On solutions of third boundary value problem for Laplace equation in a half-infinite cylinder

We study the asymptotic behavior at the infinity of solutions of the Laplace equation in a half-infinite cylinder providing that third boundary value condition is met
$$ \left.{\bigg({{{\partial u }\over{\partial\nu}}+\beta(x)u}\bigg)}\right|_{\Gamma}=0, $$
where $\Gamma$ is the lateral surface of the cylinder; $\beta(x)\geqslant 0$. We prove that any bounded solution is stabilized to some constant and its Dirichlet integral is finite. We describe a condition on boundary coefficient decrease at infinity which provides Dirichlet (dichotomy, stabilization to zero) or Neumann (trichotomy, stabilization to some constant which can be nonzero) problem type behavior of solutions. The main condition on boundary coefficient leading to Dirichlet or Neumann problem type is established in terms of divergence or convergence correspondingly of the integral $\displaystyle{\int_{\Gamma}}x_1\beta(x)\,dS,\quad $ where the variable $x_1$ corresponds to the direction of an axis of the cylinder.