Asymptotic behavior at infinity of the stationary solution to a semilinear heat equation

In this paper we investigate the asymptotic behavior at infinity of the backward self-similar solution of the differential equation ut=Δu+eu, x∈Ω,t>0, where Ω is a ball with the Dirichlet boundary or Rn, 3≤n<∞. We prove that, under some reasonable condition at infinity, every radial symmetric, nontrivial, bounded above solution of the equationωyy+(n−1y−y2)ωy+eω−1=0 tends to minus infinity as y→∞. This equation comes from the scaled ignition model. Furthermore, ω+logy2 converges to a constant for sufficiently large y. This result extends the similar one in Lacey (1993) for an arbitrary solution which is bounded above and for dimension 3≤n<∞ in space.