A remark on the blowing-up of solutions to the Cauchy problem for nonlinear Schrödinger equations
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- by O. Kavian
- Trans. Amer. Math. Soc. 299 (1987), 193-203
- DOI: https://doi.org/10.1090/S0002-9947-1987-0869407-0
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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
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Bibliographic 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