  In this paper we consider the wave equations with power type nonlinearities
including time-derivatives of unknown functions and their weakly coupled
systems. We propose a framework of test function method and give a simple proof
of the derivation of sharp upper bound of lifespan of solutions to nonlinear
wave equations and their systems. We point out that for respective critical
case, we use a family of self-similar solution to the standard wave equation
including Gauss's hypergeometric functions which are originally introduced by
Zhou (1992). However, our framework is much simpler than that. As a
consequence, we found new $(p,q)$-curve for the system $\partial_t^2u-\Delta
u=|v|^q$, $\partial_t^2v-\Delta v=|\partial_tu|^p$ and lifespan estimate for
small solutions for new region.
