Type: Article

Small solutions to the nonlinear Schrödinger equation with a nonlinear Neumann boundary condition

Hayashi, Nakao; Kaikina, Elena I.; Ogawa, Takayoshi;

Abstract:

This paper is concerned with the initial value problem for Schrödinger equations with power-nonlinearities of degree p on the n-dimensional half-space with Dirichlet boundary conditions. The main result of the paper (Theorem 2.1) applies to dimensions n?3 and the range 1+4/(n+2)<p<1+4/(n?2). It states that—under certain compatibility, regularity and smallness conditions—unique global solutions exist and satisfy dispersive and Strichartz-type estimates. The Dirichlet boundary condition leads to an unfavourable commutation relation between the vector field Jxn=xn+it?xn and the linear propagator. In order to solve this problem, a new integral representation formula (Lemma 3.1) is derived.