Global existence and exponential decay for a wave equation with logarithmic damping and polynomial source

Abstract

This paper examines a nonlinear wave equation with logarithmic damping ut ln(1+ut2 ) and a polynomial source u|u|p?2. Using the potential well method, we prove that if the initial energy is below the potential well depth and the initial data lie in the stable set, then the solution exists globally. Moreover, the energy decays exponentially in time. These findings extend previous results on logarithmic wave equations by incorporating a polynomial source and highlight the stabilizing effect of logarithmic damping.