Calderón–Zygmund estimates for parabolic $p(x, t)$-Laplacian systems

We prove local Calderón–Zygmund estimates for weak solutions of the evolutionary p(x,t) -Laplacian system ∂_t u-div\ (a(x,t){|Du|}p(x,t)-2Du) = div\ ({|F|}p(x,t)-2F) under the classical hypothesis of logarithmic continuity for the variable exponent p(x,t) . More precisely, we show that the spatial gradient Du of the solution is as integrable as the right-hand side F , i.e., |F|p(·)∈ L^q_loc \ \ |Du|p(·)∈ L^q_loc any $q>1$, together with quantitative estimates. Thereby we allow the presence of eventually discontinuous coefficients a(x,t) , requiring only a VMO condition with respect to the spatial variable x .