manuscripta mathematica

, Volume 158, Issue 1–2, pp 103–117

# Global behaviour of solutions of the fast diffusion equation

Article

## Abstract

We will extend a recent result of Choi and Daskalopoulos . 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

## Preview

Unable to display preview. Download preview PDF.

## References

1. 1.
Aronson, D.G.: The porous medium equation. CIME Lectures in Some Problems in Nonlinear Diffusion, Lecture Notes in Mathematics 1224. Springer, New York (1986)Google Scholar
2. 2.
Brendle, S.: Convergence of the Yamabe flow for arbitrary energy. J. Differ. Geom. 69, 217–278 (2005)
3. 3.
Brendle, S.: Convergence of the Yamabe flow in dimension 6 and higher. Invent. Math. 170, 541–576 (2007)
4. 4.
Choi, B., Daskalopoulos, P.: Yamabe flow: steady solutions and type II singularities. arXiv:1709.03192v1
5. 5.
Dahlberg, B.E.J., Kenig, C.: Non-negative solutions of generalized porous medium equations. Rev. Mat. Iberoam. 2, 267–305 (1986)
6. 6.
Daskalopoulos, P., del Pino, M., King, J., Sesum, N.: New type I ancient compact solutions of the Yamabe flow. arXiv:1601.05349v1
7. 7.
Daskalopoulos, P., Kenig, C.E.: Degenerate diffusion-initial value problems and local regularity theory, Tracts in Mathematics 1. European Mathematical Society (2007)Google Scholar
8. 8.
Daskalopoulos, P., Sesum, N.: On the extinction profile of solutions to fast diffusion. J. Reine Angew. Math. 622, 95–119 (2008)
9. 9.
Daskalopoulos, P., del Pino, M., King, J., Sesum, N.: Type I ancient compact solutions of the Yamabe flow. Nonlinear Anal. Theory Methods Appl. 137, 338–356 (2016)
10. 10.
del Pino, M., Sáez, M.: On the extinction profile for solutions of $$u_t=\Delta u^{(n-2)/(N+2)}$$. Indiana Univ. Math. J. 50(1), 611–628 (2001)
11. 11.
Galaktionov, V.A., Peletier, L.A.: Asymptotic behaviour near finite-time extinction for the fast diffusion equation. Arch. Rat. Mech. Anal. 139, 83–98 (1997)
12. 12.
Hsu, S.Y.: Singular limit and exact decay rate of a nonlinear elliptic equation. Nonlinear Anal. TMA 75(7), 3443–3455 (2012)
13. 13.
Hsu, S.Y.: Existence and asymptotic behaviour of solutions of the very fast diffusion. Manuscr. Math. 140(3–4), 441–460 (2013)
14. 14.
Hui, K.M., Kim, Sunghoon: Extinction profile of the logarithmic diffusion equation. Manuscr. Math. 143(3–4), 491–524 (2014)
15. 15.
Peletier, L.A.: The porous medium equation. In: Amann, H., Bazley, N., Kirchgassner, K. (eds.) Applications of Nonlinear Analysis in the Physical Sciences. Pitman, Boston (1981)Google Scholar
16. 16.
Vazquez, J.L.: Smoothing and Decay Estimates for Nonlinear Diffusion Equations, Oxford Lecture Series in Mathematics and its Applications 33. Oxford University Press, Oxford (2006)Google Scholar