Thus, it suffices to show that $C_{c}^{\infty}({\mathbb{R}};H)$ is a core for $\partial_{t,\rho}^{\ast}\mathcal{M}^{\ast}$. However, since $$\mathcal{M}^{\ast}\partial_{t,\rho}^{\ast}\subseteq(\partial_{t,\rho}\mathcal{M})^{\ast}=\partial_{t,\rho}^{\ast}\mathcal{M}^{\ast}+(\mathcal{M}')^{\ast}$$ we can follow the same lines as above and obtain the assertion.

In order to prove we adopt the idea presented in [@Trostorff2013_nonautoincl] and first prove the well-posedness of an auxiliary problem. The well-posedness of the original problem will then follow by the perturbation result . The auxiliary problem reads as follows $$(u,f)\in\overline{\partial_{t,\rho}\mathcal{M}-\mathcal{M}'+\delta+\mathcal{A}}\label{eq:aux}$$ for a suitable $\delta>0.$

\[lem:matriallaw-aux\]Let $\mathcal{L}\in L(L_{2,\rho}({\mathbb{R}};H))$ and $\delta>0$. Then $\partial_{t,\rho}\mathcal{M}+\mathcal{L}+\delta$ is $(c+\delta-\|\mathcal{L}-\mathcal{N}\|)$-maximal monotone.

For $u\in{\operatorname{dom}}(\partial_{t,\rho}\mathcal{M})$ we have that $$\begin{aligned}
{\operatorname{Re}}\langle(\partial_{t,\rho}\mathcal{M}+\mathcal{L}+\delta)u,u\rangle_{\rho} & ={\operatorname{Re}}\langle\left(\partial_{t,\rho}\mathcal{M}+\mathcal{N}\right)u,u\rangle_{\rho}+\langle(\delta+(\mathcal{L}-\mathcal{N}))u,u\rangle_{\rho}\\
 & \geq\left(c+\delta-\|\mathcal{L}-\mathcal{N}\|\right)|u|_{\rho}^{2},\end{aligned}$$ where we have used . Moreover, since $$\partial_{t,\rho}\mathcal{M}+\mathcal{L}+\delta-\left(c+\delta-\|\mathcal{L}-\mathcal{N}\|\right)=\left(\partial_{t,\rho}\mathcal{M}+\mathcal{N}-c\right)+(\|\mathcal{L}-\mathcal{N}\|+\mathcal{L}-\mathcal{N})$$ the assertion follows by .

As an immediate consequence we derive the following proposition.

\[prop:c-monotne\_aux\]Let $\delta>\|\mathcal{M}'+\mathcal{N}\|$. Then $\partial_{t,\rho}\mathcal{M}-\mathcal{M}'+\delta+\mathcal{A}$ is $c$-monotone and hence,

Thus, in order to prove that is well-posed, it suffices to prove that the domain of $\left(\partial_{t,\rho}\mathcal{M}-\mathcal{M}'+\delta+\mathcal{A}\right)^{-1}$ is dense in $L_{2,\rho}({\mathbb{R}};H).$ We will prove that for each $f\in H_{\rho}^{1}({\mathbb{R}};H)$ there exists $u\in H_{\rho}^{1}({\mathbb{R}};H)$ such that $$(u,f)\in\partial_{t,\rho}\mathcal{M}-\mathcal{M}'+\delta+\mathcal{A},$$ which in particular would give $$H_{\rho}^{1}({\mathbb{R}};H)\subseteq{\operatorname{dom}}\left(\left(\partial_{t,\rho}\mathcal{M}-\mathcal{M}'+\delta+\mathcal{A}\right)^{-1}\right)$$

and thus, the well-posedness of would follow. For doing so, let $f\in H_{\rho}^{1}({\mathbb{R}};H)$ and define $$u_{\lambda}\coloneqq\left(\partial_{t,\rho}\mathcal{M}-\mathcal{M}'+\delta+\mathcal{A}_{\lambda}\right)^{-1}(f)\quad(\lambda>0).\label{eq:u_lambda}$$ Note that $\left(\partial_{t,\rho}\mathcal{M}-\mathcal{M}'+\delta+\mathcal{A}_{\lambda}\right)^{-1}$ with $\delta>\|\mathcal{M}'+\mathcal{N}\|$ is a Lipschitz-continuous mapping on $L_{2,\rho}({\mathbb{R}};H)$ defined on the whole space by and and 