Blow-up of solutions for some non-linear wave equations of Kirchhoff type with some dissipation

The initial boundary value problem for non-linear wave equations of Kirchhoff type with dissipation in a bounded domain is considered. We prove the blow-up of solutions for the strong dissipative term -Δut-Δut and the linear dissipative term utut by the energy method and give some estimates for the life span of solutions. We also show the nonexistence of global solutions with positive initial energy for non-linear dissipative term by Vitillaro's argument.