Existence and blow-up of solutions in Hénon-type heat equation with exponential nonlinearity

Abstract

Abstract In the present article, we are concerned with the following problem: v t = Δ v + ∣ x ∣ β e v , x ∈ R N , t > 0 , v ( x , 0 ) = v 0 ( x ) , x ∈ R N , \left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{ll}{v}_{t}=\Delta v+| x{| }^{\beta }{e}^{v},\hspace{1.0em}& x\in {{\mathbb{R}}}^{N},\hspace{0.33em}t\gt 0,\\ v\left(x,0)={v}_{0}\left(x),\hspace{1.0em}& x\in {{\mathbb{R}}}^{N},\end{array}\right. where N ≥ 3 N\ge 3 , 0 < β < 2 0\lt \beta \lt 2 , and v 0 {v}_{0} is a continuous function in R N {{\mathbb{R}}}^{N} . We prove the existence and asymptotic behavior of forward self-similar solutions in the case where v 0 {v}_{0} decays at the rate − ( 2 + β ) log ∣ x ∣ -\left(2+\beta )\log | x| as ∣ x ∣ → ∞ | x| \to \infty . Particularly, we obtain the optimal decay bound for initial value v 0 {v}_{0} .