a subspace in ${{\cal H}}$ of the form $${\widehat{{{\cal H}}}}= \left\{ (v_0;u_0;u_1)\in {{\cal H}}:\; w_0\in {\widehat{H}}^2_0({{\Omega}})\right\},$$ where $w_0$ is the third component of the initial displacement vector $u_0$.

Our main result in this section is the following well-posedness theorem.

\[th:WP\] Assume that $U_0=(v_0;u_0;u_1)\in {{\cal H}}$, $G_f(t)\in L_2(0,T; V')$, $G_{sh}(t)\in L_2\big(0,T; \left[H^{-1/2}({{\Omega}})\right]^2\times H^{-1}({{\Omega}})\big)$, ${{\gamma}}\ge 0$. We also assume that ${{\varrho}}\ge0$ (in the case ${{\varrho}}=0$ the data ${\bar{u}}_1$ are not fixed). Then for any interval $[0,T]$ there exists a unique weak solution $(v(t); u(t))$ to – with the initial data $U_0$. This solution possesses the following properties:

-   In the case ${{\varrho}}>0$ we have $$\label{cont-ws}
    U(t;U_0)\equiv U(t)\equiv (v(t); u(t); u_t(t))\in C(0,T; X\times W \times Y),$$ If ${{\varrho}}=0$, then $$\label{cont-ws-0}
    U(t;U_0)\equiv U(t)\equiv (v(t); u(t); w_t(t))\in C(0,T; X\times W \times H_{{\alpha}}).$$

-   The solutions depends continuously (both in strong and weak sense) on initial data. More precisely, if ${{\varrho}}>0$ and $U_n\to U_0$ in the norm of ${{\cal H}}$ (resp. weakly in ${{\cal H}}$), then $U(t;U_n)\to U(t;U_0)$ strongly (resp. weakly) in ${{\cal H}}$ for each $t>0$. In the case ${{\varrho}}=0$ the corresponding convergence take place in $ X\times W \times H_{{\alpha}}$.

-   The energy balance equality $$\begin{gathered}
    \label{lin_energy}
    {{\cal E}}(v(t), u(t), u_t(t))+\nu \int_0^t E(v,v) d\tau 