In this paper, we discuss the solvability of the nonlinear parabolic systems associated to the nonlinear parabolic equation: \(\frac{\partial b_{i}(x,u_{i})}{\partial t} -div(a(x,t,u_{i},\nabla u_{i}))- \phi_{i}(x,t,u_{i})) +f_{i}(x,u_{1},u_{2})=0\) where the function  \(b_{i}(x,u_{i})\) verifies some regularity conditions, the term \(\Big(a(x,t,u_{i},\nabla u_{i})\Big)\) is a generalized Leray-Lions operator and \(\phi_{i}\) is a Caratheodory function assumed to be  Continuous on \(u_i\) and satisfy only a…