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

Abstract

We consider the semilinear heat equation $u_t - \Delta u = f(u)$ in $\Omega = B_R(0) \subset \mathbb{R}^n$ with super-exponential nonlinearities $f(u) = e^{u^p}u^q$ ($p>1$, $q \in \{0\}\cup [1,\infty)$), 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 $n \leq 2$, 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.