Remarks on global existence and blowup for damped nonlinear Schrödinger equations

Discrete and Continuous Dynamical Systems

Volumn 23, Issue 4, 2009, Pages 1313-1325

  Ohta, Masahito ^a   Todorova, Grozdena ^b  

^a SAITAMA UNIVERSITY   (Japan)

^b UNIVERSITY OF TENNESSEE   (United States)

Author keywords

Blowup; Damped nonlinear Schr dinger equation; Global existence

Indexed keywords

EID: 62649111158     PISSN: 10780947     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcds.2009.23.1313     Document Type: Conference Paper

References (31)

1. H. Berestycki and T. Cazenave, Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires, C. R. Acad. Sci. Paris Sér. I Math., 293 (1981), 489-492.
2. T. Cazenave, "Semilinear Schrödinger Equations," Courant Lecture Notes in Mathematics, 10, New York University Courant Institute of Mathematical Sciences, New York, 2003.
3. T. Cazenave and F. B. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation, Comm. Math. Phys., 147 (1992), 75-100.
4. G. Fibich, Self-focusing in the damped nonlinear Schrödinger equation, SIAM J. Appl. Math., 61 (2001), 1680-1705.
5. J.-M. Ghidaglia, Finite-dimensional behavior for weakly damped driven Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 5 (1988), 365-405.
6. J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case, J. Funct. Anal., 32 (1979), 1-32.
7. J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Schrödinger equation revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 309-327.
8. L. Glangetas and F. Merle, Concentration properties of blow-up solutions and instability results for Zakharov equation in dimension two. II, Comm. Math. Phys., 160 (1994), 349-389.
9. R. T. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys., 18 (1977), 1794-1797.
10. M. V. Goldman, K. Rypdal and B. Hafizi, Dimensionality and dissipation in Langmuir collapse, Phys. Fluids, 23 (1980), 945-955.
11. 2, Adv. Differential Equations, 3 (1998), 337-360.
12. T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 46 (1987), 113-129.
13. n, Arch. Rational Mech. Anal., 105 (1989), 243-266.
14. N, N ≤ 3, NoDEA Nonlinear Differential Equations Appl., 2 (1995), 357-369.
15. 2 concentration of blow-up solutions for the nonlinear Schrödinger equation with critical power nonlinearity, J. Differential Equations, 84 (1990), 205-214.
16. H. Nawa, "Mass concentration" phenomenon for the nonlinear Schrödinger equation with the critical power nonlinearity. II, Kodai Math. J., 13 (1990), 333-348.
17. H. Nawa, "Mass concentration" phenomenon for the nonlinear Schrödinger equation with the critical power nonlinearity, Funkcial. Ekvac., 35 (1992), 1-18.
18. H. Nawa, Asymptotic profiles of blow-up solutions of the nonlinear Schrödinger equation with critical power nonlinearity, J. Math. Soc. Japan, 46 (1994), 557-586.
19. M. Ohta and G. Todorova, Strong instability of standing waves for nonlinear Klein-Gordon equations, Discrete Contin. Dyn. Syst., 12 (2005), 315-322.
20. T. Passot, C. Sulem and P. L. Sulem, Linear versus nonlinear dissipation for critical NLS equation, Phys. D, 203 (2005), 167-184.
21. W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.
22. R. S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., 44 (1977), 705-714.
23. C. Sulem and P.-L. Sulem, "The Nonlinear Schrödinger Equation," Self-focusing and wave collapse, Applied Mathematical Sciences, 139, Springer-Verlag, New York, 1999.
24. M. Tsutsumi, Nonexistence and instability of solutions of nonlinear Schrödinger equations, unpublished manuscript.
25. M. Tsutsumi, Nonexistence of global solutions to the Cauchy problem for the damped non-linear Schrödinger equations, SIAM J. Math. Anal., 15 (1984), 357-366.
26. M. Tsutsumi, On global solutions to ihe initial-boundary value problem for the damped non-linear Schrödinger equations, J. Math. Anal. Appl., 146 (1990), 328-341.
27. 2-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac., 30 (1987), 115-125.
28. 2 concentration of blow-up solutions for the nonlinear Schrödinger equation with critical power, Nonlinear Anal., 15 (1990), 719-724.
29. M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1982/83), 567-576.
30. M. I. Weinstein, On the structure and formation of singularities in solutions to nonlinear dispersive evolution equations, Comm. Partial Differential Equations, 11 (1986), 545-565.
31. M. I. Weinstein, The nonlinear Schrödinger equation - singularity formation, stability and dispersion, in "The Connection Between Infinite-Dimensional and Finite-Dimensional Dynamical Systems" (Boulder, CO, 1987), Contemp. Math., 99, Amer. Math. Soc., Providence, RI, (1989), 213-232.