Asymptotic behaviour of solutions of the fast diffusion equation near its extinction time

Let n≥3, 00, β≥mρ1n−2−nm and α=2β+ρ11−m. For any λ>0, we will prove the existence and uniqueness (for β≥ρ1n−2−nm) of radially symmetric singular solution gλ∈C∞(Rn∖{0}) of the elliptic equation Δvm+αv+βx⋅∇v=0, v>0, in Rn∖{0}, satisfying lim|x|→0⁡|x|α/βgλ(x)=λ−ρ1(1−m)β. When β is sufficiently large, we prove the higher order asymptotic behaviour of radially symmetric solutions of the above elliptic equation as |x|→∞. We also obtain an inversion formula for the radially symmetric solution of the above equation. As a consequence we will prove the extinction behaviour of the solution u of the fast diffusion equation ut=Δum in Rn×(0,T) near the extinction time T>0.