  We study the Boltzmann equation in a smooth bounded domain featuring a mixed
boundary condition. Specifically, gas particles experience specular reflection
in two parallel plates, while diffusive reflection occurs in the remaining
portion between these two specular regions. The boundary is assumed to be
motionless and isothermal. Our main focus is on constructing global-in-time
small-amplitude solutions around global Maxwellians for the corresponding
initial-boundary value problem. The proof relies on the $L^2$ hypocoercivity at
the linear level, utilizing the weak formulation and various functional
inequalities on the test functions, such as Poincar\'e and Korn inequalities.
It also extends to the linear problem involving Maxwell boundary conditions,
where the accommodation coefficient can be a piecewise constant function on the
boundary, allowing for more general bounded domains. Moreover, we develop a
delicate application of the $L^2-L^\infty$ bootstrap argument, which relies on
the specific geometry of our domains, to effectively handle this mixed-type
boundary condition.
