Singular limits and properties of solutions of some degenerate elliptic and parabolic equations

TL;DR

This paper investigates the singular limits of solutions to certain degenerate elliptic and parabolic equations, establishing uniqueness, convergence properties, and existence of radially symmetric solutions as parameters tend to zero.

Contribution

It provides new results on the limits of solutions of degenerate elliptic and parabolic equations, including convergence to logarithmic equations and the existence of specific radially symmetric solutions.

Findings

Uniqueness of radially symmetric solutions near singularities.

Convergence of solutions as the parameter m approaches zero.

Existence of a unique radially symmetric solution for the logarithmic elliptic equation.

Abstract

Let $n\geq 3$, $0\le m<\frac{n-2}{n}$, $\rho_1>0$, $\beta>\beta_0^{(m)}=\frac{m\rho_1}{n-2-nm}$, $\alpha_m=\frac{2\beta+\rho_1}{1-m}$ and $\alpha=2\beta+\rho_1$. For any $\lambda>0$, we prove the uniqueness of radially symmetric solution $v^{(m)}$ of $\La(v^m/m)+\alpha_m v+\beta x\cdot\nabla v=0$, $v>0$, in $\R^n\setminus\{0\}$ which satisfies $\lim_{|x|\to 0}|x|^{\frac{\alpha_m}{\beta}}v^{(m)}(x)=\lambda^{-\frac{\rho_1}{(1-m)\beta}}$ and obtain higher order estimates of $v^{(m)}$ near the blow-up point $x=0$. We prove that as $m\to 0^+$, $v^{(m)}$ converges uniformly in $C^2(K)$ for any compact subset $K$ of $\R^n\setminus\{0\}$ to the solution $v$ of $\La\log v+\alpha v+\beta x\cdot\nabla v=0$, $v>0$, in $\R^n\bs\{0\}$, which satisfies $\lim_{|x|\to 0}|x|^{\frac{\alpha}{\beta}}v(x)=\lambda^{-\frac{\rho_1}{\beta}}$. We also prove that if the solution $u^{(m)}$ of $u_t=\Delta (u^m/m)$, $u>0$, in $(\R^n\setminus\{0\})\times (0,T)$ which blows up near $\{0\}\times (0,T)$ at the rate $|x|^{-\frac{\alpha_m}{\beta}}$ satisfies some mild growth condition on $(\R^n\setminus\{0\})\times (0,T)$, then as $m\to 0^+$, $u^{(m)}$ converges uniformly in $C^{2+\theta,1+\frac{\theta}{2}}(K)$ for some constant $\theta\in (0,1)$ and any compact subset $K$ of $(\R^n\setminus\{0\})\times (0,T)$ to the solution of $u_t=\La\log u$, $u>0$, in $(\R^n\setminus\{0\})\times (0,T)$. As a consequence of the proof we obtain existence of a unique radially symmetric solution $v^{(0)}$ of $\La \log v+\alpha v+\beta x\cdot\nabla v=0$, $v>0$, in $\R^n\setminus\{0\}$, which satisfies $\lim_{|x|\to 0}|x|^{\frac{\alpha}{\beta}}v(x)=\lambda^{-\frac{\rho_1}{\beta}}$.