Refined behavior and structural universality of the blow-up profile for the semilinear heat equation with non scale invariant nonlinearity
Loth Damagui Chabi, Philippe Souplet
TL;DR
This work analyzes the semilinear heat equation $u_t-\Delta u=f(u)$ with non-scale-invariant nonlinearities $f(u)=u^pL(u)$, where $L$ is slowly varying and $1<p<p_S$. For radially symmetric decreasing initial data, the authors derive a sharp global blow-up profile in original variables, $u(x,t)=(1+o(1))G^{-1}\Big(T-t+\frac{p-1}{8p}\frac{|x|^2}{|\log|x||}\Big)$, and obtain a sharp final and refined space-time profile, all governed by the universal ODE resolvent $G^{-1}$, largely independent of $L$. They show a structural universality: the leading blow-up structure is the same as in the pure power case, with $L$ affecting only the argument via $G^{-1}$; explicit small-$s$ asymptotics for $G^{-1}$ are given as $G^{-1}(s)\sim \kappa s^{-\beta}L^{-{\beta}}(s^{-\beta})$ with $\beta=1/(p-1)$. The paper also develops a sharp upper bound for more general nonlinearities, and a detailed center-manifold–type analysis in similarity variables to prove a matching lower bound, culminating in a comprehensive description of the refined blow-up behavior and its universality across a broad class of nonlinearities.
Abstract
We consider the semilinear heat equation $$u_t-Δu=f(u) $$ for a large class of non scale invariant nonlinearities of the form $f(u)=u^pL(u)$, where $p>1$ is Sobolev subcritical and $L$ is a slowly varying function (which includes for instance logarithms and their powers and iterates, as well as some strongly oscillating functions). For any positive radial decreasing blow-up solution, we obtain the sharp, global blow-up profile in the scale of the original variables $(x, t)$, which takes the form: $$u(x,t)=(1+o(1))\,G^{-1}\bigg(T-t+\frac{p-1}{8p}\frac{|x|^2}{|\log |x||}\bigg), \ \hbox{as $(x,t)\to (0,T)$, \quad where } G(X)=\int_{X}^{\infty}\frac{ ds}{f(s)}.$$ This estimate in particular provides the sharp final space profile and the refined space-time profile. As a remarkable fact and completely new observation, our results reveal a {\it structural universality} of the global blow-up profile, being given by the "resolvent" $G^{-1}$ of the ODE, composed with a universal, time-space building block, which is the same as in the pure power case.