Perturbed form for some nonlinear parabolic systems

Document Type : Original Article

Authors

Abdelali Daife

Elhoussine Azroul

Laboratory of Mathematical Analysis and Applications, Faculty of Sciences Dhar el Mahraz, Sidi Mohamed Ben Abdellah University, ‎B.P‎. ‎1796 Atlas‎, Fez-Morocco

10.22034/kjm.2024.434716.3104

Abstract

This paper addresses a quasilinear parabolic system in divergence form given by $\frac{\partial u}{\partial t}-\text{div} \,\sigma(x,t,Du-\Upsilon(u))=f$, which features a gradient perturbation. The source term $f$ belongs to $L^{p'}\big(0,T;W^{-1,p'}(\Omega;\mathbb{R}^m)\big)$. The existence of weak solutions $u:Q_T\to\mathbb{R}^m$, where $m\in\mathbb{N}^*$ and $Q_T=\Omega\times (0,T))$, is established through the use of Young measures and Galerkin approximations.

Keywords

Quasilinear parabolic system

Young measure

Weak solution

Galerkin method

Subjects

• 35 Partial differential equations

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