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

TL;DR

This paper proves the existence and uniqueness of radially symmetric singular self-similar solutions for specific elliptic equations related to fast diffusion and logarithmic diffusion equations, using fixed point methods.

Contribution

It introduces a new fixed point approach to establish existence and uniqueness of singular solutions for these elliptic equations, extending previous results.

Findings

Existence and uniqueness of singular solutions for the elliptic equations.

Asymptotic decay rates of solutions at infinity.

Application to self-similar solutions of diffusion equations.

Abstract

Let $n\ge 3$, $0<m<\frac{n-2}{n}$, $\rho_1>0$, $\eta>0$, $\beta>\frac{m\rho_1}{n-2-nm}$, $\alpha=\alpha_m=\frac{2\beta+\rho_1}{1-m}$, $\beta_0>0$ and $\alpha_0=2\beta_0+1$. 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 $\Delta (f^m/m)+\alpha f+\beta x\cdot\nabla f=0$, $f>0$, in $\mathbb{R}^n\setminus\{0\}$, satisfying $\displaystyle\lim_{|x|\to 0}|x|^{\alpha/\beta}f(x)=\eta$. We also prove the existence and uniqueness of radially symmetric singular solution $g$ of the equation $\Delta\log g+\alpha_0 g+\beta_0x\cdot\nabla g=0$, $g>0$, in $\mathbb{R}^n\setminus\{0\}$, satisfying $\displaystyle\lim_{|x|\to 0}|x|^{\alpha_0/\beta_0}g(x)=\eta$. Such equations arises from the study of backward singular self-similar solution of the fast diffusion equation $u_t=\Delta u^m$ and the logarithmic diffusion equation $u_t=\Delta\log u$ respectively. We will also prove the asymptotic decay rate of the function $f$ as $|x|\to\infty$.