Large-time behaviour of solutions to the dissipative nonlinear Schrödinger equation

Royal Society of Edinburgh - Proceedings A

Volumn 130, Issue 5, 2000, Pages 1029-1043

  Hayashi, N ^a   Kaikina, E I ^b   Naumkin, P I ^c  

^a TOKYO UNIVERSITY OF SCIENCE   (Japan)

^b INSTITUTO TECNOLÓGICO DE MORELIA   (Mexico)

^c UNIVERSIDAD MICHOACANA DE SAN NICOLÁS DE HIDALGO   (Mexico)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 23044522241     PISSN: 03082105     EISSN: None     Source Type: Journal    
DOI: 10.1017/s0308210500000561     Document Type: Article

References (25)

1. C. J. Amick, J. L. Bona and M. E. Schonbek. Decay of solutions of some nonlinear wave equations. J. Diff. Eqns 81 (1989), 1-49.
2. P. Biler. Asymptotic behavior in time of solutions to some equations generalizing the Korteweg-de Vries-Burgers equation. Bull. Polish Acad. Sci. Math. 32 (1984), 275-282.
3. J. L. Bona and L. Luo. More results on the decay of the solutions to nonlinear, dispersive wave equations. Discrete Continuous Dynamical Sys. 1 (1995), 151-193.
4. J. L. Bona, K. S. Promislow and C. E. Wayne. Higher-order asymptotics of decaying solutions of some nonlinear, dispersive, dissipative wave equations. Nonlinearity 8 (1995), 1179-1206.
5. D. B. Dix. Temporal asymptotic behavior of solutions of the Benjamin-Ono-Burgers equation. J. Diff. Eqns 90 (1991), 238-287.
6. D. B. Dix. The dissipation of nonlinear dispersive waves: the case of asymptotically weak nonlinearity. Commun. PDE 17 (1992), 1665-1693.
7. D. B. Dix. Large-time behavior of solutions of linear dispersive equations. Lecture Notes in Mathematics, vol. 1668 (Springer, 1997).
8. M. Escobedo and O. Kavian. Asymptotic behavior of positive solutions of a nonlinear heat equation. Houston J. Math. 13 (1987), 39-50.
9. N. J. Funct. Analysis 100 (1991), 119-161.
10. V. A. Galaktionov and J. L. Vazquez. Asymptotic behavior of nonlinear parabolic equations with critical exponents. A dynamical systems approach. J. Funct. Analysis 100 (1991), 435-462.
11. V. A. Galaktionov, S. P. Kurdyumov and A. A. Samarskij. On asymptotic eigenfunctions of the Cauchy problem for a nonlinear parabolic equation. Math. USSR Sbornik 126 (1985), 435-472.
12. Y. Giga and T. Kambe. Large time behavior of the vorticity of two-dimensional viscous flow and its application to vortex formation. Commun. Math. Phys. 117 (1988), 549-568.
13. J. Ginibre and G. Velo. The Cauchy problem in local spaces for the complex Ginzburg-Landau equation. II. Contraction methods. Commun. Math. Phys. 187 (1997), 45-79.
14. N. J. Diff. Eqns 53 (1984), 258-276.
15. N. Hayashi, E. I. Kaikina and P. I. Naumkin. On the scattering theory for the cubic nonlinear Schrödinger and Hartree type equations in one space dimension. Hokkaido Math. J. 27 (1998), 651-667.
16. S. Kamin and L. A. Peletier. Large time behavior of solutions of the porous media equation with absorption. Israel J. Math. 55 (1986), 129-146.
17. T. P. Liu. Nonlinear stability of shock waves for viscous conservation laws. Mem. Am. Math. Soc. 56 (1985), 1-108.
18. P. I. Naumkin and I. A. Shishmarev. Asymptotic relationship as t → ∞ between solutions to some nonlinear equations. I, II. Diff. Eqns 30 (1994), 806-814; 1329-1340.
19. P. I. Naumkin and I. A. Shishmarev. Nonlinear nonlocal equations in the theory of waves. Translations of Mathematical Monographs, vol. 133 (Providence, RI: American Mathematical Society, 1994).
20. P. I. Naumkin and I. A. Shishmarev. Asymptotic representation of surface waves in the form of two traveling Burgers waves. Funct. Analysis Appl. 29 (1995), 168-179.
21. M. E. Schonbek. Uniform decay rates for parabolic conservation laws. Nonlinear Analysis TMA 10 (1986), 943-953.
22. M. E. Schonbek. Lower bounds of rates of decay for solutions to the Navier-Stokes equations. J. Am. Math. Soc. 4 (1991), 423-449.
23. M. I. Vishik. Asymptotic behavior of solutions of evolutionary equations (Cambridge University Press, 1992).
24. L. Zhang. Decay estimates for the solutions of some nonlinear evolution equations. J. Diff. Eqns 116 (1995), 31-58.
25. E. Zuazua. A dynamical system approach to the self-similar large time behavior in scalar convection-diffusion equations. J. Diff. Eqns 108 (1994), 1-35.