SeMA Journal

, Volume 76, Issue 1, pp 153–180 | Cite as

Existence and uniqueness of entropy solutions to nonlinear parabolic problem with homogeneous Dirichlet boundary conditions involving variable exponent

  • Bila Adolphe KyelemEmail author
  • Arouna Ouedraogo
  • Frédéric D. Y. Zongo


In this paper, we prove the existence and uniqueness of entropy solution to nonlinear parabolic problem with homogeneous Dirichlet boundary conditions with \(L^1\)-data. The functional setting involves Lebesgue and Sobolev spaces with variable exponent. The general assumptions and the nonlinear semigroup theory are considered to prove the existence and uniqueness of mild solution satisfying the \(L^1\)-comparison principle. Moreover, under the same general assumptions and some a-priori estimations of the sequence of mild solutions, we obtain the existence and uniqueness of weak solution. Finally, we prove the existence and uniquess of the renormalized solution which is equivalent to the existence and uniqueness of entropy solution.


Parabolic equation Variable exponent Mild solution Maximal monotone graph Renormalized solution Entropy solution \(L^1\)-data 

Mathematics Subject Classification

Primary 35K55 35D30 46E35 Secondary 76D03 



This work was done within the framework of the visit of the authors at the Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy. The authors thank the mathematic’s section of ICTP for their hospitality and for financial support and all facilities.


  1. 1.
    Alt, H.W., Luckhaus, S.: Quasi-linear elliptic-parabolic differential equations. Math. Z. 183, 311–341 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Antontsev, S.N., Rodrigues, J.F.: On stationary thermo-rheological viscous flows. Ann. del Univ. de Ferrara 52, 19–36 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bendahmane, M., Wittbold, P., Zimmermann, A.: Renormalized solutions for a nonlinear parabolic equation with variable exponents and \(L^{1}-\)data. J. Diff. Equ. 249, 1483–1515 (2010)CrossRefzbMATHGoogle Scholar
  4. 4.
    Bénilan, Ph.: Equations d’évolution dans un espace de Banach quelconque et applications. Thèse d’état, Univ. Paris XI, Orsay (1972)Google Scholar
  5. 5.
    Bénilan, Ph, Boccardo, L., Gallouèt, T., Gariepy, R., Pierre, M., Vazquez, J.L.: An \(L^{1}\) theory of existence and uniqueness of nonlinear elliptic equations. Ann. Sc. Norm. Sup. Pisa 22(2), 240–273 (1995)zbMATHGoogle Scholar
  6. 6.
    Bénilan, Ph., Crandal, M.G., Pazy, A.: Evolution equations governed by accretive operators (forthcoming book) Google Scholar
  7. 7.
    Bénilan, Ph, Wittbold, P.: On mild and weak solutions of elliptic-parabolic problems. Ad. Differ. Equ. 1(6), 1053–1073 (1996)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Blanchard, D., Murat, F.: Renormalized solutions of nonlinear parabolic problems with \(L^{1}\) data: existence and uniqueness. Proc. R. Soc. Edinb Sect. A 127(6), 1137–1152 (1997)CrossRefzbMATHGoogle Scholar
  9. 9.
    Bonzi, B.K., Ouaro, S.: Entropy solutions for doubly nonlinear elliptic problems with variable exponent. J. Math. Anal. Appl. 370(2), 392–405 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Chen, Y., Levine, S., Rao, M.: Variable exponent, linear growth functionals in image restoration. SIAM. J. Appl. Math. 66, 1383–1406 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Crandall, M.C., Liggett, T.M.: Generation of semigroups of nonlinear transformations on general Banach spaces. Am. J. Math. 93(2), 265–298 (1971)CrossRefzbMATHGoogle Scholar
  12. 12.
    Diening, L.: Theoritical and numerical results for electrorheological fluids. Ph. D. thesis, University of Freiburg, Germany, (2002)Google Scholar
  13. 13.
    Diening, L., Harjulehto, P., Hästö, P., Ruzicka, M.: Lebesgue and Sobolev spaces with variable exponents. Lecture Notes in Mathematics, vol. 2017. Springer-Verlag, Heidelberg (2011)Google Scholar
  14. 14.
    DiPerna, R.J., Lions, P.L.: On the Cauchy problem for Boltzmann equations: global existence and weak stability. Ann. Math. 2(130), 321–366 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Droniou, J., Prignet, A.: Equivalence between entropy and renormalized solutions for parabolic equations with smooth measure data. Nonlinear Diff. Equ. Appl. 14(1–2), 181–205 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Fan, X., Zhao, D.: On the spaces \(L^{p(x)}(\varOmega )\) and \(W^{m, p(x)}(\varOmega )\). J. Math. Anal. Appl. 263, 424–446 (2001)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Giannetti, F., Passarelli di Napoli, A., Ragusa, M.A., Tachikawa, A., Takabayashi, H.: Partial regularity for minimizers of a class of non autonomous functionals with nonstandard growth. Calc. Var. Partial Differ. Equ. 56(6), 153 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kovacik, O., Rakosnik, J.: On spaces \(L^{p(x)}\) and \(W^{1, p(x)}\). Czech. Math. J. 41, 592–618 (1991)zbMATHGoogle Scholar
  19. 19.
    Landes, R.: On the existence of weak solutions for quasilinear parabolic initial-boundary value problems. Proc. R. Soc. Edinb. Sect. A 89(3–4), 217–237 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Ouaro, S., Ouédraogo, A.: Nonlinear parabolic problems with variable exponent and \(L^{1}\)-data. Electron. J. Differ. Equ. 2017(32), 32 (2017)zbMATHGoogle Scholar
  21. 21.
    Ouaro, S., Traoré, S.: Existence and uniqueness of entropy solutions to nonlinear elliptic problems with variable growth. Int. J. Evol. Equ. 4(4), 451–471 (2009)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Porretta, A.: Existence results for nonlinear parabolic equations via strong convergence of truncations. Ann. Mat. Pura Appl. 4(177), 143–172 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Ragusa, M.A., Tachikawa, A.: On interior regularity of minimizers of \(p(x)\)-energy functionals. Nonlinear Anal. Theory Method. Appl. 93, 162–167 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Ragusa, M.A., Tachikawa, A.: Boundary regularity of minimizers of \(p(x)\)-energy functionals. Ann. Inst. Henri Poincaré Anal. Non Linéaire 33(2), 451–476 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Rajagopal, K.R., Ruzicka, M.: Mathematical modeling of electrorheological materials. Contin. Mech. Thermodyn. 13, 59–78 (2001)CrossRefzbMATHGoogle Scholar
  26. 26.
    Ruzicka, M.: Electrorheological fluids: modelling and mathematical theory. Lecture Notes in Mathematics 1748, Springer-Verlag, Berlin (2002)Google Scholar
  27. 27.
    Simondon, F.: Etude de l’équation \(\partial \_tb(u)- \text{div} a(b(u), Du) = 0\) par la méthode des semi-groupes dans \(L^1(\varOmega )\). Publ. Math. Besançon, Anal. non linéaire 7, 1–18 (1983)Google Scholar
  28. 28.
    Wittbold, P., Zimmermann, A.: Existence and uniqueness of renormalized solutions to nonlinear elliptic equations with variable exponents and \(L^{1}\)-data. Nonlinear Anal. TMA 72, 2990–3008 (2010)CrossRefzbMATHGoogle Scholar

Copyright information

© Sociedad Española de Matemática Aplicada 2018

Authors and Affiliations

  1. 1.LAboratoire de Mathématiques et d’Informatique (LAMI), Centre Universitaire Polytechnique de Ouahigouya, Université Ouaga I-Professeur Joseph Ki ZERBOOuagadougou 03Burkina Faso
  2. 2.LAboratoire de Mathématiques et d’Informatique (LAMI), Unité de Formation et de Recherche en Sciences et Technologie, Université de KoudougouKoudougouBurkina Faso
  3. 3.LAboratoire de Mathématiques et d’Informatique (LAMI), Institut Des SciencesOuagadougou 01Burkina Faso

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