Nonautonomous Kolmogorov parabolic equations with unbounded coefficients

Abstract

Abstract. We study a class of elliptic operators A with unbounded coefficients defined in I × R d for some unbounded interval I ⊂ R. We prove that, for any s ∈ I, the Cauchy problem u(s,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.

Cores for parabolic operators with unbounded coefficients

“…As it has been proved in [32] the following Hypothesis 2.2 provides us with a sufficient criterion for the existence of a Borel measure as in the statement of Theorem 2.1.…”

“…Recently, in [32] part of the results in [17,25,26] have been extended to more general nonautonomous Kolmogorov operators with analytical tools. In particular, under rather general assumptions on the smoothness of the coefficients of the operator A and assuming the existence of a Lyapunov function of A (see the forthcoming Hypothesis 1.1(iii)), it has been shown that an evolution operator (that we still denote by P (s, r)) can be associated with the nonautonomous elliptic operator A such that, for any bounded and continuous function ϕ, P (s, r)ϕ is the value at t of the unique classical solution to (1.2).…”

“…In [32] it has been also proved that the families of operators (T (t)) and (T p (t)) (t > 0), defined as above, give rise to evolution semigroups, in C b (R N+1 ) and in L p (R N+1 , ν) (p ∈ [1, +∞)) respectively, where the measure ν, defined accordingly to (1.5), is (infinitesimally) invariant for the semigroup (T (t)), i.e.,…”

“…It is worth stressing that no information on the domain G p of (T p (t)) are obtained in [32]. The only result therein proved in this direction shows that G p extends the realization of G in the space of smooth and compactly supported functions f : R N+1 → R. But this is enough neither to deal with the Cauchy problem (1.2) in L p -spaces nor to apply the techniques in [26] to prove several interesting results such as the Poincaré inequality, the spectral gap property and the asymptotic behaviour of the operator (T p (t)).…”

“…In this paper, we keep on the investigations in [32]. In Section 2 (see Theorem 2.1), under suitable assumptions we show that, whenever a Borel probability measure ν, satisfying (1.6) and such that ν(I × R N ) < +∞ for any bounded interval I ⊂ R, exists, the set (1.8) where B R denotes the open ball centered at 0 with radius R, is a core for G p for any p ∈ [1, +∞).…”