On the stability of the positive radial steady states for a semilinear Cauchy problem

This paper is a contribution to the analysis of the Cauchy problem equation(0.1){∂u∂t=Δu+∑i=1kci|x|liupiinRn×(0,T),u(x,0)=φ(x)inRn, with initial function φ≢0φ≢0. The stability and asymptotic stability of the positive radial steady states, which are positive solutions of equation(0.2)Δu+∑i=1kci|x|liupi=0, have been discussed with φφ in weighted Banach spaces.