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Nonlinear anisotropic degenerate parabolic equations with variable exponents and irregular data

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Abstract

In this paper, we prove the existence and the regularity of weak solutions for a class of nonlinear anisotropic parabolic equations with \(p_i(\cdot )\) growth conditions, degenerate coercivity and \(L^{m(\cdot )}\) data, with \(m(\cdot )>1\) being small. The functional setting involves Lebesgue-Sobolev spaces with variable exponents.

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References

  1. Ayadi, H., Mokhtari, F.: Nonlinear anisotropic elliptic equations with variable exponents and degenerate coercivity. Electron. J. Differ. Equ. 2018, 1–23 (2018)

    Article  MathSciNet  Google Scholar 

  2. Bendahmane, M., Karlsen, K.H., Saad, M.: Nonlinear anisotropic elliptic and parabolic equations with variable exponents and \(L^1\)-data. Commun. Pure Appl. Anal. 12(3), 1201–1220 (2013)

    Article  MathSciNet  Google Scholar 

  3. Bendahmane, M., Wittbold, P., Zimmermann, A.: Renormalized solutions for a nonlinear parabolic equation with variable exponents and \(L^1\)-data. J. Differ. Equ. 249, 1483–1515 (2010)

    Article  Google Scholar 

  4. Fan, X.: Anisotropic variable exponent Sobolev spaces and Laplacian equations. Complex Var. Elliptic Equ. 56, 623–642 (2011)

    Article  MathSciNet  Google Scholar 

  5. Li, F.: Exisyence and regularity results for some parabolic equations with degenerate coercivity, Annales Academiientiarum Fennicthematica. Mathematica 37, 605–633 (2012)

    Google Scholar 

  6. Mokhtari, F.: Anisotropic parabolic problems with measure data. Differ. Equ. Appl. 2, 123–150 (2010)

    MathSciNet  MATH  Google Scholar 

  7. Mokhtari, F.: Nonlinear anisotropic parabolic equations in \(l^m\). Arab J. Math. Sci. 20(1), 1–10 (2014)

    MathSciNet  MATH  Google Scholar 

  8. Mokhtari, F.: Nonlinear anisotropic parabolic equations in \(\mathbb{R}^N\)with locally integrable data. Math. Methods Appl. Sci. (2)36, 196–207 (2013)

  9. Mokhtari, F.: Regularity of the solution to nonlinear anisotropic elliptic equations with variable exponents and irregular data. Mediterr. J. Math. (14), 141 (2017)

  10. Mokhtari, F., Mecheter, R.: Anisotropic degenerate parabolic problems in \(\mathbb{R}^{N}\) with variable exponent and locally integrable data. Mediterr. J. Math. 16(3), 1–21 (2019)

    Article  Google Scholar 

  11. Rákosník, J.: Some remarks to anisotropic Sobolev spaces II. Beitr. Anal. 15, 127–140 (1981)

    MATH  Google Scholar 

  12. Simon, J.: compact sets in the space \(L^p(0, T;B)\). Annali di Matematica 146, 65–96 (1987)

    Article  MathSciNet  Google Scholar 

  13. Troisi, M.: Theoremi di inclusione per Spazi di Sobolev non isotropi. Ricerche mat. 18, 3–24 (1969)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the referees for the useful comments and suggestions. This research is supported by Ministry of Higher Education and Scientific Research of Algeria, PRFU project C00L03UN160120190001.

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Authors and Affiliations

  1. Laboratory of Functional Analysis and Geometry of Spaces, Mohamed Boudiaf-M’sila, University, P.O. Box 166 M’sila 28000, Algeria & Department of Mathematics, ENS, P.O. Box 92, Kouba, Algiers, 16308, Algeria

    Hellal Abdelaziz & Fares Mokhtari

  2. Laboratory of Nonlinear Partial Differential Equations and History of Mathematics, ENS-Kouba, Algiers & Department of Mathematics, Algiers University, 2 Street Didouche Mourad, Algiers, 16000, Algeria

    Hellal Abdelaziz & Fares Mokhtari

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  1. Hellal Abdelaziz
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  2. Fares Mokhtari
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Correspondence to Fares Mokhtari.

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Abdelaziz, H., Mokhtari, F. Nonlinear anisotropic degenerate parabolic equations with variable exponents and irregular data. J Elliptic Parabol Equ 8, 513–532 (2022). https://doi.org/10.1007/s41808-022-00161-0

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  • DOI: https://doi.org/10.1007/s41808-022-00161-0

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