Stability and blow-up analysis for a Kirchhoff-type system with logarithmic dissipation mechanisms and logarithmic sources

This paper investigated, for the first time, a coupled Kirchhoff-type wave system incorporating logarithmic damping and logarithmic source terms (external forces). We established both energy decay and finite-time blow-up results, emphasizing the novel interaction between the two logarithmic components acting as dissipative and driving mechanisms. In the stable region, we constructed a suitable Lyapunov functional and employed refined logarithmic estimates to derive a polynomial decay rate of the total energy. Conversely, for initial data belonging to the unstable set, we proved that the corresponding solutions blow up in finite time using a concavity argument. In addition, we provided some numerical examples to illustrate the stability and blow-up theoretical results. This study presents the first comprehensive analysis of a Kirchhoff-type system with logarithmic damping, revealing how the combined effects of the nonlocal Kirchhoff tension and logarithmic nonlinearities govern the transition between global stabilization and blow-up behavior.