On the some equations inequalities in Musielak-Orlicz spaces with measure data

Ouidad Azraibi; Mohamed Bourahma; M. Mekkour; Arij Bouzelmate; Badr El Haji

doi:10.15446/recolma.v57n1.112431

Published

2024-01-11

On the some equations inequalities in Musielak-Orlicz spaces with measure data

Desigualdades de ecuaciones sobre espacios Musielak-Orlicz con condiciones iniciales medibles

DOI:

https://doi.org/10.15446/recolma.v57n1.112431

Keywords:

Musielak-Orlicz-Sobolev spaces, Unilateral parabolic problem, entropy solutions, truncations (en)
Espacios de Musielak-Orlicz-Sobolev, problemas parabólicos unilaterales, soluciones de entropía (es)

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Authors

Ouidad Azraibi Sidi Mohammed Ben Abdallah University
Mohamed Bourahma Sidi Mohammed Ben Abdallah University
M. Mekkour Sidi Mohammed Ben Abdallah University
Arij Bouzelmate Abdelmalek Essaadi University
Badr El Haji Abdelmalek Essaadi University

Abstract (en)
Abstract (es)

Our aim in this present work is to prove the existence of a solution for the nonlinear unilateral parabolic problems associated to the equation

∂u - div a(x, t, u, ∇u) - div Φ(x, t, u) = μ in Q_T = Ω × (0, T), ∂t

where the lower order term Φ satisfies a generalized natural growth condition and the datum μ belongs to L¹(Q) + E (Q).

El objetivo de este trabajo es probar la existencia de soluciones para problemas parabólicos unilaterales asociados a la ecuación

∂u - div a(x, t, u, ∇u) - div Φ(x, t, u) = μ in Q_T = Ω × (0, T), ∂t

donde el término Φ satisface una condición natural generalizada de crecimiento y la condición μ pertenece al espacio L¹(Q) + E (Q).

References

M. Kbiri Alaoui, D. Meskine, and A. Souissi, On some class of nonlinear parabolic inequalities in Orlicz spaces, Nonlinear Anal. TMA 74 (2011), no. 17, 5863-5875. DOI: https://doi.org/10.1016/j.na.2011.04.048

N. El Amarty, B. El Haji, and M. El Moumni, Existence of renomalized solution for nonlinear Elliptic boundary value problem without 2-condition, SeMA 77 (2020), 389-414, https://doi.org/10.1007/s40324-020-00224-z. DOI: https://doi.org/10.1007/s40324-020-00224-z

N. El Amarty, B. El Haji, and M. El Moumni, Entropy solutions for unilateral parabolic problems with L1-data in Musielak-Orlicz-Sobolev spaces, Palestine Journal of Mathematics 11 (2022), no. 1, 504-523.

O. Azraibi, B. EL haji, and M. Mekkour, Nonlinear parabolic problem with lower order terms in Musielak-Sobolev spaces without sign condition and with Measure data, Palestine Journal of Mathematics 11 (2022), no. 3, 474-503.

O. Azraibi, B. EL haji, and M. Mekkour, On Some Nonlinear Elliptic Problems with Large Monotonocity in MusielakâOrliczâSobolev Spaces, Journal of Mathematical Physics, Analysis, Geometry 18 (2022), no. 3, 1â18. DOI: https://doi.org/10.15407/mag18.03.332

O. Azraibi, B. El Haji, and M. Mekkour, Entropy Solution for Nonlinear Elliptic Boundary Value Problem Having Large Monotonocity in Musielak-Orlicz-Sobolev Spaces, Asia Pac. J. Math. 10 (2023), no. 7, 104-116, DOI:doi:10.28924/APJM/10-7. DOI: https://doi.org/10.28924/APJM/10-7

O. Azraibi, B. EL Haji, and M. Mekkour, Strongly nonlinear unilateral anisotropic elliptic problem with-data, Asia Mathematika 7 (2023), no. 1, 1-20, DOI: doi.org/10.5281/zenodo.8071010.

E. Azroul, M. El Lekhlifi, H. Redwane, and A. Touzani, Entropy solutions of nonlinear parabolic equations in Orlicz-Sobolev spaces, without sign condition and L1 data, J. Nonlinear Evol. Equ. Appl. (2014), 101-130.

E. Azroul, H. Redwane, and C. Yazough, Strongly Nonlinear Nonhomogeneous Elliptic Unilateral Problems with L1 data and no sign conditions, Electronic Journal of Differential Equations 79 (2012), 1-20.

P. Benilan, L. Boocardo, T. Gallouet, R. Gariepy, M. Pierre, and J. Vazquez, An L1-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Super. Pisa 22 (1995), 241-273.

A. Benkirane, N. El Amarty, B. El Haji, and M. El Moumni, Existence of solutions for a class of nonlinear elliptic problems with measure data in the setting of MusielakâOrliczâSobolev spaces, J Elliptic Parabol Equ. 9 (2023), 647â672, https://doi.org/10.1007/s41808-022-00193-6. DOI: https://doi.org/10.1007/s41808-022-00193-6

A. Benkirane, B. El Haji, and M. El Moumni, Strongly nonlinear elliptic problem with measure data in Musielak-Orlicz spaces, Complex Variables and Elliptic Equations 67, no. 6, 1447-1469, https://doi.org/10.1080/17476933.2021.1882434. DOI: https://doi.org/10.1080/17476933.2021.1882434

A. Benkirane, B. El Haji, and M. El Moumni, On the existence of solution for degenerate parabolic equations with singular terms, Pure and Applied Mathematics Quarterly 14 (2018), no. 3-4, 591-606. DOI: https://doi.org/10.4310/PAMQ.2018.v14.n3.a8

A. Benkirane, B. El Haji, and M. El Moumni, On the Existence Solutions for some Nonlinear Elliptic Problem, Bol. Soc. Paran. Mat. (3s.) 40 (2022), 1-8, DOI:10.5269/bspm.53111. DOI: https://doi.org/10.5269/bspm.53111

A. Benkirane and M. Sidi El Vally, Some approximation properties in Musielak-Orlicz-Sobolev spaces, Thai.J. Math. 10 (2012), 371-381.

A. Benkirane and M. Sidi El Vally, Variational inequalities in Musielak-Orlicz-Sobolev spaces, Bull. Belg. Math. Soc. Simon Stevin 21 (2014), 787-811. DOI: https://doi.org/10.36045/bbms/1420071854

R. Elarabi, M. Rhoudaf, and H. Sabiki, Entropy solution for a nonlinear elliptic problem with lower order term in Musielak-Orlicz spaces, Ric. Mat., 2017, https://doi.org/10.1007/s11587-017-0334-z. DOI: https://doi.org/10.1007/s11587-017-0334-z

A. Elmahi and D. Meskine, Strongly nonlinear parabolic equations with natural growth terms and L1 data in Orlicz spaces, Port. Math. (N.S.) 62 (2005), no. 2, 143-183. DOI: https://doi.org/10.1007/s10231-004-0107-7

J. P. Gossez and V. Mustonen, Variationnal inequalities in Orlicz-Sobolev spaces, Nonlinear Anal. 11 (1987), 317-492. DOI: https://doi.org/10.1016/0362-546X(87)90053-8

N. S. Hadj, H. Moussa, and M. Rhoudaf, Renormalized solution for a nonlinear parabolic problems with noncoercivity in divergence form in Orlicz spaces, Appl. Math. Comput. 249 (2014), 253-264.

B. El Haji and M. El Moumni, Entropy solutions of nonlinear elliptic equations with L1-data and without strict monotonocity conditions in weighted Orlicz-Sobolev spaces, Journal of Nonlinear Functional Analysis 2021 (2021), 1-17, Article ID 8. DOI: https://doi.org/10.23952/jnfa.2021.8

B. El Haji, M. El Moumni, and K. Kouhaila, On a nonlinear elliptic problems having large monotonocity with L1-data in weighted Orlicz-Sobolev spaces, Moroccan J. of Pure and Appl. Anal. (MJPAA) 5 (2019), no. 1, 104-116, DOI 10.2478/mjpaa-2019-0008. DOI: https://doi.org/10.2478/mjpaa-2019-0008

B. El Haji, M. El Moumni, and K. Kouhaila, Existence of entropy solutions for nonlinear elliptic problem having large monotonicity in weighted Orlicz-Sobolev spaces, LE MATEMATICHE LXXVI (2021), no. I, 37-61, https://doi.org/10.4418/2021.76.1.3.

B. El Haji, M. El Moumni, and A. Talha, Entropy solutions for nonlinear parabolic equations in Musielak Orlicz spaces without Delta2-conditions, Gulf Journal of Mathematics 9 (2020), no. 1, 1-26.

B. El Haji, M. El Moumni, and A. Talha, Entropy Solutions of Nonlinear Parabolic Equations in Musielak Framework Without Sign Condition and L1-Data, Asian Journal of Mathematics and Applications, 2021, 33 pp. DOI: https://doi.org/10.56947/gjom.v9i1.448

E. Hewitt and K. Stromberg, Real and Abstract Analysis, Springer, Berlin, 1965. DOI: https://doi.org/10.1007/978-3-642-88044-5

M. Ait Khellou and A. Benkirane, Elliptic inequalities with L1 data in Musielak-Orlicz spaces, Monatsh Math. 183 (2017), 1-33. DOI: https://doi.org/10.1007/s00605-016-1010-1

M. Ait Khellou, A. Benkirane, and S. M. Douiri, Some properties of Musielak spaces with only the log-Hölder continuity condition and application, Annals of Functional Analysis,Tusi Mathematical Research Group (TMRG), 2020, DOI: 10.1007/s43034-020-00069-7. DOI: https://doi.org/10.1007/s43034-020-00069-7

R. Landes and V. Mustonen, A strongly nonlinear parabolic initialboundary value problem, Ask. Mat. 25 (1987), 29-40. DOI: https://doi.org/10.1007/BF02384435

M. Mabdaoui, H. Moussa, and M. Rhoudaf, Entropy solutions for a nonlinear parabolic problems with lower order term in Orlicz spaces, Anal.Math.Phys. (2016), DOI 10.1007/s13324-016-0129-5. DOI: https://doi.org/10.1007/s13324-016-0129-5

D. Meskine, Parabolic equations with measure data in Orlicz spaces, J. Evol. Equ. 5 (2005), 529-543. DOI: https://doi.org/10.1007/s00028-005-0217-8

H. Moussa and M. Rhoudaf, Renormalized solution for a nonlinear parabolic problems with noncoersivity in divergence form in Orlicz Spaces, Appled Mathematics and Computation 249 (2014), 253-264. DOI: https://doi.org/10.1016/j.amc.2014.10.026

J. Musielak, Modular spaces and Orlicz spaces, Lecture Notes in Math. 1034 (1983), iii+222 pp. DOI: https://doi.org/10.1007/BFb0072210

M. L. Ahmed Oubeid, A. Benkirane, and M. Sidi El Vally, Strongly nonlinear parabolic problems in Musielak-Orlicz-Sobolev spaces, Bol. Soc. Paran. Mat. 33 (2015), no. 1, 191-230. DOI: https://doi.org/10.5269/bspm.v33i1.23083

A. Porretta, Existence and uniqueness of entropy solutions of parabolic problems with L1 - data, Non-linear Anal. 528 (1997), 1943-1954. DOI: https://doi.org/10.1016/S0362-546X(96)00030-2

A. Porretta, Existence results for strongly nonlinear parabolic equations via strong convergence of truncations, Ann. Mat. Pura Appl. 177 (1999), no. IV, 143-172. DOI: https://doi.org/10.1007/BF02505907

H. Redwane, Existence Results For a Class of Nonlinear Parabolic Equations in Orlicz Spaces, Electronic Journal of Qualitative Theory of Differential Equations 2 (2010), 19pp. DOI: https://doi.org/10.14232/ejqtde.2010.1.2

W. Rudin, Real and Complex Analysis, 3rd ed., McGraw-Hill, New York, 1974.

H. Sabiki, M. Rhoudaf, and R. Elarabi, Existence results for a non-linear obstacle parabolic semicoercive problems with lower order term, COMP. VAR. ELL. EQU., 2018, https://doi.org/10.1080/17476933.2018.1427077. DOI: https://doi.org/10.1080/17476933.2018.1427077

J. Simon, Compact sets in Lp(0; T;B), Ann. Mat. Pura Appl. 146 (1987), 65-96. DOI: https://doi.org/10.1007/BF01762360

How to Cite

APA

Azraibi, O., Bourahma, M., Mekkour, M., Bouzelmate, A. and El Haji, B. (2024). On the some equations inequalities in Musielak-Orlicz spaces with measure data. Revista Colombiana de Matemáticas, 57(1), 123–154. https://doi.org/10.15446/recolma.v57n1.112431

ACM

[1]

Azraibi, O., Bourahma, M., Mekkour, M., Bouzelmate, A. and El Haji, B. 2024. On the some equations inequalities in Musielak-Orlicz spaces with measure data. Revista Colombiana de Matemáticas. 57, 1 (Jan. 2024), 123–154. DOI:https://doi.org/10.15446/recolma.v57n1.112431.

ACS

(1)

Azraibi, O.; Bourahma, M.; Mekkour, M.; Bouzelmate, A.; El Haji, B. On the some equations inequalities in Musielak-Orlicz spaces with measure data. rev.colomb.mat 2024, 57, 123-154.

ABNT

AZRAIBI, O.; BOURAHMA, M.; MEKKOUR, M.; BOUZELMATE, A.; EL HAJI, B. On the some equations inequalities in Musielak-Orlicz spaces with measure data. Revista Colombiana de Matemáticas, [S. l.], v. 57, n. 1, p. 123–154, 2024. DOI: 10.15446/recolma.v57n1.112431. Disponível em: https://revistas.unal.edu.co/index.php/recolma/article/view/112431. Acesso em: 9 mar. 2025.

Chicago

Azraibi, Ouidad, Mohamed Bourahma, M. Mekkour, Arij Bouzelmate, and Badr El Haji. 2024. “On the some equations inequalities in Musielak-Orlicz spaces with measure data”. Revista Colombiana De Matemáticas 57 (1):123-54. https://doi.org/10.15446/recolma.v57n1.112431.

Harvard

Azraibi, O., Bourahma, M., Mekkour, M., Bouzelmate, A. and El Haji, B. (2024) “On the some equations inequalities in Musielak-Orlicz spaces with measure data”, Revista Colombiana de Matemáticas, 57(1), pp. 123–154. doi: 10.15446/recolma.v57n1.112431.

IEEE

[1]

O. Azraibi, M. Bourahma, M. Mekkour, A. Bouzelmate, and B. El Haji, “On the some equations inequalities in Musielak-Orlicz spaces with measure data”, rev.colomb.mat, vol. 57, no. 1, pp. 123–154, Jan. 2024.

MLA

Azraibi, O., M. Bourahma, M. Mekkour, A. Bouzelmate, and B. El Haji. “On the some equations inequalities in Musielak-Orlicz spaces with measure data”. Revista Colombiana de Matemáticas, vol. 57, no. 1, Jan. 2024, pp. 123-54, doi:10.15446/recolma.v57n1.112431.

Turabian

Azraibi, Ouidad, Mohamed Bourahma, M. Mekkour, Arij Bouzelmate, and Badr El Haji. “On the some equations inequalities in Musielak-Orlicz spaces with measure data”. Revista Colombiana de Matemáticas 57, no. 1 (January 11, 2024): 123–154. Accessed March 9, 2025. https://revistas.unal.edu.co/index.php/recolma/article/view/112431.

Vancouver

1.

Azraibi O, Bourahma M, Mekkour M, Bouzelmate A, El Haji B. On the some equations inequalities in Musielak-Orlicz spaces with measure data. rev.colomb.mat [Internet]. 2024 Jan. 11 [cited 2025 Mar. 9];57(1):123-54. Available from: https://revistas.unal.edu.co/index.php/recolma/article/view/112431

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1. Arij Bouzelmate, Badr El Haji, Adnan Lamtarah. (2024). Nonlinear elliptic unilateral problems with measure data in the anisotropic Sobolev space. Nonautonomous Dynamical Systems, 11(1) https://doi.org/10.1515/msds-2024-0001.