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

TL;DR

This work analyzes finite-time blow-up for the semilinear heat equation $u_t-\Delta u=f(u)$ with genuinely non-scale-invariant nonlinearities $f(u)=u^pL(u)$, where $p>1$ is subcritical and $L$ is slowly varying. By rescaling around a blow-up point and introducing an ODE-anchored profile $\psi$ solving $y'=f(y)$, the authors derive a precise local blow-up description: near blow-up time $T$, $u(a+y\sqrt{T-t},t)\sim \psi(t)$ as $t\to T$, uniformly for bounded $y$, and for a broad class of $L$ (including logs and oscillatory variations) under a crucial regular variation condition with $\alpha>\tfrac12$. A key tool is a weighted energy functional, adapted from Giga-Kohn, which acts as a Lyapunov functional despite nonautonomous perturbations from the slowly varying nonlinearities; this yields convergence to autonomous equilibria $\{0,\pm1\}$ in rescaled variables, nondegeneracy of blow-up, and the no-needle property. The paper also extends the local results to sign-changing initial data under a type I assumption and proves the compactness of the blow-up set for initial data in $C_0(\mathbb{R}^n)$, with the appendix establishing type I blow-up bounds in the subcritical regime. Overall, the work generalizes classical blow-up asymptotics to a broad, non-scale-invariant class of nonlinearities and provides robust analytical tools for understanding singularity formation in these problems.

Abstract

We characterize the asymptotic behavior near blowup points for positive solutions of the semilinear heat equation \begin{equation*} \partial_t u-Δu =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 $ψ$ 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 $a\inΩ$ is a blowup point of $u$, then \begin{equation*} \lim_{t\to T}\frac{u(a+y\sqrt{T-t},t)}{ψ(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.

Theorems & Definitions (33)

• Theorem 2.1
• Remark 2.1
• Theorem 2.2
• Proposition 2.1
• Remark 2.2
• Theorem 3.1
• Remark 3.1
• Proposition 3.1
• Remark 3.1
• proof : Proof of Proposition \ref{['LDC']}