Consider the semilinear critical parabolic PDE in $\mathbb{R}^3$ \begin{align} \partial_t u &= \Delta u + |u|^{4/(n-2)}u = \Delta u + u^5\\ u(0,x) &= u_0\in \dot{H}^1(\mathbb{R}^n). \end{align}

I am trying to understand the smoothing effects of the above flow. From the seminal work of Brezis-Cazenave, there exists $T=T(u_0)$ such that the above flow is well posed on $[0,T)$ for $u_0\in \dot{H}^1$ with the following time decay:

$$\|u(t)\|_{L^{\infty}}\leq \frac{C}{t^{1/4}}.$$

I would like to understand if one can show further gain in the regularity of solutions to the above PDE. For instance, is it true that the following integral is finite

$$\int_t^T \int_{\mathbb{R}^n} |u|^{2(n+2)/(n-2)} dx dt = \int_t^T \int_{\mathbb{R}^n} |u|^{10} dx dt < +\infty$$

for $0<t<T?$ In general I know from elliptic PDE theory that $\Delta u + u^{5}=0$ that $u$ gains two derivatives more than the nonlinearity (which is in $L^{6/5}$), so we can expect that solution to the elliptic problem is in $W^{2,6/5}$ which can then be bootstrapped to get the smoothness of the solution. What is the analogue of this principle for the above parabolic problem? Is it true that $u(t) \in W^{2,6/5}$ for all time implying that $u(t)\in W^{1,2}(\mathbb{R}^n)$ or $u(t)\in L^2(\mathbb{R}^n)$?