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 \smash{\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 $u_0\in \smash{\dot{H}}^1$ 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 $0<t<T?$.

Edit: Thanks to the comment below, when $T<+\infty$ this follows from the smoothing effect which can be used to show that $$\int_{t}^{T} |u|^{2p} dx dt \leq C \log(T/t).$$ However, when the flow is global, i.e. $T=+\infty$ I am not sure how to extend the above argument since $1/t$ is not integrable at $+\infty.$