Stability of ground states for a nonlinear parabolic equation

We consider the Cauchy-problem for the parabolic equation $$ u_t = \Delta u+ f(u,|x|), $$ where $x \in \mathbb R^n$, $n >2$, and $f(u,|x|)$ is either critical or supercritical with respect to the Joseph-Lundgren exponent. In particular, we improve and generalize some known results concerning stability and weak asymptotic stability of positive ground states