arXiv subjects
Publications and source records attributed to Paolo De Fazio.
In this paper we prove the well-posedness of non-autonomous deterministic and stochastic reaction-diffusion equations with a polynomial reaction term. Concerning the stochastic problem, we also prove a new result on the space-time regularity of the non-autonomous stochastic convolution.
In this paper we study the heat content for sets with positive reach. In details, we investigate the asymptotic behavior of the heat content of bounded subsets of the Euclidean space with positive reach. The concept of positive reach was introduced by Federer in \cite{fed_1959} and widely developed in the following years (see for instance the recent book by Rataj and Zh{ä}le \cite{rat_zah_2019}). It extends the class of sets with smooth boundaries to include certain non-smooth and singular sets while still admitting a well-defined normal geometry. For such sets $E\subseteq\Rn$, we analyze the short-time asymptotics of the heat content $\|T_t\mathbbm{1}_E\|_2$, where $T_t\mathbbm{1}_E$ is the soluzion of the heat equation in $\Rn$ with initial condition $\mathbbm{1}_E$. The present paper is in the spirit of Angiuli, Massari and Miranda Jr.\cite{ang_mas_mir_2013}, but the technique's used here are completely different and also the final result is slightly different.
In an infinite dimensional separable Hilbert space $X$, we study compactness properties and the hypercontractivity of the Ornstein-Uhlenbeck evolution operators $P_{s,t}$ in the spaces $L^p(X,γ_t)$, $\{γ_t\}_{t\in\R}$ being a suitable evolution system of measures for $P_{s,t}$. Moreover, we study the asymptotic behavior of $P_{s,t}$. Our results are produced thanks to a representation formula for $P_{s,t}$ through the second quantization operator. Among the examples, we consider the transition evolution operator associated to a non-autonomous stochastic parabolic PDE.
In an infinite dimensional separable Hilbert space $X$, we study the realizations of Ornstein-Uhlenbeck evolution operators $\pst$ in the spaces $L^p(X,\g_t)$, $\{\g_t\}_{t\in\R}$ being the unique evolution system of measures for $\pst$ in $\R$. We prove hyperconctractivity results, relying on suitable Log-Sobolev estimates. Among the examples we consider the transition evolution operator of a non autonomous stochastic parabolic PDE.
We prove smoothing properties along suitable directions of the Ornstein-Uhlenbeck evolution operator, namely the evolution operator associated to non autonomous Ornstein-Uhlenbeck equations. Moreover we use such smoothing estimates to prove Schauder type theorems, again along suitable directions, for the mild solutions of a class of evolution equations.