Journal of Dynamics and Differential Equations

, Volume 30, Issue 3, pp 1029–1051 | Cite as

Boundedness for Some Doubly Nonlinear Parabolic Equations in Measure Spaces

  • Eurica HenriquesEmail author
  • Rojbin Laleoglu


In the context of measure spaces equipped with a doubling non-trivial Borel measure supporting a Poincaré inequality, we derive local and global sup bounds of the nonnegative weak subsolutions of
$$\begin{aligned} (u^{q})_t-\nabla \cdot {(|\nabla u|^{p-2}\nabla u)}=0, \quad \mathrm {in} \ U_\tau = U \times (\tau _1, \tau _2] , \quad p>1,\quad q>1 \end{aligned}$$
and of its associated Dirichlet problem, respectively. For particular ranges of the exponents p and q, we show that any locally nonnegative weak subsolution, taken in \(Q (\subset \bar{Q}\subset U_\tau )\), is controlled from above by the \(L^\alpha (\bar{Q}) \)-norm, for \(\alpha = \max \{p, q+1\}\). As for the global setting, under suitable assumptions on the boundary datum g and on the initial datum, we obtain sup bounds for u, in \(U \times \{ t\}\), which depend on the \(\sup g\) and on the \(L^{q+1}(U \times (\tau _1, \tau _1+t])\)-norm of \((u-\sup g)_+\), for all \(t \in (0, \tau _2-\tau _1]\). On the critical ranges of p and q, a priori local and global \(L^\infty \) estimates require extra qualitative information on u.


Boundedness Singular PDE Degenerate PDE 

Mathematics Subject Classification

Primary 35B50 Secondary 35K65 35K67 35K20 



Funding was provided by Fundação para a Ciância e a Tecnologia.


  1. 1.
    Bonforte, M., Grillo, G.: Super and ultracontractive bounds for doubly nonlinear evolution equations. Rev. Mat. Iberoam. 22(1), 111–129 (2006)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Cirmi, G.R., Porzio, M.M.: \(L^\infty \)-solutions for some nonlinear degenerate elliptic and parabolic equations. Ann. Mat. Pura Appl. CLXIX(IV), 67–86 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Diaz, J.I., Thélin, F.: On a nonlinear parabolic problem arising in some models related to turbulent flows. SIAM J. Math. Anal. 25(4), 1085–1111 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    DiBenedetto, E.: Degenerate Parabolic Equations. Universitext. Springer, New York (1993)Google Scholar
  5. 5.
    Fornaro, S., Sosio, M., Vespri, V.: \(L_{loc}^r - L^{\infty }_{loc}\) estimates and expansion of positivity for a class of doubly nonlinear singular parabolic equations. Discrete Contin. Dyn. Syst. Ser. S 7(4), 737–760 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hajlasz, P., Koskela, P.: Sobolev met Poincaré. Mem. Am. Math. Soc. 145(688), x+101 (2000)zbMATHGoogle Scholar
  7. 7.
    Henriques, E., Laleoglu, R.: Local Hölder continuity for some doubly nonlinear parabolic equations in measure spaces. Nonlinear Anal. Theory Methods Appl. 79, 156–175 (2013)CrossRefzbMATHGoogle Scholar
  8. 8.
    Ivanov, A.V.: Regularity for doubly nonlinear parabolic equations. J. Math. Sci. 83(1), 22–37 (1997)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kalashnikov, A.S.: Some problems of the qualitative theory of nonlinear degenerate second order parabolic equations. Russ. Math. Surv. 42, 169–222 (1987)CrossRefzbMATHGoogle Scholar
  10. 10.
    Kinnunen, J., Kuusi, T.: Local behaviour of solutions to doubly nonlinear parabolic equations. Math. Ann. 337(3), 705–728 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kinnunen, J., Shanmugalingam, N.: Regularity of quasi-minimizers on metric spaces. Manuscr. Math. 105(3), 401–423 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kuusi, T., Laleoglu, R., Siljander, J., Urbano, J.M.: Hölder continuity for Trudinger’s equation in measure spaces. Calc. Var. Partial Differ. Equ. 45(1–2), 193–229 (2012)CrossRefzbMATHGoogle Scholar
  13. 13.
    Kuusi, T., Siljander, J., Urbano, J.M.: Hölder continuity to a doubly nonlinear parabolic equation. Indiana Univ. Math. J. 61, 399–430 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Ladyzhenskaya, 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 (1968)CrossRefGoogle Scholar
  15. 15.
    Leibenzon, L.S.: General problem of the movement of a compressible fluid in a porous medium. Izv. Akad. Nauk SSSR Geogr. Geophys. 9, 7–10 (1945). (in Russian) MathSciNetGoogle Scholar
  16. 16.
    Porzio, M.M.: \(L^{\infty }_{loc}\)-estimates for degenerate and singular parabolic equations. Nonlinear Anal. Theory Methods Appl. 17(11), 1093–1107 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Porzio, M.M.: \(L^\infty \) estimates for a class of doubly nonlinear parabolic equations with source. Rend. Mat. Ser. VI I(16), 433–456 (1996)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Porzio, M.M.: On decay estimates. J. Evol. Equ. 9, 561–591 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Trudinger, N.S.: Pointwise estimates and quasilinear parabolic equations. Commun. Pure Appl. Math. 21, 205–226 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Vespri, V.: \(L^\infty \) estimates for nonlinear parabolic equations with natural growth conditions. Rend. Sem. Mat. Univ. Padova 90, 1–8 (1993)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Vespri, V.: Harnack type inequalities for solutions of certain doubly nonlinear parabolic equations. J. Math. Anal. Appl. 181, 104–131 (1994)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Centro de Matemática: Pólo CMAT - UTADUniversidade de Trás-os-Montes e Alto Douro (UTAD)Vila RealPortugal

Personalised recommendations