A matching construction of self-similar profiles for the fast diffusion equation

Abstract

Let $n\ge 3$, $0 0$, $η_0>0$, $ρ_1>0$, $β_-(ρ_1)=-\frac{ρ_1}{2}$, $β_+(ρ_1)=\frac{mρ_1}{n-2-nm}$ and $α=\frac{2β+ρ_1}{1-m}$. For any $β_-(ρ_1)\leβ\leβ_+(ρ_1)$, we construct the unique maximal positive radial branch of \[ Δ(f^m/m)+αf+βx\cdot\nabla f=0 \] issuing from prescribed origin data $f(0)=η_0$ and $f_r(0)=0$. For any $β\leβ_+(ρ_1)$, we construct the unique maximal positive radial branch at infinity satisfying \[ \lim_{|x|\to\infty}|x|^{\frac{n-2}{m}}f(x)=η. \] The unmatched branches may reach zero at a finite radius. We formulate the shooting construction through the slope--amplitude equations of the increasing and decreasing half-branches before their first turning points. As a consequence we obtain a new proof of the existence result of Peletier and Zhang \cite{PeZ}: there exists $β\in (β_-(ρ_1),β_+(ρ_1))$ for which the equation $Δ(f^m/m)+αf+βx\cdot\nabla f=0$, $f>0$, in $\mathbb{R}^n$ has a positive radial solution $f$ satisfying \[ f(0)=η_0,\qquad f_r(0)=0, \qquad \lim_{r\to\infty}r^{\frac{n-2}{m}}f(r) =C_*ρ_1^{-\frac{n-2}{2m}} η_0^{-\frac{n-2-nm}{2m}}. \] for some constant $C_*>0$ depending on $n$, $m$, and is independent of $ρ_1$ and $η_0$. For every selected matching value of $β$, this solution is unique among positive radial solutions with the prescribed origin data. When $ρ_1=1$, the function $V(x,t)=(T-t)^αf((T-t)^βx)$ is a backward self-similar solution of $u_t=Δ(u^m/m)$ in $\mathbb{R}^n\times (-\infty,T)$.