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Existence of a renormalized solutions for parabolic–elliptic system in anisotropic Orlicz–Sobolev spaces

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Abstract

The purpose of this paper is to establish the existence of renormalized solutions to the following parabolic–elliptic system

$$\begin{aligned} \varvec{\left\{ \begin{aligned} \frac{\partial u}{\partial t}-\sum _{i=1}^d \partial _i\left( a_i\left( x, t, u, \partial _i u\right) \right)&=\kappa (u)|\nabla v|^{2}{} & {} \text{ in } Q_{T}=\Omega \times (0, T), \\ {\text {div}}(\kappa (u) \nabla v)+{\text {div}} F(u)&=0{} & {} \text{ in } Q_{T}, \\ u&=0{} & {} \text{ on } \partial \Omega \times (0, T), \\ v&=0{} & {} \text{ on } \partial \Omega \times (0, T), \\ u(\cdot , 0)&=u_{0}{} & {} \text{ in } \Omega , \end{aligned}\right. } \end{aligned}$$

We will consider the inhomogeneous anisotropic Orlicz–Sobolev spaces, without imposing the \(\Delta _{2}\)-condition on the N-functions.

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Acknowledgements

Our sincere gratitude goes to the reviewers for their valuable suggestions which contributed significantly to improving the quality of this article.

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  1. Y. Ahakkoud, J. Bennouna and M. Elmassoudi have contributed equally to this work.

Authors and Affiliations

  1. Department of Mathematics, Faculty of Sciences Dhar El Mahraz, University Sidi Mohamed Ben Abdellah, B.P 1796 Atlas, Fez, 30000, Fez, Morocco

    Y. Ahakkoud, J. Bennouna & M. Elmassoudi

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  1. Y. Ahakkoud
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Ahakkoud, Y., Bennouna, J. & Elmassoudi, M. Existence of a renormalized solutions for parabolic–elliptic system in anisotropic Orlicz–Sobolev spaces. Rend. Circ. Mat. Palermo, II. Ser (2024). https://doi.org/10.1007/s12215-024-01024-4

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