On existence of a solution periodic int to the first exterior mixed problem for the Sobolev system

For a given initial velocity $\vec{\mathcal{V}}^0(x)$ such that $\vec{\mathcal{V}}^0(x)\to0$ as $|x|\to\infty$ along all the rays parallel to the axis $x_3$ or the plane $x_1Ox_2$, an example of a solution to the exterior mixed problem for the Sobolev system is constructed such that is periodic in $t$ and belongs to $L_{\mathbb{p}}(G)$ for every fixed $t$, $\mathbb{p}=(p,p,p_3)$, $1<p_3<2$, $p>2/(1-1/p_3)$.