A note on quasilinear parabolic systems in generalized spaces

Document Type : Original Article

Authors

Elhoussine Azroul ¹

Farah Balaadich ²

¹ Department of Mathematics, Faculty of Sciences, Dhar El Mahraz FEZ

² Departement of Mathematics, Faculty of Sciences, Dhar El Mahraz FEZ

10.22034/kjm.2020.211591.1660

Abstract

We study the existence of solutions for quasilinear parabolic systems of the form
\[\partial_tu-\text{div}\,\sigma(x,t,Du)=f\quad\text{in}\;Q=\Omega\times(0,T),\]
whose right hand side belongs to $W^{-1,x}L_{\overline{M}}(Q;\R^m)$, supplemented with the conditions $u=0$ on $\partial\Omega\times(0,T)$ and $u(x,0)=u_0(x)$ in $\Omega$. By using a mild monotonicity condition for $\sigma$, namely strict quasimonotone, and the theory of Young measures, we deduce the needed result.

Keywords

Quasilinear parabolic systems

Orlicz-Sobolev spaces

Young measures

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