The global well-posedness of the relativistic Boltzmann equation with hard potentials and diffuse reflection boundary condition in bounded domains

Abstract In the present paper, assuming that the motion of single-species relativistic particles in a bounded domain is governed by the relativistic Boltzmann equation with hard potentials and diffuse reflection boundary condition of non-isothermal wall temperature of small variations around a positive constant, we construct a unique non-negative stationary solution <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>F</mml:mi> <mml:mrow> <mml:mo>∗</mml:mo> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> , and further establish the dynamical stability of such stationary solution with exponential time decay rate. Due to the relativistic effect, the key point is to control the hydrodynamic part <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi mathvariant="bold">P</mml:mi> </mml:mrow> <mml:mi>f</mml:mi> </mml:mrow> </mml:math> by choosing suitable test functions, which are different from the ones of Esposito et al (2013 Commun. Math. Phys. 323 177–239).