Advertisement

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
Article
  • 43 Downloads

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 \(\lambda _1>0\).

Mathematics Subject Classification

Primary 35B40 Secondary 35B20 35B09 

Preview

Unable to display preview. Download preview PDF.

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)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brendle, S.: Convergence of the Yamabe flow in dimension 6 and higher. Invent. Math. 170, 541–576 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  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)CrossRefzbMATHGoogle Scholar
  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)MathSciNetzbMATHGoogle Scholar
  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)MathSciNetCrossRefzbMATHGoogle Scholar
  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)MathSciNetCrossRefzbMATHGoogle Scholar
  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)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hsu, S.Y.: Singular limit and exact decay rate of a nonlinear elliptic equation. Nonlinear Anal. TMA 75(7), 3443–3455 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hsu, S.Y.: Existence and asymptotic behaviour of solutions of the very fast diffusion. Manuscr. Math. 140(3–4), 441–460 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hui, K.M., Kim, Sunghoon: Extinction profile of the logarithmic diffusion equation. Manuscr. Math. 143(3–4), 491–524 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  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

Copyright information

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

Authors and Affiliations

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

Personalised recommendations