Existence results for a class of nonlinear parabolic equations with two lower order terms

Ahmed Aberqi ¹   ; Jaouad Bennouna ¹   ; M. Hammoumi ¹   ; Mounir Mekkour ²   ; Ahmed Youssfi ³

¹ Laboratory LAMA, Department of Mathematics Faculty of Sciences Dhar El Mahraz Sidi Mohamed Ben Abdellah University P.O. Box 1796 Atlas Fez, Morocco
² Laboratory LIMAO Poly-disciplinary Faculty of Taza Sidi Mohamed Ben Abdellah University P.O. Box 1223 Taza, Morocco
³ Deptment of Mathematics Faculty of Sciences and Technology Moulay Ismail University P.O. Box 509-Boutalamine 52 000 Errachidia, Morocco

Applicationes Mathematicae, Tome 41 (2014) no. 2-3, pp. 207-219

Résumé

We investigate the existence of renormalized solutions for some nonlinear parabolic problems associated to equations of the form $$\begin{cases} \frac{\partial (e^{\beta u}-1)}{\partial t}-{\rm div}(|\nabla u|^{p-2}\nabla u)+ {\rm div}(c(x,t)|u|^{s-1}u)+b(x,t)|\nabla u|^{r}=f {\rm in}\ Q=\varOmega\times (0,T),\\ u(x,t)=0 {\rm on}\ \partial\varOmega \times(0,T),\\ (e^{\beta u}-1)(x,0)=(e^{\beta u_{0}}-1)(x) {\rm in}\ \varOmega. \end{cases} $$ with $ s =\frac{N+2}{N+p}(p-1)$, $\displaystyle c(x,t)\in (L^{\tau}(Q_{T}))^{N}$, $ \tau =\frac{N+p}{p-1}$, $ r =\frac{N(p-1)+p}{N+2}$, $ b(x,t)\in L^{N+2,1}(Q_{T})$ and $ f\in L^{1}(Q).$

Détail

DOI : 10.4064/am41-2-8

Keywords: investigate existence renormalized solutions nonlinear parabolic problems associated equations form begin cases frac partial beta partial div nabla p nabla div s nabla varomega times partial varomega times beta beta varomega end cases frac p displaystyle tau tau frac p frac p