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| The long-time behavior of the kinetic linear equations is well-known in the case where the cross-section is bounded from below by a strictly positive constant. In the case where the cross-section vanishes in a portion of a domain -- such an equation is called "degenerate" -- , the results mentioned above have no obvious extension.
The presentation is based on a series of three papers written by the authors on the subject. We show that the geometry of the portion of the domain where the cross-section vanishes is the key feature of the problem. More precisely, we first exhibit an example where the convergence to equilibrium can not be exponential. Then we give a necessary and sufficient condition on the geometry of the vanishing zone so that we have uniform convergence to equilibrium.
Eventually, we will discuss the issue of finding an explicit rate of convergence. |
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