Advertisement

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 Henriques
  • Rojbin Laleoglu
Article
  • 88 Downloads

Abstract

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.

Keywords

Boundedness Singular PDE Degenerate PDE 

Mathematics Subject Classification

Primary 35B50 Secondary 35K65 35K67 35K20 

Notes

Acknowledgements

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

References

  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