We consider the Cauchy-problem for the parabolic equation
ut=Deltau+f(u,∣x∣),
where xinmathbbRn, 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