Stable and unstable sets for damped nonlinear wave equations with variable exponent sources

In this paper, we study a class of nonlinear wave equations with variable exponent sources. By introducing a family of potential wells, we first prove the global existence of solutions with initial data in the potential wells and the finite time blow-up for solutions starting in the unstable sets. The boundedness and asymptotic behavior of global solutions are also concerned. Finally, we obtain the finite time blow-up with high energy initial data.