  We provide existence and uniqueness of renomalized solutions to a general
nonlinear parabolic equation with merely integrable data on a Lipschitz bounded
domain in $\mathbb{R}^n$. Namely we study \begin{equation*}
\left\{\begin{array}{l } \partial_t u-{\rm div} A(t,x,\nabla u)= f(t,x) \in
L^1(\Omega_T),\\ u(0,x)=u_0(x)\in L^1(\Omega). \end{array}\right.
\end{equation*} The growth of the monotone vector field $A$ is assumed to be
controlled by a generalized nonhomogeneous and anisotropic $N$-function
$M:[0,T)\times \Omega \times\mathbb{R}^n \to[0,\infty)$. Existence and
uniqueness of renormalized solutions are proven in absence of~Lavrentiev's
phenomenon. The condition we impose to ensure approximation properties of the
space is a certain type of balance of interplay between the behaviour of $M$
for large $|\xi|$ and small changes of time and space variables. Its instances
are log-H\"older continuity of variable exponent (inhomogeneous in time and
space) or optimal closeness condition for powers in double phase spaces
(changing in time). The noticeable challenge of this paper is considering the
problem in non-reflexive and inhomogeneous fully anisotropic space that changes
along time. New delicate approximation-in-time result is proven and applied in
the construction of renormalized solutions.
