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This paper is concerned with the Cauchy problem for a semilinear wave equation with a time-dependent damping. In case that the space dimension n=1 and the nonlinear power is bigger than 2, the life-span T(ε) and global existence of the classical solution to the problem has been investigated in a unified way. More precisely, with respect to different values of an index K, which depends on the time-dependent damping and the nonlinear term, the life-span T(ε) can be estimated below by ε^-p1-K , e^ε^-p or +∞, where ε is the scale of the compact support of the initial data. , number=3, journal=Journal of Partial Differential Equations, year=2023, month=Aug., pages=235–261