Existence of entropy solutions to nonlinear degenerate parabolic problems with variable exponent and L 1 -data

Abdelali Sabri; Ahmed Jamea; Hamad Talibi Alaoui

Communications in Mathematics (2020)

  • Volume: 28, Issue: 1, page 67-88
  • ISSN: 1804-1388

Abstract

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In the present paper, we prove existence results of entropy solutions to a class of nonlinear degenerate parabolic p ( · ) -Laplacian problem with Dirichlet-type boundary conditions and L 1 data. The main tool used here is the Rothe method combined with the theory of variable exponent Sobolev spaces.

How to cite

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Sabri, Abdelali, Jamea, Ahmed, and Alaoui, Hamad Talibi. "Existence of entropy solutions to nonlinear degenerate parabolic problems with variable exponent and $L^1$-data." Communications in Mathematics 28.1 (2020): 67-88. <http://eudml.org/doc/297372>.

@article{Sabri2020,
abstract = {In the present paper, we prove existence results of entropy solutions to a class of nonlinear degenerate parabolic $p(\cdot )$-Laplacian problem with Dirichlet-type boundary conditions and $L^1$ data. The main tool used here is the Rothe method combined with the theory of variable exponent Sobolev spaces.},
author = {Sabri, Abdelali, Jamea, Ahmed, Alaoui, Hamad Talibi},
journal = {Communications in Mathematics},
keywords = {Degenerate parabolic problem; entropy solution; existence; semi-discretization; Rothe's method; weighted Sobolev space},
language = {eng},
number = {1},
pages = {67-88},
publisher = {University of Ostrava},
title = {Existence of entropy solutions to nonlinear degenerate parabolic problems with variable exponent and $L^1$-data},
url = {http://eudml.org/doc/297372},
volume = {28},
year = {2020},
}

TY - JOUR
AU - Sabri, Abdelali
AU - Jamea, Ahmed
AU - Alaoui, Hamad Talibi
TI - Existence of entropy solutions to nonlinear degenerate parabolic problems with variable exponent and $L^1$-data
JO - Communications in Mathematics
PY - 2020
PB - University of Ostrava
VL - 28
IS - 1
SP - 67
EP - 88
AB - In the present paper, we prove existence results of entropy solutions to a class of nonlinear degenerate parabolic $p(\cdot )$-Laplacian problem with Dirichlet-type boundary conditions and $L^1$ data. The main tool used here is the Rothe method combined with the theory of variable exponent Sobolev spaces.
LA - eng
KW - Degenerate parabolic problem; entropy solution; existence; semi-discretization; Rothe's method; weighted Sobolev space
UR - http://eudml.org/doc/297372
ER -

References

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  1. Abassi, A., Hachimi, A. El, Jamea, A., Entropy solutions to nonlinear Neumann problems with L 1 -data, Int. J. Math. Statist, 2, 2008, 4-17, (2008) MR2348474
  2. Akdim, Y., Chakir, A., Elgorch, N., Mekkour, M., Entropy Solutions of Nonlinear p ( x ) -Parabolic Inequalities, Nonlinear Dyn. Syst. Theory, 18, 2, 2018, 107-129, (2018) MR3820826
  3. Alaoui, M.K., Meskine, D., Souissi, A., 10.1016/j.na.2011.04.048, Nonlinear Analysis: Theory, Methods & Applications, 74, 17, 2011, 5863-5875, Elsevier, (2011) MR2833359DOI10.1016/j.na.2011.04.048
  4. Azroul, E., Barbara, A., Benboubker, M.B., Haiti, K. El, Existence of entropy solutions for degenerate elliptic unilateral problems with variable exponents, Boletim da Sociedade Paranaense de Matem{á}tica, 36, 1, 2018, 79-99, (2018) MR3632473
  5. Azroul, E., Redwane, H., Rhoudaf, M., 10.4171/PM/1829, Portugaliae Mathematica, 66, 1, 2009, 29-63, (2009) MR2512819DOI10.4171/PM/1829
  6. Bénilan, Ph., Boccardo, L., Gallouët, T., Gariepy, R., Pierre, M., Vázquez, J.L., An L 1 -theory of existence and uniqueness of solutions of nonlinear elliptic equations, Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 22, 2, 1995, 241-273, (1995) MR1354907
  7. Chen, Y., Levine, S., Rao, M., 10.1137/050624522, SIAM journal on Applied Mathematics, 66, 4, 2006, 1383-1406, SIAM, (2006) Zbl1102.49010MR2246061DOI10.1137/050624522
  8. Eden, A., Michaux, B., Rakotoson, J.M., 10.1512/iumj.1990.39.39036, Indiana University mathematics journal, 1990, 737-783, JSTOR, (1990) MR1078736DOI10.1512/iumj.1990.39.39036
  9. Hachimi, A. El, Jamea, A., Nonlinear parabolic problems with Neumann-type boundary conditions and L 1 -data, Electronic Journal of Qualitative Theory of Differential Equations, 2007, 27, 2007, 1-22, University of Szeged, Hungary, (2007) MR2354156
  10. Fan, X., Zhao, D., On the spaces L p ( x ) ( Ω ) and W m , p ( x ) ( Ω ) , Journal of Mathematical Analysis and Applications, 263, 2, 2001, 424-446, Elsevier, (2001) 
  11. Ho, K., Sim, I., 10.1016/j.na.2013.12.003, Nonlinear Analysis: Theory, Methods & Applications, 98, 2014, 146-164, Elsevier, (2014) MR3158451DOI10.1016/j.na.2013.12.003
  12. Hästö, P., The p ( x ) -Laplacian and applications, J. Anal, 15, 2007, 53-62, Citeseer, (2007) MR2554092
  13. Jamea, A., 10.12988/ijma.2016.6223, International Journal of Mathematical Analysis, 10, 12, 2016, 553-564, (2016) DOI10.12988/ijma.2016.6223
  14. Jamea, A., Lamrani, A.A., Hachimi, A. El, 10.1007/s11587-018-0359-y, Ricerche di Matematica, 67, 2, 2018, 785-801, Springer, (2018) MR3864809DOI10.1007/s11587-018-0359-y
  15. Kim, Y.H., Wang, L., Zhang, C., 10.1016/j.jmaa.2010.05.058, Journal of Mathematical Analysis and Applications, 371, 2, 2010, 624-637, Elsevier, (2010) MR2670139DOI10.1016/j.jmaa.2010.05.058
  16. Ouaro, S., Ouedraogo, A., Nonlinear parabolic problems with variable exponent and L 1 -data, Electronic Journal of Differential Equations, 2017, 32, 2017, 1-32, (2017) MR3609160
  17. Růžička, M., 10.1007/BFb0104030, 2000, Springer, Berlin, (2000) MR1810360DOI10.1007/BFb0104030
  18. Sanchón, M., Urbano, J.M., 10.1090/S0002-9947-09-04399-2, Transactions of the American Mathematical Society, 361, 12, 2009, 6387-6405, (2009) MR2538597DOI10.1090/S0002-9947-09-04399-2
  19. Zhang, C., Entropy solutions for nonlinear elliptic equations with variable exponents, Electronic Journal of Differential Equations, 2014, 92, 2014, 1-14, (2014) MR3193998
  20. Zhang, C., Zhou, S., 10.1016/j.jde.2009.11.024, Journal of Differential Equations, 248, 6, 2010, 1376-1400, Elsevier, (2010) MR2593046DOI10.1016/j.jde.2009.11.024

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