A Well-Posed and Dissipative Dynamical System for the Logarithmic Schrödinger Equation

This paper investigates the global dynamics and long-term behavior of solutions to the weakly damped logarithmic Schrödinger equation (LogSE) subject to a stationary external force in H¹(ℝ). By establishing suitable energy functionals, J(u) and F(u), and employing refined energy estimates alongside Gronwall-type differential inequalities, we rigorously establish the global existence and uniqueness of solutions in H¹(ℝ). Furthermore, we prove the existence of a bounded absorbing set ℬ₀ in H¹(ℝ), thereby confirming the global dissipation of the generated dynamical semi-group S(t). Explicit estimates for the entry time t₀ into the absorbing ball and its corresponding radius M₂ are explicitly derived to account for the non- Lipschitz logarithmic singularity at the origin.