Asymptotic behavior of solutions for the damped wave equation with slowly decaying data

We consider the Cauchy problem for the damped wave equationutt−Δu+ut=|u|ρ−1u,(t,x)∈R+×RN and the heat equationϕt−Δϕ=|ϕ|ρ−1ϕ,(t,x)∈R+×RN. If the data is small and slowly decays likely c1(1+|x|)−kNc1(1+|x|)−kN, 0