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.
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.
We review some known results of local existence in the framework of Le-besgue spaces for the nonlinear heat equation with polynomial nonlinearity. Then we consider nonlinearity of exponential growth and we present a new result of local existence in the context of Orlicz spaces. We consider the Cauchy problem for the semilinear heat equation ∂ t u = ∆u + f… (More)
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)