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,\[u_t-div (F(z,∇ u)∇ u )=f(z), z=(x,t)∈ Q_T:=Ω× (0,T),\] under the homogeneous Dirichlet boundary conditions. Here,\[F(z,∇ u)=a(z)|∇ u|ᵖ⁽ᶻ⁾⁻² + b(z) |∇ u |q(z)-2\]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 QT, the space and time derivatives of a and b belong to Lᵈ(QT) with some d depending on the data. If\[f∈ L^σ(Q_T) for \ σ ∈ (2, N+2] and F((·,0),∇ u_0)\,|∇ u_0|ʳ⁺²∈ L^1(Ω),\]where \(0≤ r≤ K(N,σ,p,q)\) if \(σ<N+2\), while \(r≥0\) is arbitrary if \(σ=N+2\), 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.