An initial-boundary-value problem of 2D nonlinear Schrödinger equation in a strip

Abstract

The paper studies the initial-boundary-value problem (IBVP) of a nonlinear Schrödinger equation posed on a strip domain R×[0,1] with non-homogeneous Dirichlet boundary conditions. For 0≤s<1/2, if the initial condition φ(x,y) is in Sobolev space Hs(R×[0,1]) and the boundary condition h(x,t) is inHs(R2)={h(x,t)∈L2(R2)|(1+|λ|+|ξ|)12(1+|λ|+|ξ|2)s2hˆ(λ,ξ)∈L2(R2)} where hˆ is the Fourier transform of h with respect to t and x, the local well-posedness of the IBVP in C([0,T];Hs(R×[0,1])) is proved. For 1/2<s<5/2, because of compatibility conditions, the same local well-posedness result holds if an extra condition h(x,0)∈Hs+12(R) is added. Moreover, the global well-posedness is obtained for s=1. The basic idea used here relies on the derivation of an integral operator for the non-homogeneous boundary data and the proof of Strichartz's estimates for this operator. After the problem is transformed to finding a fixed point of an integral operator, the contraction mapping argument then yields a fixed point using the Strichartz's estimates for initial and boundary operators. The global well-posedness is proved using a-priori estimates of the solutions.