The authors prove compactness and approximation results in inhomogeneous Orlicz-Sobolev spaces and look at, as an application, the Cauchy-Dirichlet problem \[ \begin{alignedat}{2} &\frac{\partial u}{\partial t} -\text{div}\left(a(x,t,u,\nabla u)\right)+g(x,t,u,\nabla u)=f &\quad &\text{in } Q,\\ &u(x,t)=0 &\quad &\text{on } \partial\Omega\times(0,T),\\ &u(x,0)=u_{0}(x)&\quad &\text{in}\;\Omega, \end{alignedat} \] where \(f\in L^{1}(Q)\), \(Q=\Omega\times(0,T)\), \(\Omega\subset\mathbb{R}^{N}\) is a bounded open set. They also give a trace result allowing to deduce the continuity of the solutions with respect to time.