Nonexistence of global weak solutions for strongly damped <i>p</i> –Laplacian wave equations with logarithmic source terms

This paper is devoted to the study of the global behavior of solutions to a class of strongly damped wave equations involving the p–Laplacian operator and logarithmic source terms. More precisely, we consider the problem utt−Δut−Δpu=λ|u|α−2uln⁡(1+|u|)+γ|∇u|βln⁡(1+|∇u|),(x,t)∈RN×(0,+∞),where Δpu=div(|∇u|p−2∇u) with p>2, and λ,γ,α,β>0 are given constants. The presence of logarithmic nonlinearities combined with nonlinear diffusion and strong damping introduces significant analytical difficulties, since the logarithmic terms exhibit a growth behavior that lies between polynomial and exponential nonlinearities. To overcome these challenges, we employ a rescaled test function method together with suitable logarithmic inequalities. Under appropriate conditions on the spatial dimension and the nonlinear exponents, we prove the nonexistence of nontrivial global weak solutions. Furthermore, we derive an explicit upper bound for the lifespan of local weak solutions in terms of the initial data and the parameters of the problem. The obtained results illustrate how strong damping, p–Laplacian diffusion, and logarithmic source terms interact with each other. In particular, this interaction leads to the appearance of a critical threshold phenomenon that differs from the one observed in the classical power-type nonlinear case.