  Consider nonlinear wave equations in the spatially flat
Friedmann-Lema\^itre-Robertson-Walker (FLRW) spacetimes. We show blow-up in
finite time of solutions and upper bounds of the lifespan of blow-up solutions
to give the FLRW spacetime version of Glassey's conjecture for the time
derivative nonlinearity. We also show blow-up results for the space time
derivative nonlinearity.
