Aparajita Dasgupta, Shyam Swarup Mondal, Michael Ruzhansky, Abhilash Tushir
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引用次数: 0
Abstract
This article aims to investigate the semi-classical analog of the general Caputo-type diffusion equation with time-dependent diffusion coefficient associated with the discrete Schrödinger operator, \(\mathcal {H}_{\hbar ,V}:=-\hbar ^{-2}\mathcal {L}_{\hbar }+V\) on the lattice \(\hbar \mathbb {Z}^{n},\) where V is a positive multiplication operator and \(\mathcal {L}_{\hbar }\) is the discrete Laplacian. We establish the well-posedness of the Cauchy problem for the general Caputo-type diffusion equation with a regular coefficient in the associated Sobolev-type spaces. However, it is very weakly well-posed when the diffusion coefficient has a distributional singularity. Finally, we recapture the classical solution (resp. very weak) for the general Caputo-type diffusion equation in the semi-classical limit \(\hbar \rightarrow 0\).
本文旨在研究与离散薛定谔算子(\(\mathcal {H}_{\hbar ,V}.=-\hbar^{-2}\mathcal {L}_{\hbar }+V}:=-\hbar ^{-2}\mathcal {L}_{\hbar }+V\) on the lattice \(\hbar \mathbb {Z}^{n},\) where V is a positive multiplication operator and \(\mathcal {L}_{\hbar }\) is the discrete Laplacian.我们在相关的 Sobolev 型空间中建立了具有规则系数的一般 Caputo 型扩散方程的 Cauchy 问题的良好求解性。然而,当扩散系数具有分布奇异性时,该问题的良好求解程度很弱。最后,我们在半经典极限 \(\hbar\rightarrow 0\) 中重获了一般卡普托型扩散方程的经典解(即非常弱的解)。