Global gradient estimates for parabolic equations with measurable nonlinearities

We establish a global Calderón–Zygmund theory of nonlinear parabolic equations with measurable nonlinearities in divergence form by proving that the spatial gradient of a weak solution is as integrable as the inhomogeneous term. Nonlinearity a(ξ,x,t) is assumed to be only measurable in one spatial variable and has locally small BMO semi-norm in the other spatial and time variables, uniformly with respect to ξ variable. The boundary of the bounded domain can be beyond the Lipschitz category, but it is well trapped in two narrow strips at each point and at each scale.