contains a subsequence such that $$\begin{aligned}
&(v_{{{\varrho}}}; u_{{\varrho}}) \rightharpoonup (v; u) \quad \ast\mbox{-weakly in } L_\infty(0,T; X\times W),\notag  \\
 &{\partial}_t w_{{{\varrho}}} \rightharpoonup {\partial}_t w \quad \ast\mbox{-weakly in } L_\infty(0,T; H^1_0({{\Omega}})),\notag  \\
&v_{{{\varrho}}} \rightharpoonup v \quad \mbox{weakly in } L_2(0,T;V), \notag  $$ when ${{\varrho}}\to 0$, and also $$\begin{aligned}
&v_{{{\varrho}}} \rightharpoonup v \quad \mbox{weakly in } L_2(0,T; H^{1/2}(\partial{{\cal O}})),
 \notag     \\
&{\partial}_t u_{{\varrho}}\rightharpoonup {\partial}_t u \quad \mbox{weakly in } L_2\left(0,T; \left[H^{1/2}(\partial{{\cal O}})\right]^3\right)\notag $$ Therefore we have that $$\begin{aligned}
&u_{{\varrho}}\rightarrow u \quad \mbox{strongly in } C(0,T; H^{1-{\varepsilon}}_0({{\Omega}})\times H^{1-{\varepsilon}}_0({{\Omega}})\times H^{2-{\epsilon}}_0({{\Omega}}))
$$ for every ${\varepsilon}>0$. This allows us to make limit transition when ${{\varrho}}\to 0$ in and prove the existence of weak solutions for the case ${{\varrho}}=0$.

[*Step 4. Uniqueness.*]{} We use the same idea as in [@Sed_1997]. However the additional regularity of ${\bar{u}}_t$ (see Remark \[re:weak-sol\](2)) makes it possible to cover both cases ${{\varrho}}>0$ and ${{\varrho}}=0$ simultaneously.

Let $(\widehat{v}(t);\widehat{u}(t))$ and $(\widetilde{v}(t);\widetilde{u}(t))$ be two solutions to the problem in question with the same initial data. These solutions satisfy and their difference $(v(t);u(t)) =(\widehat{v}(t)-\widetilde{v}(t);\widehat{u}(t)-\widetilde{u}(t))$ possesses properties $$\begin{aligned}
      &v=(v^1;v^2;v^3)\in L_\infty(0,T;X)\bigcap L_2(0,T; V);     \\
  & u =(u^1;u^2;w)\in L_\infty(0,T; H^1_0({{\Omega}})\times H^1_0({{\Omega}})\times H^2_0({{\Omega}})); \\
 & w_t\in  L_\infty(0,T;  H_\alpha),~~~
 (u^1_t;  u^2_t) \in L_2(0,T; [H^{1/2}_*({{\Omega}})]^2);
     \end{aligned}$$ and by satisfies the relation $$\begin{gathered}
           