We prove an existence result of entropy solutions for the nonlinear parabolic problems: \(\frac{\partial b(x,u)}{\partial t} + A(u) - div(\Phi(x,t,u))+H(x,t,u,\nabla u) =f,\) and \(A(u)=-div(a(x,t,u,\nabla u))\) is a Leary-Lions operator defined on the inhomogeneous Musielak-Orlicz space, the term \(\Phi(x,t,u)\) is a Cratheodory function assumed to be continuous on u and satisfy only the growth condition \(\)\Phi(x,t,u)\leq…