Global Calderón-Zygmund estimates for irregular double-phase evolution problem with non-divergence data

We study irregular double-phase parabolic equations with variable exponents and non-divergence data,\[u_t-\operatorname{div} \left(\mathcal{F}(z,\nabla u)\nabla u \right)=f(z),\quad z=(x,t)\in Q_T:=\Omega\times (0,T),\] under the homogeneous Dirichlet boundary conditions. Here,\[\mathcal{F}(z,\nabla u)=a(z)|\nabla u|^{p(z)-2} + b(z) |\nabla u |^{q(z)-2}\]with given Lipschitz-continuous exponents $p,q$ that satisfy a suitable balance condition. The nonnegative coefficients $a(z), b(z)$ satisfy the inequality$a(z)+b(z)>0$ in $Q_T$, the space and time derivatives of $a$ and $b$ belong to $L^d(Q_T)$ with some $d$ depending on the data. If\[f\in L^\sigma(Q_T) \quad \text{for} \ \sigma \in (2, N+2] \quad \text{and} \quad \mathcal{F}((\cdot,0),\nabla u_0)\,|\nabla u_0|^{r+2}\in L^1(\Omega),\]where \(0\le r\le K(N,\sigma,p,q)\) if \(\sigma<N+2\), while \(r\ge0\) is arbitrary if \(\sigma=N+2\), then the problem has a unique strong solution., We prove the global transfer of integrability from the data to the double-phase flux in the spirit of Calder\'on-Zygmund theory, higher integrability of the gradient, and the second-order space regularity.