Suppose we have the differential inequality $$ |\partial_t{u}+\Delta{u}|\leq C(|u|+|\nabla u|) $$ in $\mathbb{R}^n\backslash B_R\times[0,1]$, where $B_R=\{x\in\mathbb{R}^n,|x|\leq R\}$. Then do we have that $$ \||u|+|\nabla u|\|_{L^2(B_{3R}\backslash B_{2R}\times[0,\frac{1}{2}])}\leq C\|u\|_{L^\infty(B_{4R}\backslash B_{R}\times[0,1])} $$ Maybe this is something standard, but I can't find a reference right now. So my question is how to prove such kind of inequalites.
Thanks for any comment and reference.