I am working on the Bidomain-Model which, during a time interval [0,T], describes the electrical behaviour of the myocardial muscle considered as $\Omega \subset \mathbb{R}^3$.
This model has partial differential equations which involve the intra- and extracellular voltages, $u$ and $v$, which are functions of this type:
$$ u=u(t,x):[0,T]\times\Omega\rightarrow\mathbb{R}$$
$$ v=v(t,x):[0,T]\times\Omega\rightarrow\mathbb{R}$$
Now, as functions of time,
$$ u,v \in X=L^2\left(0,T,H^1(\Omega)\right)$$
with norm $\| u \|_{_{X}} = \left(\int_{_{[0,T]}}\| u(t) \|^{^2}_{_{H^1(\Omega)}} dt \right)^{1/2}$.
Their time derivatives
$$ \dot u,\dot v \in Y=L^2\left(0,T,H^1(\Omega)' \right)$$
with norm $\| u \|_{_{Y}} = \left(\int_{_{[0,T]}}\| u(t) \|^{^2}_{_{H^1(\Omega)'}} dt \right)^{1/2}$.
Further, for any fixed $t_0 \in [0,T]$,
$$\| u(t_0) \|_{_{H^1(\Omega)}}^{^2} = \| \dot u(t_0,x) \|_{_{L^2(\Omega)}}^{^2} + \| u(t_0,x) \|_{_{L^2(\Omega)}}^{^2} = \langle u(t_0,x), u(t_0,x) \rangle_{_{H^1(\Omega)}}$$
but
Which is $\| u(t_0,x) \|_{_{H^1(\Omega)'}}$?
Is it $$\|u(t_0)\|_{_{H^1(\Omega)'}}^{^2} = \sup_{\xi \in H^1(\Omega)} \left\{ \langle u(t_0,x), \xi(t_0,x) \rangle_{_{H^1(\Omega)}} \, | \, \| \xi(t_0,x)\|_{_{H^1(\Omega)}}^{^2} \le 1 \right\}$$ as this post suggests?