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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
Article
  • 40 Downloads

Abstract

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.

Keywords

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 

Notes

Acknowledgements

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.

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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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