Elliptic and Parabolic Equations - Abstract

Lunardi, Alessandra

Asymptotic behavior in linear nonautonomous parabolic problems

II will present asymptotic behavior results for a class of linear nonautonomous parabolic problems, u_t = A(t)u, where the elliptic second order differential operators A(t) have possibly unbounded coefficients in R^n. The results are given in terms of a family of probability measure m_t(dx) = r(t,x)dx, that play the role of the invariant measure in the autonomous case. The simplest examples that can be studied in detail are time depending Ornstein--Uhlenbeck operators, A(t)f(x) = Trace(Q(t)D^2f(x)) + , where Q(t), B(t) are given matrices, Q(t) symmetric and uniformly positive definite, and the null solution to y'(t) = B(t)y(t) is exponentially stable.