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

Valeria Banica (2003)

Journées équations aux dérivées partielles

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 from below the blow-up rate 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 state that blow-up cannot occur on the boundary, under certain geometric conditions on the 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

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.

Remarks on the equatorial shallow water system

Chloé Mullaert (2010)

Annales de la faculté des sciences de Toulouse Mathématiques

This article recalls the results given by A. Dutrifoy, A. Majda and S. Schochet in [1] in which they prove an uniform estimate of the system as well as the convergence to a global solution of the long wave equations as the Froud number tends to zero. Then, we will prove the convergence with weaker hypothesis and show that the life span of the solutions tends to infinity as the Froud number tends to zero.

Remarks on the Fundamental Solution to Schrödinger Equation with Variable Coefficients

Kenichi Ito, Shu Nakamura (2012)

Annales de l’institut Fourier

We consider Schrödinger operators H on ℝ n with variable coefficients. Let H 0 = - 1 2 ▵ be the free Schrödinger operator and we suppose H is a “short-range” perturbation of H 0 . Then, under the nontrapping condition, we show that the time evolution operator: e - i t H can be written as a product of the free evolution operator e - i t H 0 and a Fourier integral operator W ( t ) which is associated to the canonical relation given by the classical mechanical scattering. We also prove a similar result for the wave operators. These results...

Remarks on the powers of elliptic operators.

Jan W. Cholewa, Tomasz Dlotko (2000)

Revista Matemática Complutense

Under natural regularity assumptions on the data the powers of regular elliptic boundary value problems (e.b.v.p.) are shown to be higher order regular e.b.v.p.. This result is used in description of the domains of fractional powers of elliptic operators which information is in order important in regularity considerations for solutions of semilinear parabolic equations. Presented approach allows to avoid C∞-smoothness assumption on the data that is typical in many references.

Remarks on the qualitative behavior of the undamped Klein-Gordon equation

Esquivel-Avila, Jorge A. (2017)

Proceedings of Equadiff 14

We present sufficient conditions on the initial data of an undamped Klein-Gordon equation in bounded domains with homogeneous Dirichlet boundary conditions to guarantee the blow up of weak solutions. Our methodology is extended to a class of evolution equations of second order in time. As an example, we consider a generalized Boussinesq equation. Our result is based on a careful analysis of a differential inequality. We compare our results with the ones in the literature.

Currently displaying 381 – 400 of 614