June  2013, 6(3): 761-770. doi: 10.3934/dcdss.2013.6.761

Non-hamiltonian Schrödinger systems

1. 

Dipartimento di Matematica, Università degli Studi di Bari, Via Orabona 4, 70125 Bari, Italy

2. 

Dipartimento di Matematica, Sapienza Università di Roma, Piazzale A. Moro 5, 00185 Roma

Received  April 2010 Revised  November 2010 Published  December 2012

In this paper we study local and global in time existence for the Cauchy Problem of some semilinear Schrödinger systems. In particular we do not assume that the nonlinear term guarantees conservation of charge or energy.
Citation: Sandra Lucente, Eugenio Montefusco. Non-hamiltonian Schrödinger systems. Discrete and Continuous Dynamical Systems - S, 2013, 6 (3) : 761-770. doi: 10.3934/dcdss.2013.6.761
References:
[1]

T. Cazenave, "Semilinear Schrödinger Equations," Courant Lecture Notes in Mathematics, 10, American Mathematical Society, Providence, 2003.

[2]

D. G. de Figueiredo and Y. Jianfu, Decay, symmetry and existence of solutions of semilinear elliptic systems, Nonlinear Anal., Theory Methods Appl., 33 (1998), 211-234. doi: 10.1016/S0362-546X(97)00548-8.

[3]

D. Del Santo, V. Georgiev and E. Mitidieri, Global existence of solutions and formation of singularities for a class of hyperbolic systems, in "Geometric optics and relates topics" (eds. F. Colombini and N. Lerner), Progress in Nonlinear Differential Equations and Their Applications, 32, Birkhäuser, (1997), 117-140.

[4]

M. Escobedo and M. A. Herrero, A uniqueness result for a semilinear reaction-diffusion system, Proc. Amer. Math. Soc., 112 (1991), 175-185. doi: 10.2307/2048495.

[5]

L. Fanelli, S. Lucente and E. Montefusco, Semilinear Hamiltonian Schrödinger systems, Int. J. Dyn. Syst. Differ. Equ., 3 (2011), 401-422. doi: 10.1504/IJDSDE.2011.042938.

[6]

L. Fanelli and E. Montefusco, On the blow-up threshold for weakly coupled nonlinear Schrödinger equations, J. Phys. A, 40 (2007), 14139-14150. doi: 10.1088/1751-8113/40/47/007.

[7]

F. Merle, Determination of blow-up solutions with minimal mass for nonlinear Schr\"odinger equation with critical power, Duke Math. J., 69 (1993), 427-453. doi: 10.1215/S0012-7094-93-06919-0.

[8]

E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $\mathbf R^\mathbb N$, Differ. Integral Equ., 9 (1996), 465-479.

[9]

J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden system, Differ. Integral Equ., 9 (1996), 635-653.

[10]

T. Tao, Nonlinear dispersive equations: Local and global analysis, CBMS Regional Conference Series in Mathematics, 106 (2006).

show all references

References:
[1]

T. Cazenave, "Semilinear Schrödinger Equations," Courant Lecture Notes in Mathematics, 10, American Mathematical Society, Providence, 2003.

[2]

D. G. de Figueiredo and Y. Jianfu, Decay, symmetry and existence of solutions of semilinear elliptic systems, Nonlinear Anal., Theory Methods Appl., 33 (1998), 211-234. doi: 10.1016/S0362-546X(97)00548-8.

[3]

D. Del Santo, V. Georgiev and E. Mitidieri, Global existence of solutions and formation of singularities for a class of hyperbolic systems, in "Geometric optics and relates topics" (eds. F. Colombini and N. Lerner), Progress in Nonlinear Differential Equations and Their Applications, 32, Birkhäuser, (1997), 117-140.

[4]

M. Escobedo and M. A. Herrero, A uniqueness result for a semilinear reaction-diffusion system, Proc. Amer. Math. Soc., 112 (1991), 175-185. doi: 10.2307/2048495.

[5]

L. Fanelli, S. Lucente and E. Montefusco, Semilinear Hamiltonian Schrödinger systems, Int. J. Dyn. Syst. Differ. Equ., 3 (2011), 401-422. doi: 10.1504/IJDSDE.2011.042938.

[6]

L. Fanelli and E. Montefusco, On the blow-up threshold for weakly coupled nonlinear Schrödinger equations, J. Phys. A, 40 (2007), 14139-14150. doi: 10.1088/1751-8113/40/47/007.

[7]

F. Merle, Determination of blow-up solutions with minimal mass for nonlinear Schr\"odinger equation with critical power, Duke Math. J., 69 (1993), 427-453. doi: 10.1215/S0012-7094-93-06919-0.

[8]

E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $\mathbf R^\mathbb N$, Differ. Integral Equ., 9 (1996), 465-479.

[9]

J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden system, Differ. Integral Equ., 9 (1996), 635-653.

[10]

T. Tao, Nonlinear dispersive equations: Local and global analysis, CBMS Regional Conference Series in Mathematics, 106 (2006).

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