Qualitative properties of solutions to semilinear heat equations with singular initial data

This article concerns the nonnegative solutions to the Cauchy problem $$displaylines{ u_t - Delta u + b(x,t)|u|^{p-1}u = 0 quad hbox{in } mathbb{R}^N imes (0,{infty}), cr u(x,0) = u_0(x) quad mbox{in } mathbb{R}^N ,. }$$ We investigate how the comparison principle, extinction in finite time, instantaneous shrinking of support, and existence of solutions depend on the behaviour of the coefficient $b(x,t)$.