Cauchy problem generated by oscillations of stratified fluid partially closed by ice

In the paper, we study the Cauchy problem for the differential second-order equation in Hilbert space $\mathscr{H} = H_1 \oplus H_2$ of the following form
$$\mathscr{C}\frac{d^2u}{dt^2} + \mathscr{B}_0u = f(t),\quad u(0) = u^0,\quad u^\prime(0) = u^1,$$
$$
\mathscr{C} =
\left(
\begin{matrix}
I_1& 0\\
0& A
\end{matrix},
\right),\quad
\mathscr{B}_0 =
\left[
\left(
\begin{matrix}
0& 0\\
0& N
\end{matrix}
\right)
+
\left(
\begin{matrix}
B_{11}& B_{12}\\
B_{21}& B_{22}
\end{matrix}
\right)
\right],
$$
$$u = (u_1; u_2)^t,\quad f = (f_1; f_2)^t.$$
Here $u = u(t)$ is an unknown function, $f = f(t)$ is a given function, $I_1$ is the identity operator,
$$
0 < A = A^\ast \in \mathscr{L}(H_2),\quad 0 \le \mathscr{B} = \mathscr{B}^\ast \in \mathscr{L}(\mathscr{H}),\quad
\mathscr{B} =
\left(
\begin{matrix}
B_{11}& B_{12}\\
B_{21}& B_{22}
\end{matrix},
\right)
$$
For the operator $N$, two situations are considered.
Situation 1. $N = I_2$ is the identity operator acting in $H_2$. This situation is possible in the study of the Cauchy problem generated by the oscillations of a stratified fluid partially covered with crumbled ice. Under the crumbled ice we understand that on the free surface float ponderable particles of some substance, the interaction of which one with another is negligibly small.
Situation 2. $N = N^\ast \gg 0, \overline{\mathscr{D}(N)} = H_2$. This situation is possible in the study of the Cauchy problem generated by oscillations of a stratified fluid partially covered with the elastic ice. Elastic ice is modeled by an elastic plate.
We find sufficient conditions for the existence of a strong (with respect to time variable) solution of initial Cauchy problems.

Keywords: stratification effect in ideal fluids, differential equation in Hilbert space, Cauchy problem, strong solution.