Nonlinear Parabolic Equations with Variable Exponent Growth in Nonsmooth Domains

Abstract

We study the Calderon--Zygmund theory for nonlinear parabolic problems in the setting of the variable exponent Lebesgue spaces. In particular, we prove the global $L^{s(\cdot)}$ integrability of the gradient of solutions to parabolic equations with $p(\cdot)$ growth in nonsmooth domains with $s(\cdot)>p(\cdot)$. In addition, we present precise regularity conditions on the variable exponent functions, the nonlinearity, and the boundary of a domain to enable us to obtain the desired result.