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Resumen:
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We study in this paper the asymptotic behaviour of solutions of a nonlinear Fokker-Plank equation. Such an equation describes the evolution of radiation for a gas of photons, which interacts with electrons by means of Compton scattering and Bremsstrahlung radiation. Assuming that a suitable adimensional parameter ? (which measures the strength of the Bremsstrahlung effect) is small enough, we show that the problem considered has two natural timescales. For times t = O(1), the dynamics is conducted by that of a reduced problem, corresponding to setting ? = O in the original equations. Solutions of that problem may blow up in finite time, and the total number of photons is no longer preserved after the singularity formation. Nevertheless, solutions of this problem can be continued for all times, if defined in a suitable sense. When t --> infinity, solutions of such a modified problem converge towards a Bose-Einstein distribution with a suitable (in general nonzero) chemical potential. However, at times of order t = O((?|log ?|)(-2/3)), the Bremsstrahlung term becomes dominant at low frequencies, and drives the photon distribution to approach to a Planck distribution as time goes to infinity.
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