  A half-space problem of a linear kinetic equation for gas molecules
physisorbed close to a solid surface, relevant to a kinetic model of
gas-surface interactions and derived by Aoki et al. [K.~Aoki et al., in: Phys.
Rev. E 106:035306, 2022], is considered. The equation contains a confinement
potential in the vicinity of the solid surface and an interaction term between
gas molecules and phonons. It is proved that a unique solution exists when the
incoming molecular flux is specified at infinity. This validates the natural
observation that the half-space problem serves as the boundary condition for
the Boltzmann equation. It is also proved that the sequence of approximate
solutions used for the existence proof converges exponentially fast. In
addition, numerical results showing the details of the solution to the
half-space problem are presented.
