Optimal Gradient Estimates for Parabolic Equations in Variable Exponent Spaces

Abstract

We establish an optimal |$L^{p(\cdot )}$|-regularity theory for the gradient of weak solutions for parabolic equations in divergence form with bounded measurable coefficients in rough domains beyond the Lipschitz category. With a function |$p(\cdot )=p(x,t)$| of spatial and time variables satisfying log-Hölder continuity, we prove that the spatial gradient of the weak solution is as integrable as the nonhomogeneous term in the variable exponent space |$L^{p(\cdot )}$| under the assumption that the coefficients are merely measurable in one of spatial variables while have a small bounded mean oscillation in the other spatial variables and time variable. On the other hand, the domain is assumed to be sufficiently flat in the Reifenberg sense.