Lifespan of Solutions to Wave Equations on de Sitter Spacetime

In this paper, we consider the finite time blow up of solutions for the following two kinds of nonlinear wave equations on de Sitter spacetime \begin{eqnarray*} &&\square_g=F(u),\\ &&\square_g=F(\partial_tu,\nabla u). \end{eqnarray*} This proof is based on a new blow up criterion, which generalize the blow up criterion in Sideris \cite{Sider}. Furthermore, we give the lifespan estimate of solutions for the problems.