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On well-posedness for inhomogeneous Hartree equations in the critical case

Seongyeon Kim
Dec 2022
摘要
We study the Cauchy problem for the inhomogeneous Hartree equation$i\partial_t u + \Delta u = \lambda(I_\alpha \ast|\cdot|^{-b}|u|^p)|x|^{-b}|u|^{p-2}u$ in the critical case. Until recently, itswell-posedness theory has been intensively studied, focusing on solving theproblem on Sobolev initial data with the Sobolev critical index, but the energycritical case remains unsolved. In this paper, we develop the well-posednesstheory for the energy critical case. To this end, we obtain some nonlinearestimates in Lorentz spaces which make it possible to perform a finer analysisof the nonlinearity involving the singularity $|x|^{-b}$.
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