, Volume 61, Issue 4, pp 477-494

The Cauchy problem for the nonlinear Schrödinger equation in H1

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Abstract

We consider the initial value problem for the nonlinear Schrödinger equation in H1(Rn). 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, anda priori estimates derived from the conservation of energy. In particular, we do not need any differentiability property of the nonlinearity with respect to x.

Research supported by NSF grants DMS 8201639 and DMS 8703096.