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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. Brezis–Cazenave here proved

Theorem: Assume $q>N(p-1) / 2$ (resp. $q=N(p-1) / 2$ ) and $q \geq 1$ (resp. $q>1), N \geq 1$. Given any $u_0 \in L^q(\Omega)$, there exist a time $T=T\left(u_0\right)>0$ and a unique function $u \in C\left([0,T], L^q(\Omega)\right)$ with $u(0)=u_0$, which is a classical solution of $\partial_t u = \Delta u + |u|^{p-1}u$ on $(0, T) \times\bar{\Omega}$ (in the Duhamel sense).

Moreover, we have:

(i) Smoothing effect and continuous dependence, namely

$$ \|u(t)-v(t)\|_{L^q}+t^{N / 2 q}\|u(t)-v(t)\|_{L^{\infty}} \leq C\left\|u_0-v_0\right\|_{L^q} $$

for all $t \in(0, T]$ where $T=\min \left\{T\left(u_0\right), T\left(v_0\right)\right\}$ and $C$ can be estimated in terms of $\left\|u_0\right\|_{L^a}$ and $\left\|v_0\right\|_{L^q}$.

(ii) $\lim_{t \downarrow 0} t^{N / 2 q}\|u(t)\|_{L^{\infty}}=0$.

(iii) If $u_0 \geq 0$, then $u(t) \geq 0$ for all $t \in\left[0, T\left(u_0\right)\right]$.

Furthermore, for any bounded set (resp. compact set) ${K}$ in $L^q(\Omega)$, there is a (uniform) time $T=T({K})$ such that for any $u_0 \in \mathcal{K}$ the solution of the nonlinear heat equation exists on $[0, T]$.

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 $+\infty.$

Consider the semilinear energy-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. Brezis–Cazenave here proved

Theorem: Assume $q>N(p-1) / 2$ (resp. $q=N(p-1) / 2$ ) and $q \geq 1$ (resp. $q>1), N \geq 1$. Given any $u_0 \in L^q(\Omega)$, there exist a time $T=T\left(u_0\right)>0$ and a unique function $u \in C\left([0,T], L^q(\Omega)\right)$ with $u(0)=u_0$, which is a classical solution of $\partial_t u = \Delta u + |u|^{p-1}u$ on $(0, T) \times\bar{\Omega}$ (in the Duhamel sense).

Moreover, we have:

(i) Smoothing effect and continuous dependence, namely

$$ \|u(t)-v(t)\|_{L^q}+t^{N / 2 q}\|u(t)-v(t)\|_{L^{\infty}} \leq C\left\|u_0-v_0\right\|_{L^q} $$

for all $t \in(0, T]$ where $T=\min \left\{T\left(u_0\right), T\left(v_0\right)\right\}$ and $C$ can be estimated in terms of $\left\|u_0\right\|_{L^a}$ and $\left\|v_0\right\|_{L^q}$.

(ii) $\lim_{t \downarrow 0} t^{N / 2 q}\|u(t)\|_{L^{\infty}}=0$.

(iii) If $u_0 \geq 0$, then $u(t) \geq 0$ for all $t \in\left[0, T\left(u_0\right)\right]$.

Furthermore, for any bounded set (resp. compact set) ${K}$ in $L^q(\Omega)$, there is a (uniform) time $T=T({K})$ such that for any $u_0 \in \mathcal{K}$ the solution of the nonlinear heat equation exists on $[0, T]$.

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 $+\infty.$

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 $+\infty.$

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. Brezis–Cazenave here proved

Theorem: Assume $q>N(p-1) / 2$ (resp. $q=N(p-1) / 2$ ) and $q \geq 1$ (resp. $q>1), N \geq 1$. Given any $u_0 \in L^q(\Omega)$, there exist a time $T=T\left(u_0\right)>0$ and a unique function $u \in C\left([0,T], L^q(\Omega)\right)$ with $u(0)=u_0$, which is a classical solution of $\partial_t u = \Delta u + |u|^{p-1}u$ on $(0, T) \times\bar{\Omega}$ (in the Duhamel sense).

Moreover, we have:

(i) Smoothing effect and continuous dependence, namely

$$ \|u(t)-v(t)\|_{L^q}+t^{N / 2 q}\|u(t)-v(t)\|_{L^{\infty}} \leq C\left\|u_0-v_0\right\|_{L^q} $$

for all $t \in(0, T]$ where $T=\min \left\{T\left(u_0\right), T\left(v_0\right)\right\}$ and $C$ can be estimated in terms of $\left\|u_0\right\|_{L^a}$ and $\left\|v_0\right\|_{L^q}$.

(ii) $\lim_{t \downarrow 0} t^{N / 2 q}\|u(t)\|_{L^{\infty}}=0$.

(iii) If $u_0 \geq 0$, then $u(t) \geq 0$ for all $t \in\left[0, T\left(u_0\right)\right]$.

Furthermore, for any bounded set (resp. compact set) ${K}$ in $L^q(\Omega)$, there is a (uniform) time $T=T({K})$ such that for any $u_0 \in \mathcal{K}$ the solution of the nonlinear heat equation exists on $[0, T]$.

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 $+\infty.$

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 $0<t<T?$ 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 large enough time implying that $u(t)\in W^{1,2}(\mathbb{R}^n)$ or $u(t)\in L^2(\mathbb{R}^n)$?

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 $+\infty.$