Hypercontractivity and asymptotic behaviour in nonautonomous Kolmogorov   equations
  We consider a class of nonautonomous second order parabolic equations with unbounded coefficients defined in $I\times\R^d$, where $I$ is a right-halfline. We prove logarithmic Sobolev and Poincar\'e inequalities with respect to an associated evolution system of measures $\{\mu_t: t \in I\}$, and we deduce hypercontractivity and asymptotic behaviour results for the evolution operator $G(t,s)$. 
L. Angiuli, L. Lorenzi, A. Lunardi
math.AP