Infinite-time blowup and global solutions for a semilinear Klein–Gordan equation with logarithmic nonlinearity

In this article, our focus lies in investigating the existence of global solutions and the occurrence of infinite-time blowup for a nonlinear Klein–Gordon equation characterized by logarithmic nonlinearity, specifically in the form $ \mathbf {w}\log \vert \mathbf {w}\vert ^k $ . Notably, our inquiry involves handling logarithmic functions within the reaction terms and accommodating the functions $ \mathbf {w}_0 $ and $ \mathbf {w}_1 $ in the boundary terms. Consequently, a pivotal task is to establish blowup conditions that intricately hinge on the characteristics of the domains and the boundary conditions. It is of significant note that our exploration incorporates domain and boundary information into the formulation of blowup conditions. By employing a combination of the potential well technique and energy estimation methodology, we delve into scenarios of low initial energy and critical initial energy. In doing so, we derive a set of sufficient conditions that encompass both the global existence and the potential explosive behaviour of solutions pertaining to this variant of the Klein–Gordon equation. This work contributes to enhancing our understanding of the intricate interplay between logarithmic nonlinearity, domain characteristics, and boundary conditions in shaping the behaviour of solutions in the realm of nonlinear wave equations.