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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.

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)
Full Text: DOI
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