Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-09T16:17:08.879Z Has data issue: false hasContentIssue false
  • English
  • Français

L∞C(ℝn)-decay of classical solutions for nonlinear Schrödinger equations*

Published online by Cambridge University Press:  14 November 2011

Nakao Hayashi  and
Masayoshi Tsutsumi
Nakao Hayashi
Affiliation:
Department of Applied Physics, Waseda University, Tokyo 160, Japan
Masayoshi Tsutsumi
Affiliation:
Department of Applied Physics, Waseda University, Tokyo 160, Japan
Get access Rights & Permissions [Opens in a new window]

Synopsis

We study the initial value problem for the nonlinear Schrödinger equation

Under suitable regularity assumptions on f and ø and growth and sign conditions on f, it is shown that the maximum norms of solutions to (*) decay as t→² ∞ at the same rate as that of solutions to the free Schrödinger equation.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 104 , Issue 3-4 , 1986 , pp. 309 - 327
Copyright
Copyright © Royal Society of Edinburgh 1986

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Baillon, J. B., Cazenave, T. and Figueira, M.. Equation de Schrodinger nonlineaire. C.R. Acad. Sci. Paris Sér. I. Math. 284 (1977), 869–872.Google Scholar
2Barab, J. E.. Nonexistence of asymptotic free solutions for a nonlinear Schrödinger equation. J. Math. Phys. 25 (1984), 3270–3273.Google Scholar
3Bergh, J. and Löfström, J.. Interpolation Spaces (Berlin, Heidelberg, New York: Springer, 1976).Google Scholar
4Brenner, P. and Wahl, W. von. Global classical solutions of nonlinear wave equations. Math. Z. 176 (1981), 87–121.Google Scholar
5Cazenave, T.. Equations de Schrödinger nonlineares en dimension deux. Proc. Roy. Soc. Edinburgh Sect. A 84 (1979), 327–346.Google Scholar
6Dong, C. C. and Li, S.. On the initial value problem for a nonlinear Schrödinger equation. J.Differential Equations 42 (1981), 353–365.Google Scholar
7Friedman, A.. Partial Differential Equations (New York: Holt, Rinehart and Winston, 1969).Google Scholar
8Ginibre, J. and Velo, G.. On a class of nonlinear Schrödinger equation I, II. J. Fund. Anal. 32 (1979), 1–32, 33–71; III, Ann. Inst. H. Poincari Sect. A. (N.S.) 28 (1978), 287–316.Google Scholar
9Ginibre, J. and Velo, G.. The global Cauchy problem for the nonlinear Schrödinger equation revisited. Ann. Inst. H. Poincaré Sect. A (N.S.) 2 (1985), 309–327.Google Scholar
10Glassey, R. and Pecher, H.. Time decay for nonlinear wave equations in two space dimensions. Manuscripta Math. 38 (1982), 387–400.CrossRefGoogle Scholar
11Hayashi, N.. Classical solutions of nonlinear Schrödinger equations. Manuscripta Math. 55 (1986), 171–190.CrossRefGoogle Scholar
12Hayashi, N., Nakamitsu, K. and Tsutsumi, M.. On solutions of the initial value problem for the nonlinear Schrödinger equations in one space dimension. Math. Z. 192 (1986), 637–650.CrossRefGoogle Scholar
13Hayashi, N., Nakamitsu, K. and Tsutsumi, M.. On solutions of the initial value problem for the nonlinear Schrödinger equations. Submitted (1985).Google Scholar
14Kadekawa, S.. Ph.D. Thesis, Indiana University, 1980.Google Scholar
15Kato, T.. Quasilinear equations of evolutions with applications to partial differential equations. Lecture Notes in Mathematics 448, pp. 25–70 (Berlin: Springer, 1975).Google Scholar
16Klainerman, S.. Uniform decay estimates and the Lorentz invariance of the classical wave equation. Comm. Pure Appl. Math. 38 (1985), 321–332.Google Scholar
17Klainerman, S. and Ponce, G.. Global, small amplitude solutions to nonlinear evolution equations. Comm. Pure Appl. Math. 36 (1983), 133–141.Google Scholar
18Lin, J. E. and Strauss, W.. Decay and scattering of solutions of a nonlinear Schrödinger equation. J. Fund. Anal. 30 (1978), 245–263.CrossRefGoogle Scholar
19Mizohata, S.. Theory of Partial Differential Equations (Cambridge: Cambridge University Press, 1973).Google Scholar
20Pecher, H.. Decay of solutions of nonlinear wave equations in three space dimensions. J. Fund. Anal. 46 (1982), 221–229.CrossRefGoogle Scholar
21Pecher, H.. Decay and asymptotics for higher dimensional nonlinear wave equations. J. Differential Equations 46 (1982), 103–151.CrossRefGoogle Scholar
22Pecher, H. and Wahl, W. von. Time dependent nonlinear Schrödinger equations. Manuscripta Math. 27 (1979), 125–157.CrossRefGoogle Scholar
23Reed, M.. Abstract nonlinear wave equations (Berlin-Heidelberg-New York: Springer, 1976).Google Scholar
24Segal, I. E.. Nonlinear semigroups. Ann. of Math. 78 (1963), 339–364.Google Scholar
25Shatah, J.. Global existence of small solutions to nonlinear evolution equations. J. Differential Equations 46 (1982), 409–425.CrossRefGoogle Scholar
26Strauss, W.. The nonlinear Schrödinger equation. In Proceedings of the International Symposium on Continuum Mechanics and Partial Differential Equations (Rio de Janeiro 1977), pp. 452–465 (Amsterdam, New York, Oxford: North Holland, 1978).Google Scholar
27Strauss, W.. Nonlinear scattering theory at low energy. J. Fund. Anal. 41 (1981), 110–133; 43 (1981), 281–293.Google Scholar
28Tsutsumi, M. and Hayashi, N.. Classical solutions of nonlinear Schrödinger equations in higher dimensions. Math. Z. 177 (1981), 217–234.Google Scholar
29Tsutsumi, Y.. Scattering problem for nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Sect. A. (N.S.) (to appear).Google Scholar
30Tsutsumi, Y and Yajima, K.. The asymptotic behavior of nonlinear Schrödinger equations. Bull. Amer. Math. Soc. (N.S.) 11 (1984), 186–188.CrossRefGoogle Scholar