  We are concerned with the nonexistence of sign-changing global weak solutions
for a class of semilinear parabolic differential inequalities with convection
terms in exterior domains. A weight function of the form $t^\alpha |x|^\sigma$
is considered in front of the power nonlinearity. Two types of non-homogeneous
boundary conditions are investigated: Neumann-type and Dirichlet-type boundary
conditions. Using a unified approach, for each case, we establish sufficient
criteria for the nonexistence of global weak solutions. When $\alpha=0$, the
critical exponent in the sense of Fujita is obtained. This exponent is bigger
than that found previously by Zheng and Wang (2008) in the case of homogeneous
Neumann and Dirichlet boundary conditions.
