Regularity of weak solutions for nonlinear parabolic problem with $p(x)$-growth

摘要

In this paper, we study the nonlinear parabolic problem with $p(x)$-growth conditions in the space $W^{1,x}L^{p(x)}(Q)$, and give a regularity theorem of weak solutions for the following equation $$rac{partial u}{partial t}+A(u)=0$$ where $A(u)=-mbox{div} a(x,t,u,abla u)+a_0(x,t,u,abla u)$, $a(x,t,u,abla u)$ and $a_0(x,t,u,abla u)$ satisfy $p(x)$-growth conditions with respect to $u$ and $abla u$.