In this paper, we investigate the existence of solutions to a nonlinear parabolic
system, which couples a non-homogeneous reaction-diffusion-type equation and a non-homogeneous
viscous Hamilton-Jacobi one. The initial data and right-hand sides satisfy suitable
integrability conditions and non-negative. To simplify the presentation of our results,
we will consider separately two simplified models : first, vanishing initial data,
and then, vanishing right-hand sides.