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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A remark on the blowing-up of solutions to the Cauchy problem for nonlinear Schrödinger equations


Author: O. Kavian
Journal: Trans. Amer. Math. Soc. 299 (1987), 193-203
MSC: Primary 35B30; Secondary 35B35, 35Q20
MathSciNet review: 869407
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Abstract: We consider solutions to $ i{u_t} = \Delta u + {\left\vert u \right\vert^{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\Vert {\nabla u(t)} \right\Vert _{{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\vert u \right\vert^{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\Vert {\nabla u(t)} \right\Vert _{{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.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1987-0869407-0
PII: S 0002-9947(1987)0869407-0
Article copyright: © Copyright 1987 American Mathematical Society