  We prove the existence of a critical Fujita exponent for a non-homogeneous
semilinear heat equation which involves degenerate coefficients. More
precisely, in order to give a rather complete theory, we focus on two types of
weights $w(x)=|x_1|^a$ or $w(x)=|x|^b$ where $a, b>0$ in a suitable range. The
coefficients under consideration admit either a singularity at the origin or a
line of singularities. In the latter case, the problem is related to the
fractional Laplacian.
