  The final open part of the famous Strauss conjecture on semilinear wave
equations of the form \Box u=|u|^{p}, i.e., blow-up theorem for the critical
case in high dimensions was solved by Yordanov and Zhang, or Zhou
independently. But the estimate for the lifespan, the maximal existence time,
of solutions was not clarified in both papers. Recently, Takamura and Wakasa
have obtained the sharp upper bound of the lifespan of the solution to the
critical semilinear wave equations, and their method is based on the method in
Yordanov and Zhang. In this paper, we give a much simple proof of the result of
Takamura and Wakasa by using the method in Y. Zhou for space dimensions n\geq
2. Simultaneously, this estimate of the life span also proves the last open
optimality problem of the general theory for fully nonlinear wave equations
with small initial data in the case n=4 and quadratic nonlinearity(One can see
Li and Chen for references on the whole history).
