Journal of Mathematical Physics, Analysis, Geometry
2018, vol. 14, No 1, pp. 27-53   https://doi.org/10.15407/mag14.01.027     ( to contents , go back )
https://doi.org/10.15407/mag14.01.027

Renormalized Solutions for Nonlinear Parabolic Systems in the Lebesgue{Sobolev Spaces with Variable Exponents

B. El Hamdaoui

Université Sidi Mohammed Ben Abdellah, Déepartement de Mathéematiques, Laboratoire LAMA, Facultée des Sciences Dhar-Mahrez, B.P 1796 Atlas Fès, Morocco
E-mail: elhammdaoui.bouchra@gmail.com

J. Bennouna

Université Sidi Mohammed Ben Abdellah, Déepartement de Mathéematiques, Laboratoire LAMA, Facultée des Sciences Dhar-Mahrez, B.P 1796 Atlas Fès, Morocco
E-mail: jbennouna@hotmail.com

A. Aberqi

Université Sidi Mohammed Ben Abdellah, National School of Applied Sciences, LISA, Fès, Morocco

Received January 10, 2016, revised May 6, 2016.

Abstract

The existence result of renormalized solutions for a class of nonlinear parabolic systems with variable exponents of the type

teλui(x,t) - div(|ui(x, t)|p(x)-2ui(x, t))

           + div(c(x, t)|ui(x, t)|γ (x)-2ui(x, t)) = fi(x, u1, u2) - div(Fi),

for i = 1, 2, is given. The nonlinearity structure changes from one point to other in the domain Ω. The source term is less regular (bounded Radon measure) and no coercivity is in the nondivergent lower order term div(c(x, t)|u(x, t)|γ (x)-2u(x, t)). The main contribution of our work is the proof of the existence of renormalized solutions without the coercivity condition on nonlinearities which allows us to use the Gagliardo–Nirenberg theorem in the proof.

Mathematics Subject Classification 2010: 35J70, 35D05.
Key words: parabolic problems, Lebesgue–Sobolev space, variable exponent, renormalized solutions.

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