On refined blowup estimates for the exponential reaction-diffusion equation
Philippe Souplet
TL;DR
This paper analyzes the semilinear heat equation with exponential nonlinearity in radial settings and proves a global, refined blowup bound in original variables, with $m(t)=u(0,t)$. The authors derive a global bound for $u(r,t)$ that yields sharp final and refined blowup profiles as $t\to T$, namely $u(r,t) ≤ log(|log(me^{-m}+r^2/4)|/(me^{-m}+r^2/4)) + o(1)$ and corollaries for the final profile $u(r,T) ≤ 2 log r + log|log r| + log 8 + o(1)$ and the refined space-time profile $u(ξ√{m e^{-m}},t) ≤ m - log(1+ξ^2/4) + o(1)$. The method relies on a parabolic maximum-principle approach via an auxiliary functional $J$ with a judicious multiplier $φ$, giving sharp remainder terms and a simpler route than center-manifold analyses for the exponential nonlinearity. These results advance understanding of blowup behavior for exponential reaction-diffusion equations and provide precise, implementable bounds for related radial problems.
Abstract
We consider radial decreasing solutions of the semilinear heat equation with exponential nonlinearity. We provide a relatively simple proof of the sharp upper estimates for the final blowup profile and for the refined space-time behavior. We actually establish a global, upper space-time estimate, which contains those of the final and refined profiles as special cases.