Asymptotic Blow-up Behavior for the Semilinear Heat Equation with Super-exponential Nonlinearities

We consider the semilinear heat equation $u_t-\mathrm{\Delta }u=f(u)$ in $\mathrm{\Omega }=B_R(0)\subset {\mathbb{R}}^n$ with super-exponential nonlinearities $f(u)=e^{u^p}{u}^q$ ($p>1$, $q\in \{0\}\cup [1,\mathrm{\infty })$), nonnegative bounded radially symmetric initial data and 0-Dirichlet boundary condition. In this paper, we show the asymptotic blow-up behavior for nonnegative, radial type I blow-up solution. More precisely, we prove that if $n\le 2$, then such blow-up solution satisfies \begin{equation*} \lim_{t \rightarrow T} \frac{T-t}{F(u(y\sqrt{T-t},t))} = 1, \quad $\text{where }$ F(u) = \int_{u}^{\infty} \frac{ds}{f(s)}. \end{equation*} We note that this result corresponds to the one which is proved by Liu in 1989 for the case of $f(u)=e^u$, which has the scale invariance property unlike our super-exponential case. To prove the main result, we see the equation as a perturbation of the equation with $f(u)=e^u$ through a transformation introduced by Fujishima and Ioku in 2018 and estimate the additional term which appears after the transformation.