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

This paper is contributed to the Cauchy problem{∂u∂t=Δu+K(|x|)up+μf(|x|)inRn×(0,T),u(x,0)=φ(x)inRn, with initial function φ≢0φ≢0. The monotonicity and stability of the positive radial steady states, which are positive solutions of Δu+K(|x|)up+μf(|x|)=0Δu+K(|x|)up+μf(|x|)=0, are discussed with K(|x|)K(|x|), f(|x|)f(|x|) and φ in weighted Banach spaces.