H. Hjiaj

Consider the nonlinear parabolic system $$\left\{\begin{array}{lll} \frac{\partial b_i(x,u_i)}{\partial t}-\mathop{\rm div}\Big(\mathcal{A}(x,t,u_i,\nabla u_i)+\Phi_i(x,t,u_i)\Big)+f_i(x,u_1,u_2)= 0 &\mbox{ in } Q_T \\ u_i=0 &\mbox{ on } \Gamma \\ b_i(x,u_i)(t=0)=b_i(x,u_{i,0})&\mbox{ in } \Omega,\end{array}\right.$$ where $ i=1,2$. In this paper we deal with the renormalized solution for the above system in Orlicz-Sobolev spaces where $f_i$ is a Carath\'{e}odory function satisfying some growth assumptions. The main term which contains the space derivatives and a non-coercive lower order term are considered in divergence form satisfying only the original Orlicz growths.