Cauchy problem and dependency analysis for logarithmic Schr\"odinger equation on waveguide manifold

In this paper, we develop a novel idea to study $y$-dependence for the logarithmic Schr\"odinger 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.