Asymptotic large time behavior of singular solutions of the fast diffusion equation

TL;DR

This paper constructs and analyzes singular self-similar solutions to the fast diffusion equation, providing existence proofs, asymptotic expansions near the origin, and describing the large-time behavior of solutions with specific initial conditions.

Contribution

It introduces a new fixed point method to prove the existence of singular radially symmetric self-similar solutions for the fast diffusion equation and details their asymptotic properties.

Findings

Existence of singular self-similar solutions is established.

Asymptotic expansion of solutions near the origin is derived.

Large-time behavior of solutions with specific initial decay is characterized.

Abstract

Let $n\ge 3$, $0<m<\frac{n-2}{n}$, $\alpha=\frac{2\beta-1}{1-m}$ and $\frac{2}{1-m}<\frac{\alpha}{\beta}<\frac{n-2}{m}$. We give a new direct proof using fixed point method on the existence of singular radially symmetric forward self-similar solution of the form $V(x,t)=t^{-\alpha} f(t^{-\beta}x)$ $\forall x\in\mathbb{R}^n\setminus\{0\}$, $t>0$, for the fast diffusion equation $u_t=\Delta (u^m/m)$ in $(\mathbb{R}^n\setminus\{0\})\times (0,\infty)$, where $f$ satisfies \begin{equation*} \Delta (f^m/m) + \alpha f + \beta x \cdot \nabla f =0 \quad \text{in} \; \mathbb{R}^n\setminus\{0\} \end{equation*} with $\lim_{|x| \to 0} |x|^{ \frac{\alpha}{\beta}}f(x)=A$ and $\lim_{|x| \to \infty}f(x) = D_A$ for some constants $A>0$, $D_A > 0$. We also obtain an asymptotic expansion of such singular radially symmetric solution $f$ near the origin. We will also prove the asymptotic large time behaviour of the singular solutions of the fast diffusion equation $u_t= \Delta (u^m/m)$ in $(\mathbb{R}^n\setminus\{0\})\times (0,\infty)$, $u(x,0)=u_0(x)$ in $\mathbb{R}^n\setminus\{0\}$, satisfying the condition $A_1|x|^{-\gamma}\leq u_0(x)\leq A_2|x|^{-\gamma}$ in $\mathbb{R}^n\setminus\{0\}$, for some constants $A_2>A_1>0$ and $n\le\gamma<\frac{n-2}{m}$.