  We consider the homogeneous Dirichlet problem for the parabolic equation
  \[ u_t- \operatorname{div} \left(|\nabla u|^{p(x,t)-2} \nabla u\right)=
f(x,t) + F(x,t, u, \nabla u) \] in the cylinder $Q_T:=\Omega\times (0,T)$,
where $\Omega\subset \mathbb{R}^N$, $N\geq 2$, is a $C^{2}$-smooth or convex
bounded domain. It is assumed that $p\in C^{0,1}(\overline{Q}_T)$ is a given
function, and that the nonlinear source $F(x,t,s, \xi)$ has a proper power
growth with respect to $s$ and $\xi$. It is shown that if
$p(x,t)>\frac{2(N+1)}{N+2}$, $f\in L^2(Q_T)$, $|\nabla u_0|^{p(x,0)}\in
L^1(\Omega)$, then the problem has a solution $u\in C^0([0,T];L^2(\Omega))$
with $|\nabla u|^{p(x,t)} \in L^{\infty}(0,T;L^1(\Omega))$, $u_t\in L^2(Q_T)$,
obtained as the limit of solutions to the regularized problems in the parabolic
H\"older space. The solution possesses the following global regularity
properties: \[ \begin{split} & |\nabla u|^{2(p(x,t)-1)+r}\in L^1(Q_T)\quad
\text{for any $0 < r < \frac{4}{N+2}$}, \\ & |\nabla u|^{p(x,t)-2} \nabla u \in
W^{1,2}(Q_T)^N. \end{split} \]
