Asymptotics for nonlinear heat equations

We study the Cauchy problem for the nonlinear heat equation ut-Δu+u1+σ=0,x∈Rn,t>0,u(0,x)=u0(x),x∈Rn, in the sub critical case of σ∈(0,2n). In the present paper we intend to give a more precise estimate for the remainder term in the asymptotic representation known from paper Escobedo and Kavian (1987) [5]u(t,x)=t-1σw0(xt)+o(t- 1σ) as t→∞ uniformly with respect to x∈Rn, where w0(ξ) is a positive solution of equation -Δw-ξ2·∇w+w1+σ= 1σw which decays rapidly at infinity: lim|ξ|→±∞|ξ |2σw0(ξ)=0.