Abstract
We consider the Cauchy problem for the semilinear wave equation with time-dependent damping $$ \left\{ \begin{array}{@{}ll} u_{tt} - \Delta u + b(t)u_t=f(u)\,, & (t,x) \in {\bf R}^+ \times {\bf R}^N (u,u_t)(0,x) = (u_0,u_1)(x)\,, & x \in {\bf R}^N\,. \end{array}\right. {(*)} $$ hen $b(t)=(t+1)^{-\beta}$ with $0\le \beta \rho_F(N):=1+2/N$. We also obtain some blow-up results on the solution within a finite time, so that the smallness of the data is essential to get global existence in the supercritical exponent case.
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