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This paper is devoted to the initial value problems for semilinear wave equations of derivative type with spatial weights in one space dimension. The lifespan estimates of classical solutions are quite different from those for nonlinearity of unknown function itself as the global-in-time existence can be established by spatial decay.

In this paper, we show the so-called "combined effect" of two different kinds of nonlinear terms for semilinear wave equations in one space dimension. This effect means that the lifespan, the maximal existence time, of the classical solution is shorter than the minimum of ones for each term. Such a special phenomenon has been observed in all space dimensions except for one space dimension. We succeed to pick up the combined effect in one space dimension by classifying the lifespan estimates according to whether the total integral of the initial speed is zero, or not, and it is obtained only in the first case. It is also remarkable that, including the combined effect, our results on the lifespan estimates are partially better than those of the general theory for nonlinear wave equations which was considered completed about 30 years ago.1. Introduction. We consider the initial value problems;where p, q > 1, A, B ≥ 0, f and g are given smooth functions of compact support and a parameter ε > 0 is "small enough". We are interested in the lifespan T (ε),