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Asymptotics of the Fast Diffusion Equation via Entropy Estimates

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Abstract

We consider non-negative solutions of the fast diffusion equation u t  = Δum with m ∈ (0, 1) in the Euclidean space \({{\mathbb R}^d}\), d ≧ 3, and study the asymptotic behavior of a natural class of solutions in the limit corresponding to t → ∞ for m ≧ m c  = (d − 2)/d, or as t approaches the extinction time when m < m c . For a class of initial data, we prove that the solution converges with a polynomial rate to a self-similar solution, for t large enough if m ≧ m c , or close enough to the extinction time if m < m c . Such results are new in the range m ≦ m c where previous approaches fail. In the range m c  < m < 1, we improve on known results.

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References

  • Aubin T.: Problèmes isopérimétriques et espaces de Sobolev. J. Differ. Geometry 11, 573–598 (1976)

    Article  MATH  Google Scholar 

  • Bakry, D., Émery, M.: Diffusions hypercontractives. In: Séminaire de probabilités, XIX, 1983/1984. Lecture Notes in Math., vol. 1123, pp. 177–206. Springer, Berlin, 1985

  • Barthe F., Roberto C.: Sobolev inequalities for probability measures on the real line. Studia Math. 159, 481–497 (2003) Dedicated to Professor Aleksander Pełczyński on the occasion of his 70th birthday (Polish)

    Article  MathSciNet  MATH  Google Scholar 

  • Barthe, F., Roberto, C.: Modified logarithmic Sobolev inequalities on \({\mathbb{R}}\) . Potential Anal. (to appear) (2008)

  • Beckner W.: Sobolev inequalities, the Poisson semigroup, and analysis on the sphere Sn. Proc. Nat. Acad. Sci. USA 89, 4816–4819 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Beckner W.: Sharp Sobolev inequalities on the sphere and the Moser–Trudinger inequality. Ann. Math. 138(2), 213–242 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  • Blanchet A., Bonforte M., Dolbeault J., Grillo G., Vázquez J.-L.: Hardy–Poincaré inequalities and applications to nonlinear diffusions. C. R. Math. Acad. Sci. Paris. 344, 431–436 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Bobkov S.G., Götze F.: Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 163, 1–28 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  • Boček L.: Eine Verschärfung der Poincaré-Ungleichung. Časopis Pěst. Mat. 108, 78–81 (1983)

    MATH  Google Scholar 

  • Bonforte, M., Vázquez, J.L.: Fine asymptotics near extinction and elliptic Harnack inequalities for the fast diffusion equation. Preprint, April 2006

  • Bonforte M., Vázquez J.L.: Global positivity estimates and Harnack inequalities for the fast diffusion equation. J. Funct. Anal. 240, 399–428 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Caffarelli L., Kohn R., Nirenberg L.: First order interpolation inequalities with weights. Compos. Math. 53, 259–275 (1984)

    MathSciNet  MATH  Google Scholar 

  • Carrillo J.A., Jüngel A., Markowich P.A., Toscani G., Unterreiter A.: Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities. Monatsh. Math. 133, 1–82 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • Carrillo J.A., Lederman C., Markowich P.A., Toscani G.: Poincaré inequalities for linearizations of very fast diffusion equations. Nonlinearity 15, 565–580 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Carrillo J.A., Toscani G.: Asymptotic L 1-decay of solutions of the porous medium equation to self-similarity. Indiana Univ. Math. J. 49, 113–142 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  • Carrillo J.A., Vázquez J.L.: Fine asymptotics for fast diffusion equations. Commun. Partial Differ. Equ. 28, 1023–1056 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • Catrina F., Wang Z.-Q.: On the Caffarelli–Kohn–Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions. Commun. Pure Appl. Math. 54, 229–258 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • Chen Y.Z., DiBenedetto E.: Hölder estimates of solutions of singular parabolic equations with measurable coefficients. Arch. Rational Mech. Anal. 118, 257–271 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Chua S.-K., Wheeden R.L.: Sharp conditions for weighted 1-dimensional Poincaré inequalities. Indiana Univ. Math. J. 49, 143–175 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  • Daskalopoulos P., del Pino M.: On the Cauchy problem for u t = Δ log u in higher dimensions. Math. Ann. 313, 189–206 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  • Daskalopoulos P., Sesum N. Eternal solutions to the Ricci flow on \({\mathbb{R}^{2}}\) . Int. Math. Res. Not., Art. ID 83610, pp. 20, 2006

  • Daskalopoulos, P., Sesum, N.: On the extinction profile of solutions to fast-diffusion. Preprint, 2006

  • Davies, E.B.: Heat kernels and spectral theory. Cambridge Tracts in Mathematics, vol. 92. Cambridge University Press, London, 1989

  • Davies E.B., Hinz A.M.: Explicit constants for Rellich inequalities in L p (Ω). Math. Z. 227, 511–523 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  • Del Pino M., Dolbeault J.: Best constants for Gagliardo–Nirenberg inequalities and applications to nonlinear diffusions. J. Math. Pure Appl. 81(9), 847–875 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Del Pino M., Sáez M.: On the extinction profile for solutions of u t  = Δu(N-2)/(N+2). Indiana Univ. Math. J. 50, 611–628 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • Denzler J., McCann R.J.: Fast diffusion to self-similarity: complete spectrum, long-time asymptotics, and numerology. Arch. Ration. Mech. Anal. 175, 301–342 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Friedman A., Kamin S.: The asymptotic behavior of gas in an n-dimensional porous medium. Trans. Am. Math. Soc. 262, 551–563 (1980)

    MathSciNet  MATH  Google Scholar 

  • Gross L.: Logarithmic Sobolev inequalities. Am. J. Math. 97, 1061–1083 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  • Herrero M.A., Pierre M.: The Cauchy problem for u t  = Δum when 0 < m < 1. Trans. Am. Math. Soc. 291, 145–158 (1985)

    MathSciNet  MATH  Google Scholar 

  • Hsu S.-Y.: Classification of radially symmetric self-similar solutions of u t  = Δlog u in higher dimensions. Differ. Integral Equ. 18, 1175–1192 (2005)

    MATH  Google Scholar 

  • Hsu S.-Y.: Extinction profile of solutions of a singular diffusion equation. Commun. Appl. Anal. 9, 67–93 (2005)

    ADS  MathSciNet  MATH  Google Scholar 

  • Hsu S.-Y.: Large time behavior of solutions of a singular diffusion equation in \({\mathbb{R}^{n}}\) . Nonlinear Anal. 62, 195–206 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Kim Y.J., McCann R.J.: Potential theory and optimal convergence rates in fast nonlinear diffusion. J. Math. Pure Appl. 86(1,9), 42–67 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • King J.: Self-similar behavior for the equation of fast nonlinear diffusion. Philos. Trans. R. Soc. Lond. Ser. A 343, 337–375 (1993)

    Article  ADS  MATH  Google Scholar 

  • Ladyženskaja, O.A., Solonnikov, V.A., Ural′ceva, N.N.: Linear and quasilinear equations of parabolic type. Translations of Mathematical Monographs, vol. 23. American Mathematical Society, Providence, 1967 (Translated from the Russian by S. Smith)

  • Lederman C., Markowich P.A.: On fast-diffusion equations with infinite equilibrium entropy and finite equilibrium mass. Commun. Partial Differ. Equ. 28, 301–332 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • Lieb E.H.: Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities. Ann. Math. 118(2), 349–374 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  • McCann, R.J., Slepčev, D.: Second-order asymptotics for the fast-diffusion equation. Int. Math. Res. Not., Art. ID 24947, pp. 22, 2006

  • Miclo, L.: Quand est-ce que les bornes de Hardy permettent de calculer une constante de Poincaré exacte sur la droite? Preprint, 2006

  • Nirenberg L.: On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa 13(3), 115–162 (1959)

    MathSciNet  MATH  Google Scholar 

  • Otto F.: The geometry of dissipative evolution equations: the porous medium equation. Commun. Partial Differ. Equ. 26, 101–174 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • Peletier M., Zhang H.: Self-similar solutions of a fast diffusion equation that do not conserve mass. Differ. Integral Equ. 8, 2045–2064 (1995)

    MathSciNet  MATH  Google Scholar 

  • Rodriguez A., Vazquez J.L., Esteban J.R.: The maximal solution of the logarithmic fast diffusion equation in two space dimensions. Adv. Differ. Equ. 2, 867–894 (1997)

    MathSciNet  MATH  Google Scholar 

  • Saloff-Coste L.: Precise estimates on the rate at which certain diffusions tend to equilibrium. Math. Z. 217, 641–677 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  • Talenti G.: Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 110(4), 353–372 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  • Vázquez J.L.: Asymptotic behavior for the porous medium equation posed in the whole space. J. Evol. Equ. 3, 67–118 (2003) (Dedicated to Philippe Bénilan)

    Article  MathSciNet  MATH  Google Scholar 

  • Vázquez, J.L.: The Porous Medium Equation. Mathematical Theory. Oxford Mathematical Monographs. The Clarendon Press/Oxford University Press, Oxford/New York, 2007

  • Vázquez J.L. Smoothing and decay estimates for nonlinear diffusion equations. Oxford Lecture Notes in Maths. and its Applications, vol. 33. Oxford University Press, New York, 2006

  • Vázquez J.L., Esteban J.R., Rodríguez A.: The fast diffusion equation with logarithmic nonlinearity and the evolution of conformal metrics in the plane. Adv. Differ. Equ. 1, 21–50 (1996)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. CRM, Apartat 50, 08193, Bellaterra, Spain

    Adrien Blanchet

  2. Departamento de Matemáticas, University Autónoma de Madrid, Campus de Cantoblanco, 28049, Madrid, Spain

    Matteo Bonforte & Juan Luis Vázquez

  3. Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Turin, Italy

    Matteo Bonforte & Gabriele Grillo

  4. CEREMADE, University Paris Dauphine, Pl. de Lattre de Tassigny, 75775, Paris 16, France

    Jean Dolbeault

Authors
  1. Adrien Blanchet
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  2. Matteo Bonforte
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  3. Jean Dolbeault
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  4. Gabriele Grillo
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  5. Juan Luis Vázquez
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Correspondence to Juan Luis Vázquez.

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Communicated by V. Sverak

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Blanchet, A., Bonforte, M., Dolbeault, J. et al. Asymptotics of the Fast Diffusion Equation via Entropy Estimates. Arch Rational Mech Anal 191, 347–385 (2009). https://doi.org/10.1007/s00205-008-0155-z

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