Background: Let $u:\mathbb{R}^n\times [0,T_+)\to \mathbb{R}$ be a solution to $$\partial_t u = \Delta u + |u|^{p-1}u$$ where $p=2^{*}-1=\frac{2n}{n-2}-1=\frac{n+2}{n-2}$, $n\geq 3$ and $T_+>0$ denotes the maximal time of existence given initial data $u(x,0)=u_0\in \dot{H}^{1}(\mathbb{R}^n).$
Question: I would like to estimate $\int_{\mathbb{R}^n}\int_{I} |u|^{2p}\phi_R^2$ on any time interval $I=[t_1,t_2]\subset [0,T_+)$ where $\phi_R\in C^{\infty}_c(B_{2R})$ (with $\phi\equiv 1$ on $B_R$ with $|\nabla \phi|\leq \frac{C}{R}$), ideally in terms of quantities that depend on the initial data. If it is helpful, one can also assume that $u$ is radially symmetric (maybe to use Strauss Lemma at some point).
Possible Attempt: If we test the PDE with $|u|^{p-1}u\phi^2$ then after integration by parts one gets \begin{align} \int_{\mathbb{R}^n}|u|^{2p}\phi^2 \lesssim \int \phi^2 |u|^p |\partial_t u| + 2 \int \phi |\nabla \phi| |\nabla u| |u|^p + \int \phi^2 |\nabla u|^2 |u|^{p-1} \\ \end{align} which implies an estimate of the form \begin{align} \|u\|_{L^{2p}(B_R)}^{2p} \lesssim \|\partial_t u\|_{L^2(B_{2R})}\|u\|_{L^{2p}(B_{2R})}^p + \frac{C}{R^2}\int |\nabla u|^2 + \int \phi^2 |\nabla u|^2 |u|^{p-1} \end{align} However, I am not sure how to proceed after this step. Any suggestions will be much appreciated.
