Skip to main content
SpringerLink
Log in

Parabolic Equations with Anisotropic Nonstandard Growth Conditions

  • Conference paper
  • First Online:
Free Boundary Problems

Part of the book series: International Series of Numerical Mathematics ((ISNM,volume 154))

Abstract

We study the Dirichlet problem for a class of nonlinear parabolic equations with nonstandard anisotropic growth conditions. Equations of this class generalize the evolutional p(x, t)-Laplacian equation. We prove the existence of a bounded weak solution and study its localization (vanishing) properties.

The first author was partially supported by the research project DECONT, FCT/MCES (Portugal) at the “Centro de Matemática”, Universidade da Beira Interior.

The second author was supported by the research grants MTM-2004-05417 (Spain) and HPRN-CT-2002-00274 (EC).

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. E. Acerbi and G. Mingione, Regularity results for a class of functionals with non-standard growth, Arch. Rational Mech. Anal., 156 (2001), pp. 121–140.

    Article  MathSciNet  Google Scholar 

  2. E. Acerbi and G. Mingione, Regularity results for stationary electro-rheological fluids, Arch. Rational Mech. Anal., 164 (2002), pp. 213–259.

    Article  MathSciNet  Google Scholar 

  3. E. Acerbi, G. Mingione, and G.A. Seregin, Regularity results for parabolic systems related to a class of non-Newtonian fluids, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), pp. 25–60.

    MathSciNet  MATH  Google Scholar 

  4. S. Antontsev and S. Shmarev, Elliptic equations and systems with nonstandard growth conditions: existence, uniqueness and localization properties of solutions, Nonlinear Analysis Serie A: Theory and Methods (to appear).

    Google Scholar 

  5. S. Antontsev and S. Shmarev, Existence and uniqueness of solutions of degenerate parabolic equations with variable exponents of nonlinearity, Preprint 3, Departamento de Matemática, Universidade da Beira Interior, 2004, 16 pp., www.ubi.pt/externos/noe.html. (To appear in “Fundamental and Applied Mathematics”, 2005.)

    Google Scholar 

  6. —, A model porous medium equation with variables exponent of nonlinearity: existence, uniqueness and localization properties of solutions, Nonlinear Analysis Serie A: Theory and Methods, 60 (2005), pp. 515–545.

    Google Scholar 

  7. —, On localization of solutions of elliptic equations with nonhomogeneous anisotropic degeneracy, Siberian Math. Journal, 46(5) (2005), pp. 963–984.

    Google Scholar 

  8. S. Antontsev, J.I. Díaz, and S. Shmarev, Energy Methods for Free Boundary Problems:Applications to Non-linear PDEs and Fluid Mechanics, Birkhäuser, Boston, 2002. Progress in Nonlinear Differential Equations and Their Applications, Vol. 48.

    Google Scholar 

  9. S. Antontsev and V.V. Zhikov, Higher integrability for parabolic equations of p(x, t)-Laplacian type, Advances in Differential Equations, 10(9), pp. 1053–1080, (2005).

    MathSciNet  MATH  Google Scholar 

  10. L. Diening, Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces L p(·)and W k,p(·), Preprint 22/2002-15.07.2002, Albert-Ludwigs-University Freiburg, 2002. pp.1–13.

    Google Scholar 

  11. D. Edmunds and J. Rákosnik, Sobolev embeddings with variable exponent, Studia Math., 143(3) (2000), pp. 267–293.

    Article  MathSciNet  Google Scholar 

  12. P. Harjulehto and P. Hästö, An overview of variable exponent Lebesgue and Sobolev spaces, in Future trends in geometric function theory, vol. 92 of Rep. Univ. Jyväskylä Dep. Math. Stat., Univ. Jyväskylä, Jyväskylä, 2003, pp. 85–93.

    Google Scholar 

  13. E. Henriques and J.M. Urbano, Intrinsic scaling for PDE’s with an exponential nonlinearity, Preprint 04-33, Universidade de Coimbra, Departamento de Matemática, 2004.

    Google Scholar 

  14. O. Kováčik and J. Rákosník, On spaces L p(x)and W k,p(x), Czechoslovak Math. J., 41 (1991), pp. 592–618.

    Article  MathSciNet  Google Scholar 

  15. O. A. Ladyženskaja, V.A. Solonnikov, and N.N. Uraľtseva, Linear and quasilinear equations of parabolic type, American Mathematical Society, Providence, R.I., 1967. Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23.

    Google Scholar 

  16. J. Musielak, Orlicz spaces and modular spaces, vol. 1034 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1983.

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemática, Universidade da Beira Interior, Portugal

    S. Antontsev & S. Shmarev

  2. Departaménto de Matematicas, Universidad de Oviedo, España

    S. Shmarev

Authors
  1. S. Antontsev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Shmarev
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Departamento de Matemática Faculdade de Ciências e Tecnologia, Universidade de Coimbra, Apartado 3008, 3001-454, Coimbra, Portugal

    Isabel Narra Figueiredo

  2. Universidade de Lisboa / CMAF, Av. Prof. Gama Pinto 2, 1649-003, Lisboa, Portugal

    José Francisco Rodrigues

  3. Departamento de Matemática, Universidade do Minho, Campus de Gualtar, 4710-057, Braga, Portugal

    Lisa Santos

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Birkhäuser Verlag Basel/Switzerland

About this paper

Cite this paper

Antontsev, S., Shmarev, S. (2006). Parabolic Equations with Anisotropic Nonstandard Growth Conditions. In: Figueiredo, I.N., Rodrigues, J.F., Santos, L. (eds) Free Boundary Problems. International Series of Numerical Mathematics, vol 154. Birkhäuser, Basel. https://doi.org/10.1007/978-3-7643-7719-9_4

Download citation

Publish with us

Policies and ethics

Search

Navigation