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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)
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[3] Ginibre, J.; Velo, G., On a class of nonlinear Schrödinger equations. I. The Cauchy problem, J. Funct. Anal., 32, 1-32 (1979) · Zbl 0396.35028
[4] Ginibre, J.; Velo, G., The global Cauchy problem for the nonlinear Schrödinger equation revisited, Ann. Inst. H. Poincaré Phys. Théor., 4, 309-327 (1985) · Zbl 0586.35042
[5] Glassey, R. T., On the blowing-up of solutions to the Cauchy problem for the nonlinear Schrödinger equation, J. Math. Phys., 18, 1794-1797 (1977) · Zbl 0372.35009
[6] Kato, T., On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 46, 113-129 (1987) · Zbl 0632.35038
[8] Merle, F., Limit of the solution of a nonlinear Schrödinger equation at the blow-up time, J. Funct. Anal., 84, 201-214 (1989) · Zbl 0681.35078
[10] Nirenberg, L., Remarks on strongly elliptic partial differential equations, Comm. Pure Appl. Math., 8, 648-674 (1955)
[11] Strauss, W. A., Everywhere defined wave operators, (Nonlinear Evolution Equations (1978), Academic Press: Academic Press San Diego, CA/New York), 85-102
[13] Tsutsumi, Y., \(L^2\)-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac., 30, 115-125 (1987) · Zbl 0638.35021
[14] Weinstein, M. I., Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87, 567-576 (1983) · Zbl 0527.35023
[15] Weinstein, M. I., On the structure and formation of singularities in solutions to the non-linear dispersive evolution equations, Comm. Partial Differential Equations, 11, 545-565 (1986) · Zbl 0596.35022
[16] Zakharov, V. E.; Sobolev, V. V.; Synackh, V. S., Character of the singularity and stochastic phenomena in self-focusing, Zh. Éksper. Teoret. Fiz., Pis’ma Red, 14, 390-393 (1971)
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