L_{loc}^2\left(
\mathbb{R}_{+};H_0^1\left( {\cal O}\right) \right) .$ If $u\in
L_{loc}^{\infty}\left( \mathbb{R}_{+}; L^2\left( {\cal O}\right) \,
\right) \bigcap L^2_{loc} \left( \mathbb{R}_+;  H_0^1\left( {\cal
O}\right) \right),$ one has $$\left\| u\right\| _{2,\infty ;t}\vee \left\| u\right\|
_{2^{*},2;t}\le c_1\left( \left\| u\right\| _{2,\infty ;t}^2+\left\|
\nabla u\right\| _{2,2;t}^2\right) ^{\frac 12},$$with $c_1=c_S\vee
1.$

For $d\ge 3$ and some parameter $\theta \in [0,1[$ we set: $$\Gamma_\theta =\left\{ \left( p,q\right) \in  \left[ 1,\infty
\right]^2, \,  \frac d{2p}+\frac 1q=\frac d2+\theta \right\} ,$$

$$\Gamma _\theta ^{*}=\left\{ \left( p,q\right) \in \left[
1,\infty \right] ^2/\;\frac d{2p}+\frac 1q=1-\theta \right\} ,$$ $$L_\theta ^{*}=\sum_{\left( p,q\right) \in \Gamma _\theta
^{*}}L^{p,q}\left( \left[ 0,t\right] \times {\cal O}\right)$$ $$\left\| u\right\| _{\theta ;t}^{*}:=\,\inf \left\{
\sum_{i=1}^n\left\| u_i\right\|
_{p_i,q_i;\,t}\,/\;u=\sum_{i=1}^nu_i,u_i\in L^{p_i,q_i}\left( \left[
0,t\right] \times {\cal O}\right) ,\right.$$ $$\left. \left( p_i,q_i\right) \in \Gamma _\theta ^{*},\,i=1,...n;\,n\in {\bf N}^{*}\right\} .$$ If $d=1,2.$ we put $$\Gamma _\theta =\left\{ \left( p,q\right) \in \left[ 1,\infty \right] ^2/\;\frac{2^{*}}{2^{*}-2}\frac 1p+\frac 1q=\frac{2^{*}}{2^{*}-2}+\theta
\right\} ,$$ $$\Gamma _\theta ^{*}=\left\{ \left( p,q\right) \in
\left[ 1,\infty \right] ^2/\;\frac{2^{*}}{2^{*}-2}\frac 1p+\frac
1q=1-\theta \right\}$$ and by using similar calculations with the convention $\frac{2^{*}}{2^{*}-2}=1$ if $d=1.$

We remark that $\Gamma _\theta ^{*}=I\left( \infty ,\frac 1{1-\theta },\frac
d{2\left( 1-\theta \right) },\infty \right) $ and that the norm $\left\| u\right\| _{\theta ;t}^{*}$ coincides with $\left\|
u\right\| ^{\Gamma _\theta ^{*};t}=\left\| u\right\| ^{I\left(
\infty ,\frac 1{1-\theta },\frac 