  We consider a semilinear heat equation involving a forcing term which depends
only upon the space variable. Existence of a local mild solution is proved
through an application of the Banach fixed point theorem. An upper bound for
the blow-up time of local solutions is also provided. With the help of
carefully defined test functions, we prove nonexistence results for global weak
solutions. This leads to lower bounds for a possible critical Fujita-type
exponent.
