Existence of entropy solution for nonlinear periodic parabolic problem with lower order term and $L^{1}$ data in Orlicz spaces

Document Type : Original Article

Authors

Elhoussine Azroul

Abdelilah Lamrani Alaoui‎

Ghita Erriahi Idrissi

Department of Mathematics and Computer Science, Faculty of‎ ‎Sciences ``Dhar El mahraz‎" ‎Sidi Mohamed Ben Abdellah University, ‎B.P‎. ‎1769-Atlas‎, ‎B.P‎. ‎1769-Atlas‎, ‎30000‎, ‎Morocco

10.22034/kjm.2024.467392.3246

Abstract

In the present paper, we prove the existence of an entropy solution for the following nonlinear periodic problem
\begin{equation}\label{p1}
\begin{cases}
\frac{\partial u}{\partial t}+A(u)-div(\phi(x,t,u))=f(x,t) \; \textrm{ in }
Q_T, \\
u(x,t)=0\; \textrm{on}\;\partial\Omega\times(0,T),\\
u(x,0)=u(x,T)\; \textrm{in } \Omega.
\end{cases}
\end{equation}
Where A is a Leray Lions operator having growth not necessarily of polynomial type, the lower order term $\phi$ satisfies only a growth condition and the right-hand side $f$ belongs to $L^{1}(Q_T)$.

Keywords

Parabolic periodic problem

weak solution

entropy solution

Orlicz-Sobolev spaces

lower order term

Subjects

• 35 Partial differential equations

---