We prove that a family of linear bounded evolution operators $({bf G}(t,s))_{tge sin I}$ can be associated, in the space of vector-valued bounded and continuous functions, to a class of systems of elliptic operators $bm{mathcal A}$ with unbounded coe
fficients defined in $Itimes Rd$ (where $I$ is a right-halfline or $I=R$) all having the same principal part. We establish some continuity and representation properties of $({bf G}(t,s))_{t ge sin I}$ and a sufficient condition for the evolution operator to be compact in $C_b(Rd;R^m)$. We prove also a uniform weighted gradient estimate and some of its more relevant consequence.
Using time dependent Lyapunov functions, we prove pointwise upper bounds for the heat kernels of some nonautonomous Kolmogorov operators with possibly unbounded drift and diffusion coefficients.
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 $G(t,s)$.
In this paper we study the main properties of the Ces`aro means of bi-continuous semigroups, introduced and studied by K{u}hnemund in [24]. We also give some applications to Feller semigroups generated by second-order elliptic differential operators
with unbounded coefficients in $C_b(R^N)$ and to evolution operators associated with nonautonomous second-order differential operators in $C_b(R^N)$ with time-periodic coefficients.
We consider a class of nonautonomous elliptic operators ${mathscr A}$ with unbounded coefficients defined in $[0,T]timesR^N$ and we prove optimal Schauder estimates for the solution to the parabolic Cauchy problem $D_tu={mathscr A}u+f$, $u(0,cdot)=g$.
We revisit the Near Equidiffusional Flames (NEF) model introduced by Matkowsky and Sivashinsky in 1979 and consider a simplified, quasi-steady version of it. This simplification allows, near the planar front, an explicit derivation of the front equat
ion. The latter is a pseudodifferential fully nonlinear parabolic equation of the fourth-order. First, we study the (orbital) stability of the null solution. Second, introducing a parameter $epsilon$, we rescale both the dependent and independent variables and prove rigourously the convergence to the solution of the Kuramoto-Sivashinsky equation as $epsilonto 0$.
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.
We consider a class of non-trivial perturbations ${mathscr A}$ of the degenerate Ornstein-Uhlenbeck operator in ${mathbb R}^N$. In fact we perturb both the diffusion and the drift part of the operator (say $Q$ and $B$) allowing the diffusion part to
be unbounded in ${mathbb R}^N$. Assuming that the kernel of the matrix $Q(x)$ is invariant with respect to $xin {mathbb R}^N$ and the Kalman rank condition is satisfied at any $xin{mathbb R}^N$ by the same $m<N$, and developing a revised version of Bernsteins method we prove that we can associate a semigroup ${T(t)}$ of bounded operators (in the space of bounded and continuous functions) with the operator ${mathscr A}$. Moreover, we provide several uniform estimates for the spatial derivatives of the semigroup ${T(t)}$ both in isotropic and anisotropic spaces of (Holder-) continuous functions. Finally, we prove Schauder estimates for some elliptic and parabolic problems associated with the operator ${mathscr A}$.
