Global boundedness of solutions of degenerate and non-uniform parabolic equations

Abstract

Let $2\le N\in \mathbb{N}$, $\Omega$ be a bounded open in ${\mathbb{R}}^N$, $T\in (0,\infty )$, $Q=\Omega \times (0,T)$, $u$ be a weak solution of parabolic equation $\frac{\partial u}{\partial t}-Lu=f$, where $L$ is an elliptic operator on a space of functions on $Q$. The coefficients of $L$ may not be bounded, not strictly nor uniformly elliptic, and not of Muckenhoupt type. We obtain global boundedness of $u$. Our result can be applied to $u$, which may vanish on $(A\times (0,T))\cup (\Omega \times \{0\})$ of the boundary of $Q$ and is free outside this set.