Local well-posedness for the inhomogeneous nonlinear Schrödinger equation

Abstract

<p style='text-indent:20px;'>We consider the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation <inline-formula><tex-math id="M1">\begin{document}$ 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&gt;0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ \alpha&gt;0. $\end{document}</tex-math></inline-formula> Only partial results are known for the local existence in the subcritical case <inline-formula><tex-math id="M3">\begin{document}$ \alpha&lt;(4-2b)/(N-2s) $\end{document}</tex-math></inline-formula> and much more less in the critical case <inline-formula><tex-math id="M4">\begin{document}$ \alpha = (4-2b)/(N-2s). $\end{document}</tex-math></inline-formula> In this paper, we develop a local well-posedness theory for the both cases. In particular, we establish new results for the continuous dependence and for the unconditional uniqueness. Our approach provides simple proofs and allows us to obtain lower bounds of the blowup rate and of the life span. The Lorentz spaces and the Strichartz estimates play important roles in our argument. In particular this enables us to reach the critical case and to unify results for <inline-formula><tex-math id="M5">\begin{document}$ b = 0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ b&gt;0. $\end{document}</tex-math></inline-formula></p>