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

Abstract

This paper is contributed to the Cauchy problem (0.1) ∂u ∂t = Δu+K(|x|)u p in R n×(0,T), u(x,0)=ϕ(x) in R n with initial function ϕ≢0. The stability and instability of the positive radial steady states, which are positive solutions of (0.2) Δu+K(|x|)u p=0, has been discussed with different assumption on K(| x|) and ϕ under the norm (0.3) ∥ψ∥ λ= sup x∈ R n |(1+|x| λ)ψ(x)|, where ϕ and ψ are some nonnegative continuous functions in R n , and λ is a real number.