Threshold property of a singular stationary solution for semilinear heat equations with exponential growth

Let $N\ge 3$. We are concerned with a Cauchy problem of the semilinear heat equation $$\{\begin{array}{ll}{\mathrm{\partial }}_{t}u-\mathrm{\Delta }u=f(u),& x\in {\mathbb{R}}^N,t>0,\\ u(x,0)=u_0(x),& x\in {\mathbb{R}}^N,\end{array}$$ where $f(0)=0$, $f$ is nonnegative, increasing and convex, $\log f(u)$ is convex for large $u>0$ and some additional assumptions are assumed. We establish a positive radial singular stationary solution $u^{\ast }$ such that $u^{\ast }(x)\to \mathrm{\infty }$ as $\mathrm{\mid }x\mathrm{\mid }\to 0$. Then, we prove the following: The problem has a nonnegative global-in-time solution if $0\le u_0\le u^{\ast }$ and $u_0\equiv ̸u^{\ast }$, while the problem has no nonnegative local-in-time solutions $u$ such that $u\ge u^{\ast }$ if $u_0\ge u^{\ast }$ and $u_0\equiv ̸u^{\ast }$.