Smooth solutions of parabolic

Question

Let $a_{ij}:[0,\infty) \times \mathbb{R}^{d}\to \mathbb{R} $ be smooth and bounded with bounded derivatives and let $u:[0,\infty) \times \mathbb{R}^{d}\to \mathbb{R} $ be a smooth solution of $$\partial_tu-\sum_{i,j=1}^{d}\partial_{i}(a_{ij}(t,x)\partial_{j}u)=0 $$ where $\sum_{i,j=1}^{d}a_{ij}(t,x)\xi_i \xi_j \ge \theta|\xi|^2$ and $\theta>0$. Assume also that $$u\ge0, \ Du\in L^{\infty}\left(\left(0,\infty\right),\ L^{2}\left(\mathbb{R}^{d}\right)\right), \ u\in L^{\infty}\left(\left(0,\infty\right),\ L^{1}\left(\mathbb{R}^{d}\right)\right) .$$ If $$m\left(t\right):=\int_{\mathbb{R}^{d}}u\left(t,x\right)\mathrm{d}x.\ E\left(t\right):=\int_{\mathbb{R}^{d}}u^{2}\left(t,x\right)\mathrm{d}x$$ show that $m\equiv m_0$ and $E^{\prime}\left(t\right)\le C\theta m^{-4/d}E^{\left(d+2\right)/d}$.