where $\tau_{h}u\coloneqq u(\cdot+h).$ Then, $u\in H_{\rho}^{1}({\mathbb{R}};H)$ if and only if $(D_{h}u)_{h\in]0,1]}$ is bounded in $L_{2,\rho}({\mathbb{R}};H).$ In each case we have that $$D_{h}u\to\partial_{t,\rho}u\quad(h\to0+)$$ in $L_{2,\rho}({\mathbb{R}};H).$

Besides the presented Hilbert space setting, we need the framework of maximal monotone relations. For this topic and the proofs of the subsequent results we refer to the monographs [@Brezis; @Morosanu; @papageogiou].

A binary relation $A\subseteq H\times H$ is called *monotone*, if $$\forall(u,v),(x,y)\in A:\:{\operatorname{Re}}\langle u-x,v-y\rangle\geq0.$$

$A$ is called *maximal monotone* if $A$ is monotone and for each monotone relation $B\subseteq H\times H$ we have that $$A\subseteq B\Rightarrow A=B.$$

Moreover, we say that $A$ is *$c$-(maximal) monotone* for some $c\in{\mathbb{R}},$ if $A-c\coloneqq\{(u,v-cu)\,;\,(u,v)\in A\}$ is (maximal) monotone.

We need the following lifting result.

\[lem:lifting\]Let $A\subseteq H\times H$ and $(\Omega,\Sigma,\mu)$ a measure space. We define $$A_{\mu}\coloneqq\left\{ (u,v)\in L_{2}(\mu;H)\times L_{2}(\mu;H)\,;\,(u(t),v(t))\in A\quad(t\in\Omega\text{ a.e.})\right\} .$$ If $A$ is monotone, then so is $A_{\mu}.$ Moreover, if $A$ is maximal monotone and $(0,0)\in A$ it follows that $A_{\mu}$ is maximal monotone as well.

If $\mu(\Omega)<\infty$ then the assumption $(0,0)\in A$ can be dropped. However, since we want to apply this lifting result to $\Omega=\mathbb{R}$ and $\mu={\mathrm{e}}^{-2\rho t}{\,\mathrm{d}}t$, we need to impose this condition in our main result. If one just deals with $\mathbb{R}_{\geq0}$ as 