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Energy-critical inhomogeneous nonlinear Schrödinger equation with two power-type nonlinearities (2408.10866v1)

Published 20 Aug 2024 in math.AP

Abstract: We consider the initial value problem for the inhomogeneous nonlinear Schr\"odinger equation with double nonlinearities (DINLS) \begin{equation*} i \partial_t u + \Delta u = \lambda_1 |x|{-b_1}|u|{p_1}u + \lambda_2|x|{-b_2}|u|{\frac{4-2b_2}{N-2}}u, \end{equation*} where $\lambda_1,\lambda_2\in \mathbb{R}$, $3\leq N<6$ and $0<b_1,b_2<\min{2,\frac{6-N}{2}}$. In this paper, we establish global well-posedness results for certain parameter regimes and prove finite-time blow-up phenomena under specific conditions. Our analysis relies on stability theory, energy estimates, and virial identities adapted to the DINLS model.

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