Boundary Caccioppoli's inequality for parabolic equation.

Question

Let $ \Omega $ be a $ C^1 $ bounded domain in $ \mathbb{R}^d $. $ A(x,t):\mathbb{R}^d\to\mathbb{R}^{d\times d} $ is a matrix valued function such that \begin{eqnarray} \mu|\xi|^2\leq A(x,t)\xi_i\xi_j\leq \mu^{-1}|\xi|^2\text{ for any }\xi\in\mathbb{R}^d,(x,t)\in\mathbb{R}^{d+1} \end{eqnarray} where $ \mu $ is a positive constant. For $ 0<r<1/8\operatorname{diam}(\Omega) $, $ x_0\in\partial\Omega $ ana $ t_0\in\mathbb{R} $, consider the boundary localization of parabolic equation \begin{eqnarray} \left\{\begin{matrix} \partial_tu-\operatorname{div}(A(x,t)\nabla u)=\operatorname{div}(f)&\text{in}&[B(x_0,2r)\cap\Omega]\times(t_0-4r^2,t_0),\\ u=g&\text{on}&[B(x_0,2r)\times\partial\Omega]\times(t_0-4r^2,t_0), \end{matrix}\right. \end{eqnarray} where $ f\in L^2(\Omega) $ and $ g $ is in some space to be determined. For the case that $ g=0 $, By using the same arguments in the proof of the interior Caccioppoli's inequality, I can prove that \begin{eqnarray} \int_{\Omega_{r}(x_0,t_0)}|\nabla u|^2dxdt\leq \frac{C}{r^2}\int_{\Omega_{2r}(x_0,t_0)}|u|^2dxdt+\int_{\Omega_{2r}(x_0,t_0)}|f|^2dxdt, \end{eqnarray} where $ \Omega_r(x_0,t_0)=[B(x_0,r)\cap\Omega]\times(t_0-r^2,t_0) $. Now I want to consider the case that $ g\not\equiv 0 $, I want to imitate the trick in the proof for elliptic equations, that is choose $ G $ such that $ G|_{[B(x_0,2r)\times\partial\Omega]\times(t_0-4r^2,t_0)}=g $ and examine the equation of $ u-G $. However I am not familiar with the boundary condition here and know little about the choice of $ G $. Can you give me some references or hints?