We study the homogeneous Dirichlet problem for the double-phase evolution equation \( u_t-\operatorname {div} \left( a(z)|\nabla u|^{p(z)-2} \nabla u + b(z)|\nabla u|^{q(z)-2} \nabla u\right) \nabla u=f(z) \) \(z=(x,t)\in Q_T=\Omega \times (0,T)\) . The non-differentiable coefficients a(z), b(z) and the variable exponents p(z), q(z) are given functions. The coefficients a, b are nonnegative and bounded, with \(|\nabla a|, |\nabla b|, a_t, b_t \in L^d(Q_T)\) , \(d>2\) , and such that \(a(z)+b(z)\ge \alpha >0\) . It is shown that if \(u_0\in W_0^{1,r}(\Omega )\) with a sufficiently large r and \(f\in L^{N+2}(Q_T)\) , then \(u(\cdot ,t)\in W^{1,r}_0(\Omega )\) for a.e. \(t\in (0,T)\) , \(|\nabla u|^{\min \{p(z),q(z)\}+s+r} \in L^1(Q_T)\) for any \(s \in (0, \frac{4}{N+2})\) , and \(a(z)|\nabla u|^{\frac{p(z)+r-2}{2}} + b(z)|\nabla u|^{\frac{q(z)+r-2}{2}} \in L^2(0,T;W^{1,2}(\Omega )).\) The case \(f\in L^\sigma (Q_T)\) with \(\sigma \in (2,N+2)\) is also studied.
