Remarks on the blow-up for the Schrödinger equation with critical mass on a plane domain

Valeria Banica

Displaying similar documents to “Remarks on the blow-up for the Schrödinger equation with critical mass on a plane domain”

Remarks on the blow-up for the Schrödinger equation with critical mass on a plane domain

Valeria Banica (2004)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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In this paper we concentrate on the analysis of the critical mass blowing-up solutions for the cubic focusing Schrödinger equation with Dirichlet boundary conditions, posed on a plane domain. We bound the blow-up rate from below, for bounded and unbounded domains. If the blow-up occurs on the boundary, the blow-up rate is proved to grow faster than ${(T - t)}^{- 1}$ , the expected one. Moreover, we show that blow-up cannot occur on the boundary, under certain geometric conditions on the domain. ...

Global well-posedness and scattering for the mass-critical NLS

Benjamin Dodson (2011)

Journées Équations aux dérivées partielles

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Blow up dynamic and upper bound on the blow up rate for critical nonlinear Schrödinger equation

Frank Merle, Pierre Raphael (2002)

Journées équations aux dérivées partielles

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We consider the critical nonlinear Schrödinger equation $i u_{t} = - Δ u - {| u |}^{\frac{4}{N}} u$ with initial condition $u (0, x) = u_{0}$ in dimension $N$ . For $u_{0} \in H^{1}$ , local existence in time of solutions on an interval $[0, T)$ is known, and there exists finite time blow up solutions, that is $u_{0}$ such that ${lim}_{t \to T < + \infty} {| u_{x} (t) |}_{L^{2}} = + \infty$ . This is the smallest power in the nonlinearity for which blow up occurs, and is critical in this sense. The question we address is to understand the blow up dynamic. Even though there exists an explicit example of blow up solution and a class of initial...

Changing blow-up time in nonlinear Schrödinger equations

Rémi Carles (2003)

Journées équations aux dérivées partielles

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Solutions to nonlinear Schrödinger equations may blow up in finite time. We study the influence of the introduction of a potential on this phenomenon. For a linear potential (Stark effect), the blow-up time remains unchanged, but the location of the collapse is altered. The main part of our study concerns isotropic quadratic potentials. We show that the usual (confining) harmonic potential may anticipate the blow-up time, and always does when the power of the nonlinearity is $L^{2}$ -critical....

Blow-up on the boundary: a survey

Marek Fila, Ján Filo (1996)

Banach Center Publications

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Existence globale et diffusion pour l’équation de Schrödinger non linéaire répulsive cubique sur $m a t h b b R^{3}$ en dessous l’espace d’énergie

J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao (2002)

Journées équations aux dérivées partielles

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We sketch a proof of global existence and scattering for the defocusing cubic nonlinear Schrödinger equation in $H^{s} (ℝ^{3})$ for $s > \frac{4}{5}$ . The proof uses a new estimate of Morawetz type.

Energy Critical nonlinear Schrödinger equations in the presence of periodic geodesics

Sebastian Herr (2010)

Journées Équations aux dérivées partielles

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This is a report on recent progress concerning the global well-posedness problem for energy-critical nonlinear Schrödinger equations posed on specific Riemannian manifolds $M$ with small initial data in $H^{1} (M)$ . The results include small data GWP for the quintic NLS in the case of the $3 d$ flat rational torus $M = 𝕋^{3}$ and small data GWP for the corresponding cubic NLS in the cases $M = ℝ^{2} \times 𝕋^{2}$ and $M = ℝ^{3} \times 𝕋$ . The main ingredients are bi-linear and tri-linear refinements of Strichartz estimates which obey the critical scaling,...

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