Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

A remark on the blowing-up of solutions to the Cauchy problem for nonlinear Schrödinger equations
HTML articles powered by AMS MathViewer

by O. Kavian PDF
Trans. Amer. Math. Soc. 299 (1987), 193-203 Request permission

Abstract:

We consider solutions to $i{u_t} = \Delta u + {\left | u \right |^{p - 1}}u$, $u(0) = {u_0}$, where $x$ belongs to a smooth domain $\Omega \subset {{\mathbf {R}}^N}$, and we prove that under suitable conditions on $p$, $N$ and ${u_0} \in {H^2}(\Omega ) \cap H_0^1(\Omega )$, ${\left \| {\nabla u(t)} \right \|_{{L^2}}}$ blows up in finite time. The range of $p$’s for which blowing-up occurs depends on whether $\Omega$ is starshaped or not. Examples of blowing-up under Neuman or periodic boundary conditions are given. On considère des solutions de $i{u_t} = \Delta u + {\left | u \right |^{p - 1}}u$, $u(0) = {u_0}$, où la variable d’espace $x$ appartient à un domaine régulier $\Omega \subset {{\mathbf {R}}^N}$, et on prouve que sous des conditions adéquates sur $p$, $N$ et ${u_0} \in {H^2}(\Omega ) \cap H_0^1(\Omega )$, ${\left \| {\nabla u(t)} \right \|_{{L^2}}}$ explose en temps fini. Les valeurs de $p$ pour lesquelles l’explosion a lieu dépend de la forme de l’ouvert $\Omega$ (en fait $\Omega$ étoilé ou non). On donne également des exemples d’explosion sous des conditions de Neuman ou périodiques au bord.
References
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC: 35B30, 35B35, 35Q20
  • Retrieve articles in all journals with MSC: 35B30, 35B35, 35Q20
Additional Information
  • © Copyright 1987 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 299 (1987), 193-203
  • MSC: Primary 35B30; Secondary 35B35, 35Q20
  • DOI: https://doi.org/10.1090/S0002-9947-1987-0869407-0
  • MathSciNet review: 869407