Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity. (English) Zbl 0707.35021

Merle, Frank

Commun. Math. Phys. 129, No. 2, 223-240 (1990).

The paper deals with the nonlinear Schrödinger equation \[ (1)\quad i \partial u/\partial t=-\Delta u-| u|^{4/N}u,\quad u(0,)=\phi (), \] where u: [0,T)\(\times {\mathbb{R}}^ N\to {\mathbb{C}}\). The author constructs for any given points \(x_ 1,x_ 2,...,x_ k\) in \({\mathbb{R}}^ N\) a solution u(t) of (1), which blows up in a finite time T at exactly \(x_ 1,x_ 2,...,x_ k\). He also describes the behaviour of the solution u(t) when t goes to T at the blow-up points and outside them.

Reviewer: V.Mustonen

MSC:

35B40	Asymptotic behavior of solutions to PDEs
35K60	Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35Q55	NLS equations (nonlinear Schrödinger equations)

Keywords:

blow-up; critical nonlinearity

References:

[1]	Berestycki, H., Lions, P. L., Peletier, L. A.: An ODE approach to the existence of positive solutions for semilinear problems in \(\mathbb{R}\) N . Ind. Univ. Math. J.30, 141–157 (1981) · Zbl 0522.35036
[2]	Berestycki, H., Lions, P. L.: Existence d’ondes solitaires dans des problèmes nonlinéaires du type Klein-Gordon, C. R. Paris287, 503–506 (1978);288, 395–398 (1979) · Zbl 0391.35055
[3]	Berestycki, H., Cazenave, T.: Instabilité des états stationnaies dans les équations de Schrödinger et de Klein-Gordon non-linéaires. C. R. Paris293, 489–492 (1981) · Zbl 0492.35010
[4]	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
[5]	Ginibre, J., Velo, G.: The global Cauchy problem for the nonlinear Schrödinger equation revisited, Ann. Inst. Henri Poincaré, Physique Théorique4, 309–327 (1985) · Zbl 0586.35042
[6]	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
[7]	Kato, T.: On nonlinear Schrödinger equations. Ann. Inst. Henri Poincaré. Phys. Théorique46, 113–129 (1987)
[8]	Landman, M., Papanicolaou, G. C., Sulem, C., Sulem, P. L.: Rate of blow-up for solutions of the nonlinear Schrödinger equation in critical dimension. Phys. Review A (to appear) · Zbl 0728.35116
[9]	Lernesurier, B., Papanicolaou, G., Sulem, C., Sulem, P. L.: The focusing singularity of the nonlinear Schrödinger equation, preprint · Zbl 0659.35020
[10]	Chen, X., Matano, H.: Convergence, asymptotic periodicity, and finite-point blow-up in one dimensional semilinear heat equations. J. Diff. Equ.78, 160–190 (1989) · Zbl 0692.35013
[11]	Merle, F.: Limit of the solution of the nonlinear Schrödinger equation at the blow-up time. J. Funct. Analysis (to appear) · Zbl 0681.35078
[12]	Merle, F.: Sur la dépendance continue de la solution de l’équation de Schrödinger non linéaire près du temps d’explosion. C. R. Paris16, 479–482 (1987) · Zbl 0619.35023
[13]	Merle, F., Tsutsumi, Y.:L 2-concentration of blow-up solutions for the nonlinear Schrödinger equation with the critical power nonlinearity, preprint · Zbl 0722.35047
[14]	Schoen, R. M.: The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation. Commun. Pure Appl. Math. V16,3, 317–392 (1988) · Zbl 0674.35027
[15]	Tsutsumi, Y.:L 2-solutions for nonlinear Schrödinger equations and nonlinear groups. Funk. Ekva.30, 115–125 (1987) · Zbl 0638.35021
[16]	Weinstein, M. I.: Nonlinear Schrödinger equations and sharp interpolation estimates. Commun. Math. Phys.87, 567–576 (1983) · Zbl 0527.35023
[17]	Weinstein, M. I.: On the structure and formation of singularities in solutions to the nonlinear dispersive evolution equations. commun. Partial Differential Equations11, 545–565 (1986) · Zbl 0596.35022
[18]	Zakharov, V. E., Sobolev, V. V., Synach, V. S.: Character of the singularity and stochastic phenomena in self-focusing, Zh. Eksp. Teor. Fiz., Pis’ma Red14, 390–393 (1971)