We investigate the time-asymptotic behavior for rarefied gases in the spherical domain with variable boundary temperature in ℝd, d = 1, 2, 3, under the diffuse reflection boundary condition. First, we obtain an optimal convergence rate of (1 + t)-d to the steady state for free molecular flow. Next, we use this to construct the steady state solution of the Boltzmann equation for sufficiently large Knudsen number and small boundary temperature variation. We also obtain an exponential convergence to the steady state for the Boltzmann equation for small perturbation.
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