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We prove that Gevrey regularity is propagated by the Boltzmann equation with Max-wellian molecules, with or without angular cutoff. The proof relies on the Wild expansion of the solution to the equation and on the characterization of Gevrey regularity by the Fourier transform.
This paper is devoted to the grazing collision limit of the inelastic Kac model introduced in [PT04], when the equilibrium distribution function is a heavy-tailed Lévy-type distribution with infinite variance. We prove that solutions in an appropriate domain of attraction of the equilibrium distribution converge to solutions of a Fokker-Planck equation with… (More)
∂tu = ∆u + f(u) u(0) = u0 where u(t, x) : R × R → R, and f ∈ C(R,R) is a given function with f(0) = 0. It is well-known that if the initial data u0 belong to L ∞(Rn) then there exist T (u0) > 0 and a unique solution u ∈ C ([0, T [,L ∞(Rn)). In this paper we will consider initial data u0 which do not belong to L ∞(Rn). At first we will review the known… (More)
For the equation −∆u = ||x| − 2| α u p−1 , 1 < |x| < 3, we prove the existence of two solutions for α large, and of two additional solutions when p is close to the critical Sobolev exponent 2 * = 2N/(N − 2). A symmetry– breaking phenomenon appears, showing that the least–energy solutions cannot be radial functions.