q_{N} = 0, 1, 2, \ldots) $

Space S is a perfect space (complete countably normed space, in which the bounded sets are compact). Space $\overrightarrow{TS}$ of vector-functions $\vec{\varphi}$ is a direct sum of N perfect spaces S (N = 3) $\cite{VT80}, \cite{RN72}$ :\

$\overrightarrow{TS} = S \oplus S \oplus S$.

To define topology in the space $\overrightarrow{TS}$ let us introduce countable system of norms

$$\label{eqn201}
\|\vec{\varphi}\|_{p}\;=\; \sum_{i = 1}^{N}\|\varphi_{i}\|_{p}\; = \sum_{i = 1}^{N}\mathop{sup_{\;x}}_{\mid k \mid ,\; \mid q \mid \;\leq \;p}\mid x^{k}D^{q} \varphi_{i}(x)\mid\;\;\;(p = 0, 1, 2,...),\;\;(N = 3)$$

The Fourier transform maps the space S onto the whole space S and maps the space $\overrightarrow{TS}$ onto the whole space $\overrightarrow{TS}$ $\cite{gS01},\;\cite{GC68}$.

Equivalence of Cauchy problem in differential form $\textbf {(\ref{eqn1})}$ - $\textbf {(\ref{eqn3})}$ and in the form of an integral equation
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Let us denote solution of the problem $(\ref{eqn1})$ - $(\ref{eqn3})$ as {$\vec{u}(x_{1}, x_{2}, x_{3}, t)$, p($x_{1}, x_{2}, x_{3},$ t)}, in other words let us consider infinitely differentiable by t $\in$ \[0,$\infty$) vector-function $\vec{u}(x_{1}, x_{2}, x_{3}, t) \in \overrightarrow{TS}$ and infinitely differentiable function p($x_{1}, x_{2}, x_{3},$ t) $\in$ S, that turn equations $(\ref{eqn1})$ , $(\ref{eqn2})$ into identities. Vector-function $\vec{u}(x_{1}, x_{2}, x_{3}, t)$ also satisfies the initial 