Well-posedness, asymptotic stability and numerical experiments for a weakly logarithmically damped wave equation

This paper addresses the analysis of a nonlinear wave equation subject to logarithmic damping of the form sgn(u_t)ln⁡(1+|u_t|). We establish, for the first time, the global existence of strong solutions for this class of problems through a rigorous framework based on the Faedo–Galerkin approximation. This provides a solid foundation for further investigations of wave models with nonstandard damping mechanisms. In addition to the well-posedness, we demonstrate that the system energy decays at an exponential rate in the one dimensional case and in a polynomial rate for higher dimensions. The proof relies on the multiplier method, complemented by delicate estimates that overcome the analytical challenges introduced by the logarithmic term. These results not only enhance the theoretical understanding of stabilization in nonlinear wave equations but also shed new light on the long-time behavior of solutions under weak logarithmic dissipation. We also perform some numerical tests to illustrate our theoretical results.