Asymptotic blow-up behavior for the semilinear heat equation with non scale invariant nonlinearity

TL;DR

This paper analyzes the asymptotic behavior of solutions near blowup points for a class of semilinear heat equations with non scale-invariant nonlinearities, extending understanding beyond the standard power case.

Contribution

It characterizes blowup behavior for nonlinearities of the form u^p times a slowly varying function, broadening the class of equations with known blowup profiles.

Findings

Blowup profile converges to the solution of an associated ODE.

Results apply to nonlinearities including logarithmic and oscillating functions.

Establishes compactness of the blow-up set for certain initial data.

Abstract

We characterize the asymptotic behavior near blowup points for positive solutions of the semilinear heat equation \begin{equation*} \partial_t u-\Delta u =f(u), \end{equation*} for nonlinearities which are genuinely non scale invariant, unlike in the standard case $f(u)=u^p$. Indeed, our results apply to a large class of nonlinearities of the form $f(u)=u^pL(u)$, where $p>1$ is Sobolev subcritical and $L$ is a slowly varying function at infinity (which includes for instance logarithms and their powers and iterates, as well as some strongly oscillating functions). More precisely, denoting by $\psi$ the unique positive solution of the corresponding ODE $y'(t)=f(y(t))$ which blows up at the same time $T$, we show that if $a\in\Omega$ is a blowup point of $u$, then \begin{equation*} \lim_{t\to T}\frac{u(a+y\sqrt{T-t},t)}{\psi(t)}= 1,\quad \text{uniformly for $y$ bounded.} \end{equation*} Additional blow-up properties are obtained, including the compactness of the blow-up set for the Cauchy problem with decaying initial data.