Local Boundedness of Weak Solutions for Nonlinear Parabolic Problem with p(x)-Growth

Abstract

We study the nonlinear parabolic problem with p(x)-growth conditions in the space W1,xLp(x)(Q) and give a local boundedness theorem of weak solutions for the following equation (∂u/∂t)+A(u)=0, where A(u)=−diva(x,t,u,∇u)+a0(x,t,u,∇u),a(x,t,u,∇u) and a0(x,t,u,∇u) satisfy p(x)-growth conditions with respect to u and ∇u.