Blow‐up and energy decay for a class of wave equations with nonlocal Kirchhoff‐type diffusion and weak damping

The purpose of this paper is to study the following equation driven by a nonlocal integro‐differential operator : with homogeneous Dirichlet boundary condition and initial data, where is the Gagliardo seminorm, and with , is the space dimension. By virtue of a differential inequality technique, an upper bound of the blow‐up time is obtained with a bounded initial energy if and some additional conditions are satisfied. For , in particular, the blow‐up result with high initial energy also is showed by constructing a new control functional and combining energy inequalities with the concavity argument. Moreover, an estimate for the lower bound of the blow‐up time is investigated. Finally, the energy decay estimate is proved as well. These results improve and complement some recent works.