On the non-autonomous hopf bifurcation problem: Systems with rapidly varying coefficients

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.