Cauchy problem and dependency analysis for logarithmic Schrödinger equation on waveguide manifold

In this paper, we develop a novel idea to study $y$-dependence for the logarithmic Schrödinger equation on ${\mathbb{R}}^d\times {\mathbb{T}}^n$. Unlike \cite{STNT2014}(Analysis & PDE, 2014) and \cite{HHYL2024}(SIAM J. Math. Anal., 2024), the heart of the matter is that the scaling argument is invalid. Moreover, we also consider the Cauchy problem, which transforms the variational analysis into dynamical stability results.