On a nonlinear parabolic problem: Stability properties of Ground States

We consider the Cauchy-problem for the following parabolic equation: \begin{equation*} \displaystyle u_t = \Delta u+ f(u,|x|), \end{equation*} where $x \in \mathbb{R}^n$, $n >2$, and $f=f(u,|x|)$ is either critical or supercritical with respect to the Joseph-Lundgren exponent. Using a new unifying approach we extend to a larger class of nonlinear potentials $f$, some known results concerning stability and weak asymptotic stability of positive Ground States.