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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 orsupercritical with respect to the Joseph-Lundgren exponent. Using a newunifying approach we extend to a larger class of nonlinear potentials $f$, someknown results concerning stability and weak asymptotic stability of positiveGround States.

On a nonlinear parabolic problem: Stability properties of Ground States