Let $A$ be a positive definite, constant coefficients matrix. Suppose that a Sobolev function $u$ solves $$ \partial_t u - \text{div}(A\nabla u)=0 $$ in the distributional sense on $B_1\times (-1,0)$. Is it the case that the following inequality holds?
$$\int_{B_r\times(-r^2,0)} |\nabla u|^2 \le Cr^{n+2} \int_{B_1\times (-1,0)} |\nabla u|^2 $$ for $0<r<1$, $B_r$ the ball of radius $r$.