Non-existence of radially symmetric singular self-similar solutions of the fast diffusion equation
Shu-Yu Hsu
TL;DR
This work addresses the nonexistence of radially symmetric singular self-similar solutions to the fast diffusion equation $u_t=\Delta(u^m/m)$ that blow up at the origin. By reducing the problem to an elliptic equation $\Delta(f^m/m)+\alpha f+\beta x\cdot\nabla f=0$ in $\mathbb{R}^n\setminus\{0\}$ with a prescribed blow-up rate $|x|^{-\gamma}$, the authors derive precise constraints on the admissible blow-up exponents and the associated parameters. The main results characterize when such singular solutions can exist: either $\gamma=\frac{2}{1-m}$ with $\alpha>\frac{2\beta}{1-m}$ or $\gamma>\frac{2}{1-m}$ with $\beta\neq 0$ and $\gamma=\alpha/\beta$, plus a compatibility condition for the critical case $\gamma=\frac{2}{1-m}$, and strong nonexistence when $\gamma\neq\alpha/\beta$. The findings, including sign restrictions on $\alpha,\beta$ and the role of forward/backward self-similar forms, provide a rigorous obstruction to radially symmetric self-similar blow-up in broad parameter regimes, advancing the understanding of singular self-similar behavior in fast diffusion.
Abstract
Let $n\ge 3$, $0<m<\frac{n-2}{n}$, $γ>0$ and $η>0$. Suppose either (i) $α\ne 0$ and $β=0$ or (ii) $α\in\mathbb{R}$ and $β\ne 0$ holds. We will study the elliptic equation $Δ(f^m/m)+αf+βx\cdot\nabla f=0$, $f>0$, in $\mathbb{R}^n\setminus\{0\}$ with $\underset{\substack{r\to 0}}{\lim}\,r^γf(r)=η$. This equation arises from the study of the singular self-similar solutions of the fast diffusion equation which blow up at the origin. We will prove that if there exists a radially symmetric singular solution of the above elliptic equation, then either $γ=\frac{2}{1-m}$ and $α>\frac{2β}{1-m}$ or $γ>\frac{2}{1-m}$, $β\ne 0$ and $γ=α/β$. As a consequence we obtain the non-existence of radially symmetric self-similar solution of the fast diffusion equation $u_t=Δ(u^m/m)$, $u>0$, which blows up at the origin with rate $|x|^{-γ}$ when either $0<γ\ne\frac{2}{1-m}$ and $γ\neα/β$, $α\in\mathbb{R}$ and $β\ne 0$ or $γ=\frac{2}{1-m}$ and $\left(α-\frac{2β}{1-m}\right)η\ne\frac{2(n-2-nm)}{(1-m)^2}$ holds.