  We study a general nonlinear parabolic equation on a Lipschitz bounded domain
in $\mathbb{R}^N$, \begin{equation*} \left\{\begin{array}{l l} \partial_t
u-\mathrm{div} A(t,x,\nabla u)= f(t,x)&\text{in}\ \ \Omega_T,\\ u(t,x)=0 &\
\mathrm{ on} \ (0,T)\times\partial\Omega,\\ u(0,x)=u_0(x)&\text{in}\ \Omega,
\end{array}\right. \end{equation*} with $f\in L^\infty(\Omega_T)$ and $u_0\in
L^\infty(\Omega)$. The growth of the monotone vector field $A$ is controlled by
a generalized fully anisotropic $N$-function
$M:[0,T)\times\Omega\times\mathbb{R}^N\to[0,\infty)$ inhomogeneous in time and
space, and under no growth restrictions on the last variable. It results in the
need of the integration by parts formula which has to be formulated in an
advanced way. Existence and uniqueness of solutions are proven when the
Musielak-Orlicz space is reflexive OR in absence of Lavrentiev's phenomenon. To
ensure approximation properties of the space we impose natural assumption that
the asymptotic behaviour of the modular function is sufficiently balanced. Its
instances are log-H\"older continuity of variable exponent or optimal closeness
condition for powers in double phase spaces.
  The noticeable challenge of this paper is cosidering the problem in
non-reflexive and inhomogeneous fully anisotropic space that changes along
time.
