Existence and uniqueness of the singular self-similar solutions of the fast diffusion equation and logarithmic diffusion equation

TL;DR

The paper addresses the existence and uniqueness of backward self-similar singular solutions for the fast diffusion equation $u_t=\Delta(u^m/m)$ with $0<m<\frac{n-2}{n}$ and the logarithmic diffusion equation $u_t=\Delta\log u$. It develops a fixed-point framework in transformed variables to construct radially symmetric singular profiles $f$ and $g$ that blow up at the origin with rates $|x|^{-\alpha/\beta}$ and $|x|^{-{\alpha_0}/{\beta_0}}$, respectively, under precise relations $\alpha=\alpha_m=\frac{2\beta+\rho_1}{1-m}$ and $\alpha_0=2\beta_0+\rho_1$. The results include existence, uniqueness, positivity, and detailed asymptotics: for large $|x|$, $f(r)^{1-m} r^2$ tends to a calculable constant $a_3$, yielding $f(r) \sim a_3^{1/(1-m)} r^{-2/(1-m)}$, and analogous properties for $g$ in the logarithmic case. The methods provide a robust, fixed-point alternative to shooting methods and extend the analysis to a broad class of nonlinear diffusion singular solutions, clarifying their blow-up rates and long-distance decay. The work connects backward self-similar solutions to the underlying diffusion equations and supplies rigorous asymptotics essential for understanding singular blow-up in nonlinear diffusion.

Abstract

Let $n\ge 3$, $0<m<\frac{n-2}{n}$, $ρ_1>0$, $η>0$, $β>\frac{mρ_1}{n-2-nm}$, $α=α_m=\frac{2β+ρ_1}{1-m}$, $β_0>0$ and $α_0=2β_0+1$. We use fixed point argument to give a new proof for the existence and uniqueness of radially symmetric singular solution $f=f^{(m)}$ of the elliptic equation $Δ(f^m/m)+αf+βx\cdot\nabla f=0$, $f>0$, in $\mathbb{R}^n\setminus\{0\}$, satisfying $\displaystyle\lim_{|x|\to 0}|x|^{α/β}f(x)=η$. We also prove the existence and uniqueness of radially symmetric singular solution $g$ of the equation $Δ\log g+α_0 g+β_0x\cdot\nabla g=0$, $g>0$, in $\mathbb{R}^n\setminus\{0\}$, satisfying $\displaystyle\lim_{|x|\to 0}|x|^{α_0/β_0}g(x)=η$. Such equations arises from the study of backward singular self-similar solution of the fast diffusion equation $u_t=Δu^m$ and the logarithmic diffusion equation $u_t=Δ\log u$ respectively. We will also prove the asymptotic decay rate of the function $f$ as $|x|\to\infty$.

Theorems & Definitions (31)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Theorem 2.2
• proof
• Corollary 2.3
• Corollary 2.4
• proof