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

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
Full Text:

### 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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.