  This paper is devoted to the lifespan estimates of small classical solutions
of the initial value problems for one dimensional wave equations with
semilinear terms of the spatial derivative of the unknown function. It is
natural that the result is same as the one for semilinear terms of the
time-derivative. But there are so many differences among their proofs.
Moreover, it is meaningful to study this problem in the sense that it may help
us to investigate its blow-up boundary in the near future.
