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arXiv · 2404.11863

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
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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.

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Loth Damagui Chabi, Philippe Souplet. 2024-09-29. Refined behavior and structural universality of the blow-up profile for the semilinear heat equation with non scale invariant nonlinearity. https://arxiv.org/abs/2404.11863

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