Abstract
In this paper, we consider the solutions of the nonlinear Schrödinger equations ∂u/∂t−iΔu+|u|p u=f andu(x,0)=u 0(x), whereu is defined onR +×R 2. We prove the existence and uniqueness of global weak solutions of the above equations. Lastly, we consider the special case:p=2, and we obtain the strong solutions.
Similar content being viewed by others
References
J.L. Lions. Quelques Méthodes de Résolution des Problèm aux Limites Nonlinéaires. DONOD, GAUTHIER-VILLARS, Paris 1969.
Ding Xiaxi, Luo Peizhu and Li Yanyan. The Solvability of Nonlinear Parabolic Equation in Ba Space.J. of Central Teachers College (Natural Science Edition), 1985, 2: 1–9.
Li Congming. Global Properties of The Solutions of Nonlinear Schrödinger Equations on a Bounded Domain.J. Sys. Sci. & Math. Scis., 1984, 4(2): 160–164.
Li Daqian. Nonlinear Evolutional Equations. Science Press, Beijing, 1989.
Yao Jingqi. Time Decay of Solutions to a Nonlinear Schrödinger Equation in an Exterior Domain inR 2.Nonlinear Analysis, 1992, 19: 563–571.
M. Tsutsumi. On Smooth Solutions to the Initial-boundary Value Problem for the Nonlinear Schrödin -ger Equation in Two Space Dimensions.Nonlinear Analysis, 1989, 13: 1051–1056.
L. Segal. Nonlinear Semi-groups.Ann. Math., 1963, 78: 339–364.
H. Brézis and T. Gallouet. Nonlinear Schrödinger Evolution Equations.Nonlinear Analysis, 1980, 4: 677–681.
Yao Jingqi. The Solutions of One Type of Nonlinear Schrödinger Equations.Chinese Annals of Mathematics, 1986, 7A(4): 413–422 (in Chinese).
R. Temam. Navier-Stokes Equations. Elasevier Science Publishers B.V., Netherlands, 1985.
Author information
Authors and Affiliations
Additional information
An erratum to this article is available at http://dx.doi.org/10.1007/BF02683820.
Rights and permissions
About this article
Cite this article
Jiu, Q., Liu, J. Existence and uniqueness of global solutions of onR 2 . Acta Mathematicae Applicatae Sinica 13, 414–424 (1997). https://doi.org/10.1007/BF02009551
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02009551