Well-posedness of parabolic equations in the non-reflexive and anisotropic Musielak-Orlicz spaces in the class of renormalized solutions

We prove existence and uniqueness of renormalized solutions to general nonlinear parabolic equation in Musielak-Orlicz space avoiding growth restrictions. Namely, we consider $${\mathrm{\partial }}_{t}u-\mathrm{d}\mathrm{i}\mathrm{v}A(x,\mathrm{\nabla }u)=f\in L^1({\mathrm{\Omega }}_T),$$ on a Lipschitz bounded domain in ${\mathbb{R}}^n$. The growth of the weakly monotone vector field $A$ is controlled by a generalized nonhomogeneous and anisotropic $N$-function $M$. The approach does not require any particular type of growth condition of $M$ or its conjugate $M^{\ast }$ (neither ${\mathrm{\Delta }}_2$, nor ${\mathrm{\nabla }}_2$). The condition we impose on $M$ is continuity of log-H"older-type, which results in good approximation properties of the space. However, the requirement of regularity can be skipped in the case of reflexive spaces. The proof of the main results uses truncation ideas, the Young measures methods and monotonicity arguments. Uniqueness results from the comparison principle.