On a non-homogeneous and non-linear heat equation

We consider the Cauchy-problem for a parabolic equation of the following type: \begin{align*} \frac{\partial u}{\partial t}= \Delta u+ f(u,|x|), %\\ \end{align*} 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.