This paper deals with a weakly damped wave equation with the mean curvature operator and logarithmic nonlinearity, subject to the Dirichlet initial boundary conditions. Based on the technique of Faedo–Galerkin approximation, we establish the local well-posedness and regularity of weak solutions. Furthermore, by analyzing the Nehari manifold and combining Nakao’s inequality with the modified differential inequality technique, we obtain threshold results on the global existence and nonexistence of solutions with low initial energy, as well as energy decay estimates. Moreover, for the linear weak damping case (q = 2), we derive blow-up results for solutions with arbitrary initial energy and also give both upper and lower bounds for the lifespan, including the allowance of negativity in the commonly imposed positivity condition (u0, u1) 0; whereas for the nonlinear weak damping case (q ≠ 2), we reveal new blow-up phenomena corresponding to different regimes of initial energy.
Lv et al. (Mon,) studied this question.