  We are interested in the so-called "combined effect" of two different kinds
of nonlinear terms for semilinear wave equations in one space dimension.
Recently, the first result with the same formulation as in the higher
dimensional case has been obtained if and only if the total integral of the
initial speed is zero, namely Huygens' principle holds. In this paper, we
extend the nonlinear term to the general form including the product type. Such
model equations are extremely meaningful only in one space dimension because
the most cases in higher dimensions possess the global-in-time existence of a
classical solution in the general theory for nonlinear wave equations. It is
also remarkable that our results on the lifespan estimates are partially better
than those of the general theory. This fact tells us that there is a
possibility to improve the general theory which was expected complete more than
30 years ago.
