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

TL;DR

This work addresses the asymptotic blow-up behavior for the semilinear heat equation $u_t-\Delta u=f(u)$ with a super-exponential nonlinearity $f(u)=e^{|u|^p}u^q$ in a ball, for $p>1$ and $q\in\{0\}\cup[1,\infty)$, in the radially symmetric, type I blow-up regime and $n\le2$. The authors adapt a quasi-scaling transformation $u_\lambda(x,t)=-\log F(u(\lambda x,\lambda^2 t))+2\log\lambda$ (with $F(u)=\int_u^{\infty}\frac{ds}{f(s)}$) to recast the problem as a perturbation of the exponential case, controlling the remainder term $|\nabla u_\lambda|^2\,(f'(u)F(u)-1)$ since $f'(u)F(u)\to1$ as $u\to\infty$. An energy framework for the transformed variable $v(y,s)$ is developed, yielding a decay inequality with an integrable error, and a compactness argument shows any blow-up limit must be the constant profile $v_\infty\equiv1$ in $n\le2$, yielding the sharp asymptotic rate. Consequently, for radial type I blow-up, the paper proves $\lim_{t\to T} \frac{T-t}{F(u(y\sqrt{T-t},t))}=1$ uniformly on compact sets, extending Liu's classical exponential-case result to the broad super-exponential nonlinearity considered. This provides a robust mechanism to transfer exponential-type asymptotics to a wider nonlinear class lacking exact scale invariance. The results enhance understanding of blow-up profiles and rates for nonlinear heat equations with fast-growing nonlinearities and sharpen insights into the localized blow-up structure.

Abstract

We consider the semilinear heat equation $u_t - Δu = f(u)$ in $Ω= B_R(0) \subset \mathbb{R}^n$ 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.

Theorems & Definitions (25)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Lemma 2.1
• proof
• Remark 2.2
• Lemma 2.3
• proof
• Remark 2.4
• Proposition 2.5