We consider the spatially inhomogeneous Boltzmann equation for inelastic hard-spheres, with constant restitution coefficient $ \alpha\in(0,1) $, under the thermalization induced by a host medium with fixed $ e\in(0,1] $ and a fixed Maxwellian distribution. When the restitution coefficient $ \alpha $ is close to 1 we prove existence and uniqueness of global solutions considering the close-to-equilibrium regime. We also study the long-time behaviour of these solutions and prove a convergence to equilibrium with an exponential rate.

中文翻译：

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由粒子热浴驱动的非弹性玻尔兹曼方程

我们考虑非弹性硬球的空间非均匀玻尔兹曼方程，具有恒定的恢复系数 $ \alpha\in(0,1) $，在由具有固定 $ e\in(0,1] $ 和一个固定的麦克斯韦分布。当恢复系数 $ \alpha $ 接近 1 时，我们证明了考虑接近平衡状态的全局解的存在性和唯一性。我们还研究了这些解的长期行为并证明了收敛到以指数速率平衡。