+ {{\gamma}}\int_0^t ||w_t||^2_{H_\alpha} d\tau \\ ={{\cal E}}(v_0, u_0, u_1)
    +\int_0^t(G_f,  v)_{{\cal O}}d\tau +\int_0^t (G_{sh},u_t)_{{\Omega}}d\tau\end{gathered}$$ is valid for every $t>0$, where the energy functional ${{\cal E}}$ is defined by the relation $$\label{en-def}
    {{\cal E}}(v, u, u_t)=\frac12\left[\|v\|^2_{{\cal O}}+ \|M_{{\alpha}}^{1/2}w_t\|^2_{{\Omega}}+
    {{\varrho}}\|{\bar{u}}_t\|^2_{{\Omega}}+\| {\Delta}w\|_{{\Omega}}^2 +Q(u)\right]$$ with $$\begin{aligned}
    \label{Q-def}
    Q(u)=& \frac 2{(1-\mu)}\int_{{\Omega}}\left( {\varepsilon}_{11}^2 + {\varepsilon}_{22}^2 +2\mu {\varepsilon}_{11}{\varepsilon}_{22} + {{\frac12}}(1-\mu){\varepsilon}_{12}^2 \right)
    \notag \\
    = &\frac 1{2(1+\mu)}\int_{{\Omega}}\left( N_{11}^2 + N_{22}^2 -2\mu N_{11}N_{22} + 2(1+\mu)N_{12}^2 \right).\end{aligned}$$

Proof of Theorem \[th:WP\] {#proof-of-theoremthwp .unnumbered}
--------------------------

We start with the following elliptic type property of the functional $Q(u)$.

\[pr:Korn\] There exists a positive constant $C$ such that for every $u=(u^1;u^2;w)\in W= H_0^1({{\Omega}})\times H_0^1({{\Omega}})\times H_0^2({{\Omega}})$ we have that $$||u^1||_{1,{{\Omega}}}^2+||u^2||_{1,{{\Omega}}}^2 \le C\left[ Q(u)
+\|w\|^2_{{\Omega}}+\|w\|^4_{3/2,{{\Omega}}}\right].
\label{Q_best}$$

One can see that $$\left[u^i_{x_i}\right]^2\le 2{\varepsilon}^2_{ii}+2\left[k_iw +{{\frac12}}\left(w_{x_i}\right)^2\right]^2,\quad
\left[u^1_{x_2}+u^2_{x_1} \right]^2\le 2{\varepsilon}^2_{12}+
2\left[w_{x_1}w_{x_2} \right]^2.$$ Therefore using the embedding $H^{1/2}({{\Omega}})\subset L_4({{\Omega}})$ we obtain that $$\int_{{\Omega}}\left[ \left(u^1_{x_1}\right)^2+\left(u^1_{x_1}\right)^2
+\left(u^1_{x_2}+u^2_{x_1} \right)^2\right]dx'\le C\left[ Q(u)
+\|w\|^2_{{\Omega}}+\|w\|^4_{3/2,{{\Omega}}}\right].$$ Thus, property follows from the Korn inequality (see, e.g., Theorem 3.1 in Chapter 3 of [@DuLions]).

Now we use the compactness method and split the argument into several steps.

[*Step 1. Existence of an approximate solution for the case ${{\varrho}}>0$.*]{} For the construction of Galerkin’s approximations in this case we use the same idea as in [@ChuRyz2011] (which was inspired by 