  In this paper we consider the critical exponent problem for the semilinear
wave equation with space-time dependent damping. When the damping is effective,
it is expected that the critical exponent agrees with that of only space
dependent coefficient case. We shall prove that there exists a unique global
solution for small data if the power of nonlinearity is larger than the
expected exponent. Moreover, we do not assume that the data are compactly
supported. However, it is still open whether there exists a blow-up solution if
the power of nonlinearity is smaller than the expected exponent.
