×

zbMATH — the first resource for mathematics

The Cauchy problem for the nonlinear Schrödinger equation in \(H^ 1\). (English) Zbl 0696.35153
Summary: We consider the initial value problem for the nonlinear Schrödinger equation in \(H^ 1({\mathbb{R}}^ n)\). We establish local existence and uniqueness for a wide class of subcritical nonlinearities. The proofs make use of a truncation argument, space-time integrability properties of the linear equation, and a priori estimates derived from the conservation of energy. In particular, we do not need any differentiability property of the nonlinearity with respect to x.

MSC:
35Q99 Partial differential equations of mathematical physics and other areas of application
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
47J25 Iterative procedures involving nonlinear operators
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] BERGH,J, LÖFSTRÖM,J: Interpolation spaces. New York: Springer 1976 · Zbl 0344.46071
[2] BREZIS,H: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. Amsterdam: North-Holland Publ. 1973 · Zbl 0252.47055
[3] CAZENAVE,T., WEISSLER,F.B.: Some remarks on the nonlinear Schrödinger equation in the critical case. Proceedings of the Second Howard University Symposium on Nonlinear Semigroups, Partial Differential Equations, and Attractors. Washington, D.C., August 1987. Springer, to appear · Zbl 0694.35170
[4] CAZENAVE,T., WEISSLER,F.B.: Some remarks on the nonlinear Schrödinger equation in the subcritical case. Proceedings of the Symposium on Nonlinear Fields. Bielefeld, July 1987. Springer, to appear · Zbl 0699.35217
[5] CAZENAVE,T., WEISSLER,F.B.: The Cauchy problem for the critical nonlinear Schrödinger equation in Hs. To appear · Zbl 0706.35127
[6] GINIBRE,J., VELO,G.: On a class of nonlinear Schrödinger equations. J. Funct. Anal.32, 1–71 (1979) · Zbl 0396.35028
[7] GINIBRE, J., VELO, G.: On a class of nonlinear Schrödinger equations. Special theories in dimensions 1, 2 and 3. Ann. Inst. Henri Poincaré28, 287–316 (1978) · Zbl 0397.35012
[8] GINIBRE, J., VELO, G.: On a class of nonlinear Schrödinger equations with nonlocal interaction. Math. Z.170, 109–136 (1980) · Zbl 0415.35065
[9] GINIBRE, J., VELO, G.: The global Cauchy problem for the nonlinear Schrödinger equation revisited. Ann. Inst. Henri Poincaré, Analyse Non Linéaire2, 309–327 (1985) · Zbl 0586.35042
[10] KATO, T.: On nonlinear Schrödinger equations. Ann. Inst. Henri Poincaré, Physique Théorique46, 113–129 (1987)
[11] PAZY, A.: Semigroups of linear operators and applications to partial differential equations. Applied Math. Series44, New York: Springer 1983 · Zbl 0516.47023
[12] WEISSLER, F.B.: Semilinear evolution equations in Banach spaces. J. Funct. Anal.32, 277–296 (1979) · Zbl 0419.47031
[13] YAJIMA, K.: Existence of solutions for Schrödinger evolution equations. Comm. Math. Phys.110, 415–426 (1987) · Zbl 0638.35036
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.