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

Abstract

Let $n\ge 3$, $0<m<\frac{n-2}{n}$, $\eta >0$, ${\eta }_0>0$, ${\rho }_1>0$, ${\beta }_{-}({\rho }_1)=-\frac{{\rho }_1}{2}$, ${\beta }_{+}({\rho }_1)=\frac{m{\rho }_1}{n-2-nm}$ 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 \[ Δ(f^m/m)+αf+βx\cdot\nabla f=0 \] 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 \[ \lim_{|x|\to\infty}|x|^{\frac{n-2}{m}}f(x)=η. \] 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 $\beta \in ({\beta }_{-}({\rho }_1),{\beta }_{+}({\rho }_1))$ 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 \[ f(0)=η_0,\qquad f_r(0)=0, \qquad \lim_{r\to\infty}r^{\frac{n-2}{m}}f(r) =C_*ρ_1^{-\frac{n-2}{2m}} η_0^{-\frac{n-2-nm}{2m}}. \] for some constant $C_{\ast }>0$ 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 ${\mathbb{R}}^n\times (-\infty ,T)$.

49 pages, I have rewritten the whole papers correcting various mistakes in the paper