Radial solutions of initial boundary value problems of nonlinear Schrödinger equations in ℝ

The article studies an initial boundary valueproblem (ibvp) for the radial solutions of the nonlinear Schrödinger (NLS) equation in a radially symmetric region $\Omega\in \mathbb R^n$ with boundaries. All such regions can be classified into three types: a ball Ω 0 centred at origin, a region Ω 1 outside a ball, and an n -dimensional annulus Ω 2 . To study the well-posedness of those ibvps, the function spaces for the boundary data must be specified in terms of the solutions in appropriate Sobolev spaces. It is shown that when $\Omega = \Omega_1$ , the ibvp for the NLS equation is locally well-posed in $ C( [0, T^*]; H^s(\Omega_1))$ if the initial data is in $H^s(\Omega_1)$ and boundary data is in $ H^{\frac{2s+1}{4}}(0, T)$ with $s \geq 0$ . This is the optimal regularity for the boundary data and cannot be improved. When $\Omega = \Omega_2$ , the ibvp is locally well-posed in $ C( [0, T^*]; H^s(\Omega_2))$ if the initial data is in $ H^s(\Omega_2)$ and boundary data is in $ H^{\frac{s+1}{2}}(0, T)$ with $s \geq 0$ . In this case, the boundary data requires $1/4$ more derivative compared to the case when $\Omega = \Omega_1$ . When $\Omega = \Omega_0$ with n = 2 (the case with n > 2 can be discussed similarly), the ibvp is locally well-posed in $ C( [0, T^*]; H^s(\Omega_0))$ if the initial data is in $ H^s(\Omega_0)$ and boundary data is in $ H^{\frac{s+1}{2}}(0, T)$ with s > 1 (or $s \gt n/2$ ). Due to the lack of Strichartz estimates for the corresponding boundary integral operator with $ 0 \leq s \leq 1$ , the local well-posedness can only be achieved for s > 1. It is noted that the well-posedness results on Ω 0 and Ω 2 are the first ones for the ibvp of NLS equations in bounded regions of higher dimension.