We consider a class of nonautonomous second order parabolic equations with unbounded coefficients defined in $ItimesR^d$, where $I$ is a right-halfline. We prove logarithmic Sobolev and Poincare inequalities with respect to an associated evolution sy
stem of measures ${mu_t: t in I}$, and we deduce hypercontractivity and asymptotic behaviour results for the evolution operator $G(t,s)$.
We study asymptotic behavior in a class of non-autonomous second order parabolic equations with time periodic unbounded coefficients in $mathbb Rtimes mathbb R^d$. Our results generalize and improve asymptotic behavior results for Markov semigroups h
aving an invariant measure. We also study spectral properties of the realization of the parabolic operator $umapsto {cal A}(t) u - u_t$ in suitable $L^p$ spaces.
We study a class of elliptic operators $A$ with unbounded coefficients defined in $ItimesCR^d$ for some unbounded interval $IsubsetCR$. We prove that, for any $sin I$, the Cauchy problem $u(s,cdot)=fin C_b(CR^d)$ for the parabolic equation $D_tu=Au$
admits a unique bounded classical solution $u$. This allows to associate an evolution family ${G(t,s)}$ with $A$, in a natural way. We study the main properties of this evolution family and prove gradient estimates for the function $G(t,s)f$. Under suitable assumptions, we show that there exists an evolution system of measures for ${G(t,s)}$ and we study the first properties of the extension of $G(t,s)$ to the $L^p$-spaces with respect to such measures.
