on R n with polynomially bounded coefficients.

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“…It is well known, see [7], that the condition (3) allows to prove that if the coefficients of S belong to the Gevrey space G s 0 , s 0 < 1 1−σ , then the Cauchy problem (1) is globally in time well-posed in Gevrey spaces G s for s 0 ≤ s < 1 1−σ . In the critical case s = 1 1−σ , one has local in time well-posedness of the Cauchy problem (1), only.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…In the critical case s = 1 1−σ , one has local in time well-posedness of the Cauchy problem (1), only. The Cauchy problem (1) is not well-posed, neither in H ∞ nor in G s for s > 1 1−σ . Here, we refer to the necessity results from [4] and [3].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The interplay between decay of the data and regularity of the solution in Schrödinger equations

Ascanelli

Cicognani

Reissig

2019

Annali di Matematica

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We deal with the following Cauchy problem for a Schrödinger equation: D t u − Δu + n j=1 a j (t, x)D x j u + b(t, x)u = 0, u(0, x) = g(x). We assume a decay condition of type |x| −σ , σ ∈ (0, 1), on the imaginary part of the coefficients a j of the convection term for large values of |x|. This condition is known to produce a unique solution with Gevrey regularity of index s ≥ 1 and loss of an infinite number of derivatives with respect to the data for every s ≤ 1 1−σ. In this paper, we consider the case s > 1 1−σ , where, in general, Gevrey ill-posedness holds. We explain how the space where a unique solution exists depends on the decay and regularity of an initial data in H m , m ≥ 0. As a by-product, we show that a decay condition on data in H m produces a solution with (at least locally) the same regularity as the data, but with an expected different behavior as |x| → ∞.

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Weighted energy estimates for p-evolution equations in SG classes

Cited by 13 publications

References 24 publications

Schrödinger-type equations in Gelfand-Shilov spaces

Schrödinger-type equations in Gelfand-Shilov spaces

Fourier integral operators algebra and fundamental solutions to hyperbolic systems with polynomially bounded coefficients on $$\mathbb {R}^n$$ R n

The interplay between decay of the data and regularity of the solution in Schrödinger equations

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