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Integral Solution for a Parabolic Equation Driven by the p(x)-Laplacian Operator with Nonlinear Boundary Conditions and \(L^{1}\) Data

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Abstract

We study the solvability of a class of parabolic equations having p(x)-growth structure and involving nonlinear boundary conditions with irregular data. We tackle our problem in a suitable functional setting by considering the so-called Lebesgue and Sobolev spaces with variable exponents. We establish two existence and uniqueness results of solutions to the studied problem. In the first one, we use a subdifferential approach to investigate the existence and uniqueness of a weak solution when the data are regular enough. Second, we assume that the data belong only to the space \(L^1\) and we show the existence and uniqueness of an integral solution to the considered model. Our main idea relies essentially on the use of accretive operator theory and involves some new techniques.

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Authors and Affiliations

  1. Laboratory LAMAI, Faculty of Science and Technology of Marrakech, B.P. 549, Av. Abdelkarim Elkhattabi, 40000, Marrakech, Morocco

    Nour Eddine Alaa & Malika El Ghabi

  2. Interdisciplinary Research Laboratory in Sciences, Education and Training, Higher School of Education and Training Berrechid (ESEFB), Hassan First University, Route de Casablanca Km 3.5, BP 539 , 26100, Berrechid, Morocco

    Abderrahim Charkaoui

  3. Laboratory LaR2A, Faculty of Science Tetouan, University Abdelmalek Essaâdi, Av. Sebta, Mhannech II, B.P. 2121 , 93002, Tétouan, Morocco

    Mohamed El Hathout

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  1. Nour Eddine Alaa
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  2. Abderrahim Charkaoui
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  3. Malika El Ghabi
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  4. Mohamed El Hathout
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Correspondence to Abderrahim Charkaoui.

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Alaa, N.E., Charkaoui, A., El Ghabi, M. et al. Integral Solution for a Parabolic Equation Driven by the p(x)-Laplacian Operator with Nonlinear Boundary Conditions and \(L^{1}\) Data. Mediterr. J. Math. 20, 244 (2023). https://doi.org/10.1007/s00009-023-02446-7

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  • DOI: https://doi.org/10.1007/s00009-023-02446-7

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