We consider the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation $ i\partial_t u +\Delta u = \mu |x|^{-b}|u|^\alpha u,\; u(0)\in H^s({\mathbb R}^N),\; N\geq 1,\; \mu\in {\mathbb C},\; \; b>0 $ and $ \alpha>0. $ Only partial results are known for the local existence in the subcritical case $ \alpha<(4-2b)/(N-2s) $ and much more less in the critical case $ \alpha = (4-2b)/(N-2s). $ In this paper, we develop a local well-posedness theory for the both cases. In particular, we establish new results for the continuous dependence and for the unconditional uniqueness. Our approach provides simple proofs and allows us to obtain lower bounds of the blowup rate and of the life span. The Lorentz spaces and the Strichartz estimates play important roles in our argument. In particular this enables us to reach the critical case and to unify results for $ b = 0 $ and $ b>0. $

中文翻译：

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非齐次非线性薛定谔方程的局部适定性

我们考虑非齐次非线性薛定谔方程的柯西问题 $ i\partial_t u +\Delta u = \mu |x|^{-b}|u|^\alpha u,\; u(0)\in H^s({\mathbb R}^N),\; N\geq 1,\; \mu\in {\mathbb C},\; \; b>0 $ 和 $ \alpha>0。$ 亚临界情况下局部存在只知道部分结果 $ \alpha<(4-2b)/(N-2s) $ 在临界情况下更少 $ \alpha = (4-2b)/(N -2s）。$ 在本文中，我们为这两种情况开发了一个局部适定性理论。特别是，我们为连续依赖和无条件唯一性建立了新的结果。我们的方法提供了简单的证明，并允许我们获得爆破率和寿命的下限。Lorentz 空间和 Strichartz 估计在我们的论证中起着重要作用。特别是这使我们能够达到临界情况并统一 $ b = 0 $ 和 $ b>0 的结果。$