On the existence of infinite energy solutions for nonlinear Schrödinger equations

Silva, Pablo Braz e; Ferreira, Lucas C. F.; Villamizar-Roa, Élder J.

doi:10.1090/s0002-9939-09-09773-1

Cited by 13 publications

(9 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…If this condition is verified, then (3.35) follows from Theorem 3.10. Furthermore, notice that v is self-similar because v 0 is homogeneous of degree −d/α (see, for instance, [CW1,BP,SFR]). The proof of the corollary is thus completed.…”

Section: Resultsmentioning

confidence: 99%

On the two-power nonlinear Schrödinger equation with non-local terms in Sobolev–Lorentz spaces

Barros

Ferreira

Pastor

2019

Nonlinear Differ. Equ. Appl.

View full text Add to dashboard Cite

We are concerned with the two-power nonlinear Schrödinger-type equations with non-local terms. We consider the framework of Sobolev-Lorentz spaces which contain singular functions with infinite-energy. Our results include global existence, scattering and decay properties in this singular setting with fractional regularity index. Solutions can be physically realized because they have finite local L 2 -mass. Moreover, we analyze the asymptotic stability of solutions and, although the equation has no scaling, show the existence of a class of solutions asymptotically self-similar w.r.t. the scaling of the single-power NLS-equation. Our results extend and complement those of [F. Weissler, ADE 2001], particularly because we are working in the larger setting of Sobolev-weak-L p spaces and considering non-local terms. The two nonlinearities of power-type and the generality of the non-local terms allow us to cover in a unified way a large number of dispersive equations and systems.

show abstract

Section: Resultsmentioning

confidence: 99%

On the two-power nonlinear Schrödinger equation with non-local terms in Sobolev–Lorentz spaces

Barros

Ferreira

Pastor

2019

Nonlinear Differ. Equ. Appl.

View full text Add to dashboard Cite

show abstract

“…), the local well-posedness for initial data in the modulation space strengthens the local well-posedness result in Ḣ1 (R d ). Previous results on infinite energy solutions to nonlinear Schrödinger equations are due to Braz e Silva et al [7] with initial data in weak L p -spaces. The results in [7] do not cover the energy critical equations though; see also [8].…”

Section: Introductionmentioning

confidence: 99%

Infinite-energy solutions to energy-critical nonlinear Schrödinger equations in modulation spaces

Schippa¹

2022

Preprint

View full text Add to dashboard Cite

We prove new well-posedness results for energy-critical nonlinear Schrödinger equations in modulation spaces, which are larger than the energy space. First, we remove the ε-derivative loss in L p -smoothing estimates for the linear Schrödinger equation, if p is larger than the Tomas-Stein exponent. Next, we show local well-posedness results for nonlinear Schrödinger equations in modulation spaces containing the scaling critical L 2 -based Sobolev space. The proof is carried out via bilinear refinements and adapted function spaces.

show abstract

“…The approach employed here follows the same spirit of the one used in [3,6] (see also [10]), where the authors proved global-in-time existence results for the corresponding NLS equation for all dimensions n ≥ 1 by considering small data in L r and weak-L r , respectively. However, our study shows that the presence of the delay provokes two differences in comparison with NLS equations, namely the restriction on the dimension n < 6 (see Remark 2.2 (b)) and the possibility of considering large data v 0 after making a dynamical rescaling over the delay-parameter μ.…”

Section: Introductionmentioning

confidence: 99%

Global existence for Schrödinger–Debye system for initial data with infinite 𝐿²-norm

Corcho

Ferreira

2015

Quart. Appl. Math.

Self Cite

View full text Add to dashboard Cite

In this paper we study global-in-time existence for the Cauchy problem associated to the Schrödinger-Debye system for a class of initial data with infinite L 2norm, namely weak-L p spaces. This model appears in nonlinear optics as a perturbation of the classical nonlinear Schrödinger equation (NLS). Our results exhibit differences between both models in that setting, e.g. the Debye perturbation imposes restrictions in the spatial dimension. We also analyze the asymptotic stability of the solutions.

show abstract

On the existence of infinite energy solutions for nonlinear Schrödinger equations

Abstract: We derive new results about existence and uniqueness of local and global solutions for the nonlinear Schrödinger equation, including self-similar solutions. Our analysis is performed in the framework of weak-L p spaces.

Cited by 13 publications

References 16 publications

On the two-power nonlinear Schrödinger equation with non-local terms in Sobolev–Lorentz spaces

On the two-power nonlinear Schrödinger equation with non-local terms in Sobolev–Lorentz spaces

Infinite-energy solutions to energy-critical nonlinear Schrödinger equations in modulation spaces

Global existence for Schrödinger–Debye system for initial data with infinite 𝐿²-norm

Contact Info

Product

Resources

About