Regularity and existence of solutions to parabolic equations with nonstandard p(x,t),q(x,t)-growth conditions

Abstract

We study the Cauchy-Dirichlet problem for a class of nonlinear parabolic equations driven by nonstandard $p(x,t),q(x,t)$-growth condition. We prove theorems of existence and uniqueness of weak solutions in suitable Orlicz-Sobolev spaces, derive global and local in time $L^{\infty}$ bounds for the weak solutions.