Opuscula Math. 41, no. 1 (2021), 25-53
https://doi.org/10.7494/OpMath.2021.41.1.25

Opuscula Mathematica

# Nonlinear parabolic equation having nonstandard growth condition with respect to the gradient and variable exponent

Abderrahim Charkaoui
Houda Fahim
Nour Eddine Alaa

Abstract. We are concerned with the existence of solutions to a class of quasilinear parabolic equations having critical growth nonlinearity with respect to the gradient and variable exponent. Using Schaeffer's fixed point theorem combined with the sub- and supersolution method, we prove the existence results of a weak solutions to the considered problems.

Keywords: variable exponent, quasilinear equation, Schaeffer's fixed point, subsolution, supersolution, weak solution.

Mathematics Subject Classification: 35D30, 35K59, 35A01, 35K93, 35A16, 47H10.

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• Abderrahim Charkaoui (corresponding author)
• https://orcid.org/0000-0003-1425-7248
• Laboratory LAMAI, Faculty of Science and Technology of Marrakech, B.P. 549, Av. Abdelkarim Elkhattabi, 40 000, Marrakech, Morocco
• Houda Fahim
• Laboratory LAMAI, Faculty of Science and Technology of Marrakech, B.P. 549, Av. Abdelkarim Elkhattabi, 40 000, Marrakech, Morocco
• Nour Eddine Alaa
• https://orcid.org/0000-0001-8169-8663
• Laboratory LAMAI, Faculty of Science and Technology of Marrakech, B.P. 549, Av. Abdelkarim Elkhattabi, 40 000, Marrakech, Morocco
• Communicated by Vicentiu D. Radulescu.
• Revised: 2020-12-09.
• Accepted: 2020-12-09.
• Published online: 2021-02-08.