Moving boundary i.b.v.p.s for Schrödinger equations: solution bounds and related results
type: Item
instance of: scholarly article
title: Moving boundary i.b.v.p.s for Schrödinger equations: solution bounds and related results
zbMATH Open document ID: 1200.35339
author: J. N. Flavin
published in: Journal of Mathematical Analysis and Applications
publication date: 2010-09-01T00:00:00Z
review text: The paper is concerned with initial boundary value problems for Schrödinger equations in one spatial dimension, linear (\(iu_t=u_{xx}\)) and nonlinear (\(iu_t=u_{xx}-f(|u|^2)u\)), in a straight line region with prescribed, moving boundaries, upon which (time-dependent) Dirichlet conditions are specified. Bounds, in terms of data, are obtained for the \(L_{2}\) norm of the spatial derivative of the solutions, or for a measure related thereto: in the context of expanding boundaries, pointwise bounds for the solution may be inferred both in the linear case and in some nonlinear cases (e.g. the defocusing case). Asymptotic properties of the bounds for the aforementioned norm are discussed in the linear case. The methodology of the paper is based on a particular compact formula for the aforementioned norm of an arbitrary, complex-valued function whose values are assigned, as functions of time, on the assigned, moving boundaries of a straight line region.
reviewed by: Michael Perelmuter
Mathematics Subject Classification ID: 35R37
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 35B45
zbMATH DE Number: 5777330
zbMATH Keywords: solution bounds
zbMATH Keywords: moving boundary problems
zbMATH Keywords: Schrödinger equations
zbMATH Keywords: pointwise bounds
MaRDI profile type: Publication
full work available at URL: j.jmaa.2010.05.047
OpenAlex ID: W1991840636
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DOI: 10.1016/J.JMAA.2010.05.047
description: scientific article; zbMATH DE number 5777330