Non-homogeneous Problems for Nonlinear Schrödinger Equations in a Strip Domain

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This 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 any s≥0, if the initial data φ(x,y) is in Sobolev space Hˢ(R×[0,1]) and the boundary data h(x,t) is in Hˢ (R) = { h (x, t) ∈ L² (R²) | (1 + |λ| + |ξ|)12 (1+ |λ| + |ξ|²)ˢ/² h (λ, ξ) ∈ L² (R²) } where h 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. The global well-posedness is also 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 the series version 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.