Global behaviour of solutions of the fast diffusion equation

Abstract

We will extend a recent result of Choi and Daskalopoulos [4]. For any $$n\ge 3, 0<m<\frac{n-2}{n}, m\ne \frac{n-2}{n+2}$$ , $$\beta >0$$ and $$\lambda >0$$ , we prove the higher order expansion of the radially symmetric solution $$v_{\lambda ,\beta }(r)$$ of $$\frac{n-1}{m}\Delta v^m+\frac{2\beta }{1-m} v+\beta x\cdot \nabla v=0$$ in $$\mathbb {R}^n$$ , $$v(0)=\lambda $$ , as $$r\rightarrow \infty $$ . As a consequence for any $$n\ge 3$$ and $$0<m<\frac{n-2}{n}$$ if u is the solution of the equation $$u_t=\frac{n-1}{m}\Delta u^m$$ in $$\mathbb {R}^n\times (0,\infty )$$ with initial value $$0\le u_0\in L^{\infty }(\mathbb {R}^n)$$ satisfying $$u_0(x)^{1-m}=\frac{2(n-1)(n-2-nm)}{(1-m)\beta |x|^2}\left( \log |x|-\frac{n-2-(n+2)m}{2(n-2-nm)}\log (\log |x|)+K_1+o(1))\right) $$ as $$|x|\rightarrow \infty $$ for some constants $$\beta >0$$ and $$K_1\in \mathbb {R}$$ , then as $$t\rightarrow \infty $$ the rescaled function $$\widetilde{u}(x,t)=e^{\frac{2\beta }{1-m}t}u(e^{\beta t}x,t)$$ converges uniformly on every compact subsets of $$\mathbb {R}^n$$ to $$v_{\lambda _1,\beta }$$ for some constant $$\lambda _1>0$$ .