the shell. In this case the (elastic) stress tensor $\{N_{ij}\}$ is given by the formulas $$\begin{aligned}
\label{Hooke}
N_{11} =\frac 2{1-\mu}\left({\varepsilon}_{11}+\mu {\varepsilon}_{22}\right),~~
N_{22} =\frac 2{1-\mu}\left({\varepsilon}_{22}+\mu {\varepsilon}_{11}\right),~~ N_{12}={\varepsilon}_{12},\end{aligned}$$ where the deformation tensor $\{{\varepsilon}_{ij}\}$ has the form $$\begin{aligned}
{\varepsilon}_{11} &= u^1_{x_1}+ k_1 w+ \frac 12 (w_{x_1})^2, \\
{\varepsilon}_{22} &=u^2_{x_2}+ k_2 w+ \frac 12 (w_{x_2})^2,\\
{\varepsilon}_{12} &=u^1_{x_2}+u^2_{x_1} + w_{x_1}w_{x_2}.\end{aligned}$$ Here $k_1$ and $k_2$ are curvatures of the initial form of the shell which are sufficiently smooth functions of $x'\in{{\Omega}}$.

After an appropriate rescaling of the parameters and functions we can model shell dynamics by the following equations $$\begin{gathered}
\label{u3-eq}
M_\alpha (w_{tt} +{{\gamma}}w_t)  + {\Delta}^2 w +k_1N_{11}+k_2N_{22} \\  -{\partial}_{x_1}(N_{11}w_{x_1}+N_{12}w_{x_2}) -
{\partial}_{x_2}(N_{12}w_{x_1}+N_{22}w_{x_2}) \\
=  G_3(t)- 2\nu {\partial}_{x_3}v^3 +p  ~~{\rm in}~~ \Omega \times (0, \infty),\end{gathered}$$ where $M_\alpha=1-\alpha{\Delta}$, and $$\begin{aligned}
\label{u1u2-eq}
    {{\varrho}}u_{tt}^1 & ={\partial}_{x_1} N_{11}+{\partial}_{x_2} N_{12}+ G_1(t)-\nu (v^1_{x_3}+v^3_{x_1}), \notag \\
      {{\varrho}}u_{tt}^2 & ={\partial}_{x_1} N_{12}+{\partial}_{x_2} N_{22}+ G_2(t)-\nu (v^2_{x_3}+v^3_{x_2}),\end{aligned}$$ where $G_{sh}(t)\equiv (G_1; G_2; G_3)(t)$ is a given body force applied to the shell. $\alpha> 0$ and ${{\varrho}}\ge 0$ are constants which take into account rotational inertia and in-plane inertia of the shell, respectively, ${{\gamma}}$ is a non-negative parameter which describes intensity of the viscous damping of the shell material.

We impose the clamped boundary conditions on the shell $$u^1|_{{\partial}{{\Omega}}}=u^2|_{{\partial}{{\Omega}}}=w|_{{\partial}{{\Omega}}}=\left.\frac{{\partial}w}{{\partial}n} \right|_{{\partial}{{\Omega}}}=0 