Global Calderon-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, $$u_t-\mathrm{div}(\mathcal{F}(z,\mathrm{\nabla }u)\mathrm{\nabla }u)=f(z),z=(x,t)\in Q_T:=\mathrm{\Omega }\times (0,T),$$ under the homogeneous Dirichlet boundary conditions. Here, $\mathrm{\Omega }\subset {\mathbb{R}}^N$, $N\ge 2$, is a bounded domain, $T>0$, $$\mathcal{F}(z,\mathrm{\nabla }u)=a(z)\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{p(z)-2}+b(z)\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{q(z)-2}$$ 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 $$f\in L^{\sigma }(Q_T)\text{for }\sigma \in (2,N+2]\text{and}\mathcal{F}((\cdot ,0),\mathrm{\nabla }{u}_0)\mathrm{\mid }\mathrm{\nabla }{u}_0{\mathrm{\mid }}^{r+2}\in L^1(\mathrm{\Omega }),$$ where $0\le r\le K(N,\sigma ,p,q)$ if $\sigma <N+2$, while $r\ge 0$ is arbitrary if $\sigma =N+2$, then the problem has a unique strong solution, for which we prove the global transfer of integrability from the initial data and the forcing term to the double-phase flux in the spirit of Calderón-Zygmund theory, higher integrability of the gradient, and the second-order space regularity: $$\begin{array}{rl}{\displaystyle }& {\displaystyle \mathcal{F}((\cdot ,t),\mathrm{\nabla }u(\cdot ,t))\mathrm{\mid }\mathrm{\nabla }u(\cdot ,t){\mathrm{\mid }}^{r+2}\in L^1(\mathrm{\Omega })\text{ for a.e. }t\in (0,T),}\\ {\displaystyle }& {\displaystyle \mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{2(\min \{p(z),q(z)\}-1)+r+s}\in L^1(Q_T)\text{ for every }s\in (0,\frac{4}{N+2}),}\\ {\displaystyle }& {\displaystyle \mathcal{F}(z,\mathrm{\nabla }u)\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{\frac{r+2}{2}}\in L^2(0,T;W^{1,2}(\mathrm{\Omega }))\mathrm{.}}\end{array}$$ The results improve and complement the results in \cite{Arora-Shmarev-JGA-2026} and extend them to the full range $r\ge 0$.