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We study the homogeneous Dirichlet problem for the equation \[u_t-\operatorname{div}\left((a(z)\vert \nabla u\vert ^{p(z)-2}+b(z)\vert \nablau\vert ^{q(z)-2})\nabla u\right)=f\quad \text{in $Q_T=\Omega\times (0,T)$}, \]where $\Omega\subset \mathbb{R}^N$, $N\geq 2$, is a bounded domain with$\partial\Omega \in C^2$. The variable exponents $p$, $q$ and the nonnegativemodulating coefficients $a$, $b$ are given Lipschitz-continuous functions ofthe argument $z=(x,t)\in Q_T$. It is assumed that $\frac{2N}{N+2}<p(z),\ q(z)$and that the modulating coefficients and growth exponents satisfy the balanceconditions \[ \text{$a(z)+b(z)\geq \alpha>0$ in $\overline{Q}_T$},\;\alpha=const;\qquad \text{$\vert p(z)-q(z)\vert <\frac{2}{N+2}$ in$\overline{Q}_T$}. \] We find conditions on the source $f$ and the initial data$u(\cdot,0)$ that guarantee the existence of a unique strong solution $u$ with$u_t\in L^2(Q_T)$ and $a\vert \nabla u\vert ^{p}+b\vert \nabla u\vert ^q\inL^\infty(0,T;L^1(\Omega))$. The solution possesses the property of globalhigher integrability of the gradient, \[ \vert \nabla u\vert^{\min\{p(z),q(z)\}+r}\in L^1(Q_T)\quad \text{with any $r\in\left(0,\frac{4}{N+2}\right)$}, \] which is derived with the help of newinterpolation inequalities in the variable Sobolev spaces. The second-orderdifferentiability of the strong solution is proven: \[D_{x_i}\left(\left(a\vert \nabla u\vert ^{p-2}+b\vert \nabla u\vert^{q-2}\right)^{\frac{1}{2}}D_{x_j}u\right)\in L^2(Q_T),\quad i,j=1,2,\ldots,N.\]

Existence and regularity results for a class of parabolic problems with double phase flux of variable growth