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

Abstract

In the present article, we are concerned with the following problem: vt=Δv+∣x∣βev,x∈RN,t>0,v(x,0)=v0(x),x∈RN,\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≥3N\ge 3, 0<β<20\lt \beta \lt 2, and v0{v}_{0} is a continuous function in RN{{\mathbb{R}}}^{N}. We prove the existence and asymptotic behavior of forward self-similar solutions in the case where v0{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 v0{v}_{0}.

Keywords

• hénon type equation
• heat equation
• self-similar solutions
• blow-up solutions
• 35k58
• 35j15
• 35j67
• 35j61