Skip to main content
SpringerLink
Log in

Pointwise estimates of solutions to the weighted porous medium equation and the fast diffusion one via weighted Riesz potentials

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

For the weighted parabolic equation v(x)ut −  div (ω(x)um − 1 ∇ u) = f(x, t), u ≥ 0, m ≠ 1, we prove the local boundedness of weak solutions in terms of the weighted Riesz potential on the right-hand side of the equation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. B. Abdellaoui and I. Peral Alonso, “Hölder reqularity and Harnack inequality for degenerate parabolic equations related to Caffarelli–Kohn–Nirenberg inequalities,” Nonlin. Anal., 57(7-8), 971-1003 (2004).

    Article  MATH  Google Scholar 

  2. M. Aizerman and B. Simon, “Brownian motion and Harnack inequality for Schr¨odinger operators,” Comm. Pure Appl. Math., 35, 209-273 (1982).

    Article  MathSciNet  Google Scholar 

  3. D. G. Aronson and J. Serrin, “Local behavior of solutions of quasilinear parabolic equations,” Arch. Rat. Mech. Anal., 25, 81-122 (1967).

    Article  MathSciNet  MATH  Google Scholar 

  4. D. G. Aronson, “The porous medium equations,” in: Nonlinear Diffusion Problems, Springer, New York, 1986, pp. 1-46.

  5. V. Bögelein, F. Duzaar, and U. Gianazza, “Continuity estimates for porous medium type equations with measure data,” J. Funct. Anal., 267, 3351-3396 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  6. V. Bögelein, F. Duzaar, and U. Gianazza, “Porous medium type equations with measure data and potential estimates,” SIAM J. Math. Anal., 45(6), 3283-3330 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  7. V. Bögelein, F. Duzaar, and U. Gianazza, “Sharp boundedness and continuity results for the singular porous medium equation,” Israel J. Math., 214, 259-314 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  8. S. Bonafede and I. I. Skrypnik, “On Hölder continuity of solutions of doubly nonlinear parabolic equations with weight,” Ukr. Math. J., 51(7), 996-1012 (1999).

    Article  MATH  Google Scholar 

  9. M. Bonforte, J. Dolbeanlt, M. Muratori, and B. Nazaret, “Weighted fast diffusion equations (Part I): Sharp asymptotic rates without symmetry breaking in Caffarelli–Kohn–Nirenberg inequalities,” Kin. Rel. Mod., 10, 33-59 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  10. M. Bonforte, J. Dolbeanlt, M. Muratori, and B. Nazaret, “Weighted fast diffusion equations (Part II): Sharp asymptotic rates of convergence in relative error by entropy methods,” Kin. Rel. Mod., 10, 61-91 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  11. M. Bonforte and N. Simonov, Quantitative a priori estimates for fast diffusion equations with Caffarelli–Kohn–Nirenberg weights. Harnack inequalities and Hölder continuity, arXiv: 1804.03537.2018.

  12. K. O. Buryachenko and I. I. Skrypnik, “Riesz potentials and pointwise estimates of solutions to anisotropic porous medium equation,” Nonlin. Anal., 178, 56-85 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  13. L. A. Caffarelli and C. L. Evans, “Continuity of the temperature in the two phase Stefan problem,” ARMA, 81, 199-220 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  14. L. A. Caffarelli and A. Friedman, “Regularity of the free boundary of a fas flow in an n-dimensional porous medium,” Indiana Univ. J., 29, 361-391 (1980).

    Article  MATH  Google Scholar 

  15. S. Chanillo and R. L. Wheeden, “Harnack’s inequality and mean-value inequalities for solution of degenerate elliptic equations,” Comm. Partial Diff. Equat., 11(10), 1111-1134 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  16. S. Chanillo and R. L. Wheeden, “Weighted Poincaré and Sobolev enequalities and estimates for weighted Peano maximal functions,” Amer. J. Math., 107(5), 1191-1226 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  17. V. Chiado Piat and F. Serra Cassano, “Relaxation of degenerate variational integrals,” Nonlin. Anal., 22(4), 409-424 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  18. V. Chiado Piat and F. Serra Cassano, “Some remarks about the density of smooth function in weighted Sobolev space,” J. Convex Anal., 2, 135-142 (1994).

    MathSciNet  MATH  Google Scholar 

  19. F. M. Chiarenza and M. Frasca, “A note on a weighted Sobolev inequality,” Proc Amer. Soc., 93(4), 703-704 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  20. F. Chiarenza, E. Fabes, and N. Garofalo, “Harnack’s inequality for Schrödinger operators and the continuity of solutions,” Proc. Amer. Math. Soc., 98, 415-425 (1986).

    MathSciNet  MATH  Google Scholar 

  21. F. M. Chiarenza and M. Frasca, “Boundedness for the solutions of a degenerate parabolic equations,” Appl. Anal., 17, 243-261 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  22. F. M. Chiarenza and R. P. Separioni, “A Harnack inequality for degenerate parabolic equations,” Comm. Part. Diff. Equat., 9(8), 719-749 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  23. F. M. Chiarenza and R. P. Separioni, “A Harnack inequality for degenerate parabolic equations,” Rend. Sem. Math. Univ. Padova, 73, 179-190 (1985).

    MathSciNet  Google Scholar 

  24. F. M. Chiarenza and R. P. Separioni, “Degenerate parabolic equations and Harnack inequality,” Ann. Mat. Pura Appl. 139-162 (1984).

  25. F. M. Chiarenza and R. P. Separioni, “Pointwise estimates for degenerate parabolic equations,” Appl. Anal., 23(4), 287-299 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  26. A. Dall’Aglio, D. Giachetti, and I. Peral, “Results on parabolic equations related to some Caffarelli–Kohn– Nirenberg inequalities,” SIAM J. Math. Anal., 36(3), 691-716 (2004/2005).

    Article  MathSciNet  MATH  Google Scholar 

  27. E. De Giorgi, “Sulla differenziabilita e l’analiticita delle estremali degli integrali multipli regolary,” Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur., 3, 25-43 (1957).

    MATH  Google Scholar 

  28. E. Di Benedetto, Degenerate Parabolic Equations, Springer, New York, 1993.

    Book  Google Scholar 

  29. E. Di Benedetto and A. Friedman, “Hölder estimates for nonlinear degenerate parabolic systems,” J. Reine Angew. Math., 357, 1-22 (1985).

    MathSciNet  Google Scholar 

  30. G. Di Fazio, M. Stella Fanciullo, and P. Zamboni, “Harnack inequality and regularity for degenerate quasilinear elliptic equations,” Math. Zeitschr., 264(3), 679-695 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  31. E. B. Fabes, C. E. Kening, and R. P. Serapioni, “The local regularity of solutions of degenerate elliptic equations,” Comm. Part. Diff. Equat., 7(1), 77-116 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  32. J. C. Fernandes and B. Franchi, “Existence and properties of the Green function for a class of degenerate parabolic equations,” Rev. Mat. Iberoam., 12(2), 491-524 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  33. J. C. Fernandes, “Mean value and Harnack inequalities for a certain class of degenerate parabolic equations,” Rev. Mat. Iberoam., 7(3), 247-286 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  34. F. Ferrari, “Harnack inequality for two-weight subelliptic p-Laplace operator,” Math. Nachr., 279(8), 815-830 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  35. J. Garcia Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland, Amsterdam, 1985.

    MATH  Google Scholar 

  36. G. Grillo, M. Muratori, and M. M. Porzio, “Porous media equations with two weights: smoothing and decay properties of energy solutions via Poincaré inequalities.” Discr. Contin. Dyn. Syst., 33, 3599-3640 (2013).

    Article  MATH  Google Scholar 

  37. C.E. Gutierrez, “Harnack’s inequality for degenerate Schrödinger operators.” Trans. Amer. Math. Soc., 312(1), 403-419 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  38. C. Gutierrez and F. Nelson, “Bounds for the fundamental solution of degenerate parabolic equations,” Comm. Part. Diff. Equat., 13(5), 635-649 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  39. C. E. Gutierrez and R. L. Wheeden, “Bounds for the fundamental solution of degenerate parabolic equations,” Comm. Part. Diff. Equat., 17(7), 1287-1307 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  40. C. E. Gutierrez and R. L. Wheeden, “Harnack’s inequalities for degenerate parabolic equations,” Comm. Part. Diff. Equat., 16(4-5), 745-770 (1991).

    Article  MATH  Google Scholar 

  41. C. E. Gutierrez and R. L. Wheeden, “Mean value and Harnack inequalities for degenerate parabolic equations,” Colloq. Math., 60/61(1), 157-194 (1990).

  42. C. E. Gutierrez and R. L. Wheeden, “Sobolev interpolation inequalities with weights,” Trans. Amer. Math. Soc., 323, 263-281 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  43. J. Heinonen, T. Kilpeläinen, and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Clarendon Press, Oxford, 1993.

    MATH  Google Scholar 

  44. S. Kamin and P. Rosenau, “Nonlinear diffusion in a finite mass medium,” Comm. Pure Appl. Math., 35(1), 113-127 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  45. S. Kamin and P. Rosenau, “Propagatio of thermal waves in an inhomogeneous medium,” Comm. Pure Appl. Math., 34(6), 831-852 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  46. T. Kilpeläinen and J. Malý, “The Wiener test and potential estimates for quasilinear elliptic equations,” Acta Math., 172, 137-161 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  47. K. Kurata, “Continuity and Harnack’s inequality for solutions of elliptic partial differential equations of second order,” Indiana Univ. Math. J., 43, 411-440 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  48. D. Labutin, “Potential estimates for a class of fully nonlinear elliptic equations,” Duke Math. J., 111(1), 1-49 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  49. O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968.

    MATH  Google Scholar 

  50. O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence, RI, 1968.

    Book  MATH  Google Scholar 

  51. V. Liskevich and I. I. Skrypnik, “Harnack’s inequality and continuity of solutions to quasi-linear degenerate parabolic equations with coefficients from Kato-type classes,” J. Diff. Equat., 247(10), 2740-2777 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  52. V. Liskevich and I. I. Skrypnik, “Pointwise estimates for solutions to the porous medium equation with measure as a forcing term,” Israel J. Math., 194, 259-275 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  53. A. Mohamed, “Harnack’s inequalities for solutions of some degenerate elliptic equations,” Rev. Mat. Iberoam., 18, 325-354 (2002).

    Article  Google Scholar 

  54. J. Moser, “A Harnack inequality for parabolic differential equations,” Comm. Pure Appl. Math., 17(1), 101-134 (1964).

    Article  MathSciNet  MATH  Google Scholar 

  55. J. Moser, “On Harnack’s theorem for elliptic differential equations,” Comm. Pure Appl. Math., 14(3), 577-591 (1961).

    Article  MathSciNet  MATH  Google Scholar 

  56. M. Muratori, Weighted Functional Inequalities and Nonlinear Diffusions of Porous Medium Type, Ph.D. thesis, Politec. di Milano and Univ. Paris, 2015.

  57. J. Nash, “Continuity of solutions of parabolic and elliptic equations,” Amer. J. Math., 80, 931-954 (1958).

    Article  MathSciNet  MATH  Google Scholar 

  58. F. Paronetto, “Harnack’s inequalities for mixed type evolution equations,” J. Diff. Equat., 260(6), 5259- 5355 (2016).

    Article  MATH  Google Scholar 

  59. F. Paronetto, “A time regularity result for forward-backward parabolic equations,” Boll. Union Math. Ital., 4(9), 69-77 (2011).

    MathSciNet  MATH  Google Scholar 

  60. F. Paronetto, “Local boundednss for forvard- backward parabolic De Giorgi classes with coefficients depending on time,” Nonlin. Anal., 158, 168-198 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  61. J. Serrin, “Local behaviour of solutions of quasilinear equations,” Acta Math., 111, 302-347 (1964).

    Article  MATH  Google Scholar 

  62. I. I. Skrypnik, “Continuity of solutions to singular parabolic equations with coefficients from Kato-type classes,” Annali Mat. Pura Appl., 195(4), 1158-1176 (2016).

    Article  MathSciNet  Google Scholar 

  63. S. Sturm, “Pointwise estimates for porous medium type equations with low order terms and measure data,” Electron J. Diff. Equat., 215(101), 1-25 (2015).

    MathSciNet  MATH  Google Scholar 

  64. M. Surnachev, “A Harnack inequality for weighted degenerate parabolic equations,” J. Diff. Equat., 248(8), 2092-2129 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  65. M. Surnachev, “Regularity of solutions of parabolic equations with a double nonlinearity and a weight,” Trans. Moscow Math. Soc., 75, 259-280 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  66. N. Trudinger, “Pointwise estimates and quasilinear parabolic equations,” Comm. Pure Appl. Math., 21(3), 205-226 (1968).

    Article  MathSciNet  MATH  Google Scholar 

  67. N. Trudinger and X.-J. Wang, “On the weak continuity of elliptic operators and applications to potential theory,” Amer. J. Math., 124(2), 369-410 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  68. J. L. Vazquez, The Porous Medium Equation, Oxford Univ. Press, Oxford, 2007.

    Google Scholar 

  69. Y. Wang, P. Nin, and X. Cui, “Harnack estimates for a quasi-linear parabolic equations with a singular weight,” Nonlin. Anal., 74(17), 6265-6286 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  70. Q. Zhang, “A Harnack’s inequality for the equation ∇(a∇u)+b∇u = 0 when |b| ∈ Kn+1,” Manuscr. Math., 89(1), 61-77 (1996).

    Article  MathSciNet  Google Scholar 

  71. Q. Zhang, “On a parabolic equation with a singular lower order term,” Trans. Amer. Math. Soc., 348, 2811-2844 (1996).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Slavyansk, Ukraine

    Yevhen Zozulia

Authors
  1. Yevhen Zozulia
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yevhen Zozulia.

Additional information

Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 17, No. 1, pp. 116–144 January–March, 2020.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zozulia, Y. Pointwise estimates of solutions to the weighted porous medium equation and the fast diffusion one via weighted Riesz potentials. J Math Sci 248, 233–254 (2020). https://doi.org/10.1007/s10958-020-04872-4

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-020-04872-4

Keywords

Search

Navigation