On a non-homogeneous and non-linear heat equation

We consider the Cauchy-problem for a parabolic equation of the following type: $$ racpartial upartial t= Delta u+ f(u,|x|), $$ where $x in RR^n$, $n >2$, $f=f(u,|x|)$ is supercritical. We supplement this equation by the initial condition $u(x,0)=phi$, and we allow $phi$ to be either bounded or unbounded in the origin but smaller than stationary singular solutions. We discuss local existence and long time behaviour for the solutions $u(t,x;phi)$ for a wide class of non-homogeneous non-linearities $f$. We show that in the supercritical case, ground states with slow decay lie on the threshold between initial data corresponding to blow-up solutions, and the basin of attraction of the null solution. Our results extend previous ones in that we allow $f$ to be a Matukuma-type potential and in that we allow it to depend on $u$ in a more general way. We explore such a threshold in the subcritical case too, and we obtain a result which is new even for the model case $f(u)=u|u|^q-2$. We find a family of initial data $psi(x)$ which have fast decay (i.e. $sim |x|^2-n$), are arbitrarily small in $L^infty$- norm, but which correspond to blow-up solutions.