Complete Blow-up for a semi-linear parabolic problem with a localized nonlinear term

We study existence of blow-up and blow-up sets for a (degenerate or not) Heat-like equation with a uniform source term nonlinear function of the instantaneous temperature at a given point of the domain. The techniques are relevant from either classical analysis (Green function, eigenfunction expansion, maximum principle) or function analysis (semi-group of linear operators).