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We prove the continuity in norm of the translation operator in the Musielak-Orlicz $L_M$ spaces. An application to the convergence in norm of approximate identities is given, whereby we prove density results of the smooth functions in $L_M$, in both the modular and norm topologies. These density results are then applied to obtain basic topological properties.

We investigate the existence of renormalized solutions for some nonlinear parabolic problems associated to equations of the form ⎧$\partial (e^{\beta u}-1)/\partial t-{div\left(\right|\nabla u|}^{p-2}\nabla u)+div(c(x,t){\left|u\right|}^{s-1}u)+b(x,t){\left|\nabla u\right|}^r=f$ in Q = Ω×(0,T), ⎨ u(x,t) = 0 on ∂Ω ×(0,T), ⎩ $(e^{\beta u}-1)(x,0)=(e^{\beta u₀}-1)\left(x\right)$ in Ω. with s = (N+2)/(N+p) (p-1), $c(x,t)\in {\left(L^{\tau }\left(QT\right)\right)}^N$, τ = (N+p)/(p-1), r = (N(p-1) + p)/(N+2), $b(x,t)\in L^{N+2,1}\left(QT\right)$ and f ∈ L¹(Q).

We consider the anisotropic quasilinear elliptic Dirichlet problem $$\left\{\begin{array}{cc}{\displaystyle -\sum _{i=1}^N{D}^i{a}_i(x,u,\nabla u)+{\left|u\right|}^{s\left(x\right)-1}u=f+\lambda \frac{{\left|u\right|}^{p_0\left(x\right)-2}u}{{\left|x\right|}^{p_0\left(x\right)}}}\hfill & \text{in}\Omega ,\hfill \\ u=0\hfill & \text{on}\partial \Omega ,\hfill \end{array}\right.$$ where $\Omega$ is an open bounded subset of ${ℝ}^N$ containing the origin. We show the existence of entropy solution for this equation where the data $f$ is assumed to be in $L^1\left(\Omega \right)$ and $\lambda$ is a positive constant.