CAUCHY PROBLEM FOR THE SPATIALLY HOMOGENEOUS LANDAU EQUATION WITH SHUBIN CLASS INITIAL DATUM AND GELFAND-SHILOV SMOOTHING EFFECT

In this work, we study the nonlinear spatially homogeneous Landau equation with Maxwellian molecules; by using the spectral analysis, we show that the nonlinear Landau operator is almost linear, and we prove the existence of a weak solution for the Cauchy problem with the initial datum belonging to the Shubin space of the negative index which contains the probability measures. Based on this spectral decomposition, we prove also that the Cauchy problem enjoys the S-1/2(1/2)-Gelfand-Shilov smoothing effect, meaning that the weak solution of the Cauchy problem with the Shubin class initial datum is ultra-analytics and exponential decay for any positive time.