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

Abstract

Let $n\ge 3$, $0 0$, $\eta_0>0$, $\rho_1>0$, $\beta_-(\rho_1)=-\frac{\rho_1}{2}$, $\beta_+(\rho_1)=\frac{m\rho_1}{n-2-nm}$ and $\alpha=\frac{2\beta+\rho_1}{1-m}$. For any $\beta_-(\rho_1)\le\beta\le\beta_+(\rho_1)$, we construct the unique maximal positive radial branch of \[ \Delta(f^m/m)+\alpha f+\beta x\cdot\nabla f=0 \] issuing from prescribed origin data $f(0)=\eta_0$ and $f_r(0)=0$. For any $\beta\le\beta_+(\rho_1)$, we construct the unique maximal positive radial branch at infinity satisfying \[ \lim_{|x|\to\infty}|x|^{\frac{n-2}{m}}f(x)=\eta. \] 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 $\beta\in (\beta_-(\rho_1),\beta_+(\rho_1))$ for which the equation $\Delta(f^m/m)+\alpha f+\beta x\cdot\nabla f=0$, $f>0$, in $\mathbb{R}^n$ has a positive radial solution $f$ satisfying \[ f(0)=\eta _0,\qquad f_r(0)=0, \qquad \lim_{r\to\infty}r^{\frac{n-2}{m}}f(r) =C_*\rho _1^{-\frac{n-2}{2m}} \eta _0^{-\frac{n-2-nm}{2m}}. \] for some constant $C_*>0$ depending on $n$, $m$, and is independent of $\rho_1$ and $\eta_0$. For every selected matching value of $\beta$, this solution is unique among positive radial solutions with the prescribed origin data. When $\rho _1=1$, the function $V(x,t)=(T-t)^\alpha f((T-t)^\beta x)$ is a backward self-similar solution of $u_t=\Delta(u^m/m)$ in $\mathbb{R}^n\times (-\infty,T)$.