support*\
In the second step, we build a convenient decomposition of $B_{2R}$, depending on a parameter $\gamma>0$ to be chosen in Step 3. Namely, we set $r_{0}=0$ and claim that there exists an $N \in \N $ and an increasing collection of radii $\{r_{k}\}_{k=1}^{N}$ (with $r_{N}=R$) such that $$\label{eq:kthsmallDiv}
\|(\Div u)_{-}\|_{L^{q}([0,T] \times A_{r_{k-1},r_{k}})}\leq \gamma.$$ Indeed, if $\|(\Div u)_{-}\|_{L^{q}([0,T] \times B_{R})} \leq \gamma$, then we simply set $N=1$ and $r_{1}=R$. Otherwise, we use that for any $r \geq 0$ fixed, the function $s \in (r,R) \to \|(\Div u)_{-}\|_{L^{q}([0,T] \times A_{r,s})}$ is non-decreasing and continuous to select a strictly increasing sequence $\{r_{k}\}_{k=1}^{N-1}$ taking values in $(0,R)$ such that $$\|(\Div u)_{-}\|_{L^{q}([0,T] \times A_{r_{k-1},r_{k}})}=\gamma.$$ The integer $N>1$ and the radius $r_{N-1}$ are selected in order that $$\|(\Div u)_{-}\|_{L^{q}([0,T] \times A_{r_{N-1},R})}\leq \gamma,$$ and the procedure necessarily terminates in a finite $N$ because $\Div u \in L^{q}([0,T] \times B_{R})$.\
*Step 3: Auxilliary non-linear parabolic problems*\
Given $\{r_{k}\}_{k=0}^{N}$ from Step $2$ we define additionally $r_{N+1}=2R$, $r_{N+2}=3R$, and $r_{N+3}=4R$. The next step is to build a collection $\{\varphi_{k,\epsilon}\}_{k=1}^{N}$ of solutions to non-linear parabolic boundary value problems which will be used to construct our sub-solution in Step 4. By Theorem 9.1 and Remark 2.3 in [@MR1789919], we may build 