manuscripta mathematica

, Volume 158, Issue 1–2, pp 103–117 | Cite as

Global behaviour of solutions of the fast diffusion equation

  • Shu-Yu HsuEmail author


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 \(\lambda _1>0\).

Mathematics Subject Classification

Primary 35B40 Secondary 35B20 35B09 


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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsNational Chung Cheng UniversityChia-YiTaiwan, ROC

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