Abstract
The main purpose of the present paper is to study the blow-up problem of the wave equation with space-dependent damping in the scale -invariant case and time derivative nonlinearity with small initial data. Under appropriate initial data which are compactly supported, by using a test function method and taking into account the effect of the damping term ( mu root 1+|x|2ut), we show that in higher dimensions the blow-up region is given by p is an element of (1, pG(N+ mu)] where pG(N) is the Glassey exponent. Furthermore, we shall establish a blow-up region, independent of mu given by p is an element of (1,1+ 2N ), for appropriate initial data in the energy space with noncompact support.