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

Abstract

Let $n\geq 3$, $0\le m<\frac{n-2}{n}$, $ρ_1>0$, $β>β_0^{(m)}=\frac{mρ_1}{n-2-nm}$, $α_m=\frac{2β+ρ_1}{1-m}$ and $α=2β+ρ_1$. For any $λ>0$, we prove the uniqueness of radially symmetric solution $v^{(m)}$ of $\La(v^m/m)+α_m v+βx\cdot\nabla v=0$, $v>0$, in $\R^n\setminus\{0\}$ which satisfies $\lim_{|x|\to 0}|x|^{\frac{α_m}β}v^{(m)}(x)=λ^{-\frac{ρ_1}{(1-m)β}}$ 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 $\La\log v+αv+βx\cdot\nabla v=0$, $v>0$, in $\R^n\bs\{0\}$, which satisfies $\lim_{|x|\to 0}|x|^{\fracαβ}v(x)=λ^{-\frac{ρ_1}β}$. We also prove that if the solution $u^{(m)}$ of $u_t=Δ(u^m/m)$, $u>0$, in $(\R^n\setminus\{0\})\times (0,T)$ which blows up near $\{0\}\times (0,T)$ at the rate $|x|^{-\frac{α_m}β}$ satisfies some mild growth condition on $(\R^n\setminus\{0\})\times (0,T)$, then as $m\to 0^+$, $u^{(m)}$ converges uniformly in $C^{2+θ,1+\fracθ{2}}(K)$ for some constant $θ\in (0,1)$ 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 $\La \log v+αv+βx\cdot\nabla v=0$, $v>0$, in $\R^n\setminus\{0\}$, which satisfies $\lim_{|x|\to 0}|x|^{\fracαβ}v(x)=λ^{-\frac{ρ_1}β}$.