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L$${}^ 2$$ concentration of blow-up solutions for the nonlinear Schrödinger equation with critical power nonlinearity. (English) Zbl 0722.35047
The authors consider solutions of the nonlinear Schrödinger equation $iu_ t+\Delta u=f(u),\quad (\underline x,t)\in {\mathbb{R}}^ n\times [0,T),$ where f has critical growth at infinity, i.e., $$f(z)\sim -| z|^{4/n}z$$ as $$z\to \infty$$. Assuming that u(t) blows up in the $$H^ 1$$ norm, as $$t\to T$$, they show that u(t) fails to have a strong $$L^ 2$$ limit as $$t\to T$$. Further, they show that if the initial data is spherically symmetric, then the origin is a blow-up point, i.e., there exists $$L^ 2$$ concentration at the origin.

##### MSC:
 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35B40 Asymptotic behavior of solutions to PDEs 35Q55 NLS equations (nonlinear Schrödinger equations)
##### Keywords:
critical power nonlinearity; blow-up point
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##### References:
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