A note on Krylov-Tso’s parabolic inequality

Abstract:

We show that if $u$ is a solution to $\sum \nolimits _{i,j = 1}^n {{a_{ij}}(x,t){D_{ij}}u(x,t) - {D_t}u(x,t) = \phi (x)}$ on a cylinder ${\Omega _T} = \Omega \times (0,T)$, where $\Omega$ is a bounded open set in ${{\mathbf {R}}^n},T > 0$, and $u$ vanishes continuously on the parabolic boundary of ${\Omega _T}$. Then the maximum of $u$ on the cylinder is bounded by a constant $C$ depending on the ellipticity of the coefficient matrix $({a_{ij}}(x,t))$, the diameter of $\Omega$, and the dimension $n$ times the ${L^n}$ norm of $\phi$ in $\Omega$.