  This paper addresses the questions of existence and uniqueness of strong
solutions to the homogeneous Dirichlet problem for the double phase equation
with operators of variable growth:
  \[ u_t - div \left(|\nabla u|^{p(z)-2} \nabla u+ a(z) |\nabla u|^{q(z)-2}
\nabla u \right) = F(z,u) \quad \text{in $Q_T=\Omega \times (0,T)$} \] where
$\Omega \subset \mathbb{R}^N$, $N \geq 2$, is a bounded domain with the
boundary $\partial\Omega\in C^2$, $z=(x,t)\in Q_T$, $a:\bar Q_T \mapsto
\mathbb{R}$ is a given nonnegative coefficient, and the nonlinear source term
has the form \[ F(z,v)=f_0(z)+b(z)|v|^{\sigma(z)-2}v. \] The variable exponents
$p$, $q$, $\sigma$ are given functions defined on $\bar{Q}_T$, $p$, $q$ are
Lipschitz-continuous and \[ \dfrac{2N}{N+2}<p^-\leq p(z) \leq q(z) < p(z) +
{\frac{r}{2}} \ \ \text{with $0<r<r^\ast=\frac{4p^-}{2N + p^-(N+2)}$,\quad
$p^-=\min_{\bar{Q}_T}p(z)$}. \] We find conditions on the functions $f_0$, $a$,
$b$, $\sigma$ and $ u_0$ sufficient for the existence of a unique strong
solution with the following global regularity and integrability properties: \[
\begin{split} u_t \in L^{2}(Q_T),\quad & \text{$|\nabla u|^{s(z)} \in
L^{\infty}(0,T;L^1(\Omega))$ with $ s(z)=\max\{2,p(z)\}$}, & |\nabla
u|^{p(z)+\delta}\in L^1(Q_T)\quad \text{for every $0<\delta< r^*$}. {split} \]
The same results are established for the equation with the regularized flux
  \[ (\epsilon^2+|\nabla u|^2)^{\frac{p(z)-2}{2}}\nabla u + a(z)
(\epsilon^2+|\nabla u|^2)^{\frac{q(z)-2}{2}}\nabla u, \qquad \epsilon>0. \]
