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On the nonlinear Schrödinger equation in spaces of infinite mass and low regularity (2011.03289v1)
Published 6 Nov 2020 in math.AP
Abstract: We study the nonlinear Schr\"odinger equation with initial data in $\mathcal{Z}s_p(\mathbb{R}d)=\dot{H}s(\mathbb{R}d)\cap Lp(\mathbb{R}d)$, where $0<s<\min{d/2,1}$ and $2<p<2d/(d-2s)$. After showing that the linear Schr\"odinger group is well-defined in this space, we prove local well-posedness in the whole range of parameters $s$ and $p$. The precise properties of the solution depend on the relation between the power of the nonlinearity and the integrability $p$. Finally, we present a global existence result for the defocusing cubic equation in dimension three for initial data with infinite mass and energy, using a variant of the Fourier truncation method.