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Luca Lorenzi

Publications and source records attributed to Luca Lorenzi.

35 records · Page 2Linked to original sources
arXiv2017

On the estimates of the derivatives of solutions to nonautonomous Kolmogorov equations and their consequences

We consider evolution operators $G(t,s)$ associated to a class of nonautonomous elliptic operators with unbounded coefficients, in the space of bounded and continuous functions over $\mathbb{R}^d$. We prove some new pointwise estimates for the spatial derivatives of the function $G(t,s)f$, when $f$ is bounded and continuous or much smoother. We then use these estimates to prove smoothing effects of the evolution operator in $L^p$-spaces. Finally, we show how pointwise gradient estimates have been used in the literature to study the asymptotic behaviour of the evolution operator and to prove summability improving results in the $L^p$-spaces related to the so-called tight evolution system of measures.

math.AP↗
arXiv2016

Hypercontractivity, supercontractivity, ultraboundedness and stability in semilinear problems

We study the Cauchy problem associated to a family of nonautonomous semilinear equations in the space of bounded and continuous functions over R^d and in L^p-spaces with respect to tight evolution systems of measures. Here, the linear part of the equation is a nonautonomous second-order elliptic operator with unbounded coefficients defined in IxR^d, (I being a right-halfline). To the above Cauchy problem we associate a nonlinear evolution operator, which we study in detail, proving some summability improving properties. We also study the stability of the null solution to the Cauchy problem.

math.AP↗
arXiv2015

Strong convergence of solutions to nonautonomous Kolmogorov equations

We study a class of nonautonomous, linear, parabolic equations with unbounded coefficients on $\mathbb R^{d}$ which admit an evolution system of measures. It is shown that the solutions of these equations converge to constant functions as $t\to+\infty$. We further establish the uniqueness of the tight evolution system of measures and treat the case of converging coefficients.

math.AP↗
arXiv2015

$L^p$-estimates for parabolic systems with unbounded coefficients coupled at zero and first order

We consider a class of nonautonomous parabolic first-order coupled systems in the Lebesgue space $L^p({\mathbb R}^d;{\mathbb R}^m)$, $(d,m \ge 1)$ with $p\in [1,+\infty)$. Sufficient conditions for the associated evolution operator ${\bf G}(t,s)$ in $C_b({\mathbb R}^d;{\mathbb R}^m)$ to extend to a strongly continuous operator in $L^p({\mathbb R}^d;{\mathbb R}^m)$ are given. Some $L^p$-$L^q$ estimates are also established together with $L^p$ gradient estimates.

math.AP↗
arXiv2014

Non autonomous parabolic problems with unbounded coefficients in unbounded domains

Given a class of nonautonomous elliptic operators $\A(t)$ with unbounded coefficients, defined in $\overline{I \times \Om}$ (where $I$ is a right-halfline or $I=\R$ and $\Om\subset \Rd$ is possibly unbounded), we prove existence and uniqueness of the evolution operator associated to $\A(t)$ in the space of bounded and continuous functions, under Dirichlet and first order, non tangential homogeneous boundary conditions. Some qualitative properties of the solutions, the compactness of the evolution operator and some uniform gradient estimates are then proved.

math.AP↗
arXiv2013

On the Dirichlet and Neumann evolution operators in R^d_+

We prove some uniform and pointwise gradient estimates for the Dirichlet and the Neumann evolution operators $G_{\mathcal{D}}(t,s)$ and $G_{\mathcal{N}}(t,s)$ associated with a class of nonautonomous elliptic operators $\A(t)$ with unbounded coefficients defined in $I\times \Rd_+$ (where $I$ is a right-halfline or $I=\R$). We also prove the existence and the uniqueness of a tight evolution system of measures $\{μ_t^{\mathcal{N}}\}_{t \in I}$ associated with $G_{\mathcal{N}}(t,s)$, which turns out to be sub-invariant for $G_{\mathcal{D}}(t,s)$, and we study the asymptotic behaviour of the evolution operators $G_{\mathcal{D}}(t,s)$ and $G_{\mathcal{N}}(t,s)$ in the $L^p$-spaces related to the system $\{μ_t^{\mathcal{N}}\}_{t \in I}$.

math.AP↗
arXiv2012

Asymptotic Analysis in a Gas-Solid Combustion Model with Pattern Formation

We consider a free interface problem which stems from a solid-gas model in combustion with pattern formation. We derive a third-order, fully nonlinear, self-consistent equation for the flame front. Asymptotic methods reveal that the interface approaches a solution of the Kuramoto-Sivashinsky equation. Numerical results are presented which illustrate the dynamics.

math.AP↗
arXiv2012

Two-dimensional stability analysis in a HIV model with quadratic logistic growth term

We consider a Human Immunodeficiency Virus (HIV) model with a logistic growth term and continue the analysis of the previous article [6]. We now take the viral diffusion in a two-dimensional environment. The model consists of two ODEs for the concentrations of the target T cells, the infected cells, and a parabolic PDE for the virus particles. We study the stability of the uninfected and infected equilibria, the occurrence of Hopf bifurcation and the stability of the periodic solutions.

math.AP↗
arXiv2012

A strongly ill-posed problem for a degenerate parabolic equation with unbounded coefficients in an unbounded domain $Ω\times {\mathcal O}$ of $\R^{M+N}$

In this paper we deal with a strongly ill-posed second-order degenerate parabolic problem in the unbounded open set $Ω\times {\mathcal O}\subset \mathbb R^{M+N}$, related to a linear equation with unbounded coefficients, with no initial condition, but endowed with the usual Dirichlet condition on $(0,T)\times \partial(Ω\times {\mathcal O})$ and an additional condition involving the $x$-normal derivative on $Γ\times {\mathcal O}$, $Γ$ being an open subset of $Ω$. The task of this paper is twofold: determining sufficient conditions on our data implying the uniqueness of the solution $u$ to the boundary value problem as well as determining a pair of metrics with respect of which $u$ depends continuously on the data. The results obtained for the parabolic problem are then applied to a similar problem for a convolution integrodifferential linear parabolic equation.

math.AP↗
arXiv2012

On improvement of summability properties in nonautonomous Kolmogorov equations

Under suitable conditions, we obtain some characterization of supercontractivity, ultraboundedness and ultracontractivity of the evolution operator $G(t,s)$ associated to a class of nonautonomous second order parabolic equations with unbounded coefficients defined in $I\times\R^d$, where $I$ is a right-halfline. For this purpose, we establish an Harnack type estimate for $G(t,s)$ and a family of logarithmic Sobolev inequalities with respect to the unique tight evolution system of measures $\{μ_t: t \in I\}$ associated to $G(t,s)$. Sufficient conditions for the supercontractivity, ultraboundedness and ultracontractivity to hold are also provided.

math.AP↗
arXiv2012

On Schroedinger type operators with unbounded coefficients: Generation and heat kernel estimates

We consider the Schrödinger type operator ${\mathcal A}=(1+|x|^α)Δ-|x|^β$, for $α\in [0,2]$ and $β\ge 0$. We prove that, for any $p\in (1,\infty)$, the minimal realization of operator ${\mathcal A}$ in $L^p(\R^N)$ generates a strongly continuous analytic semigroup $(T_p(t))_{t\ge 0}$. For $α\in [0,2)$ and $β\ge 2$, we then prove some upper estimates for the heat kernel $k$ associated to the semigroup $(T_p(t))_{t\ge 0}$. As a consequence we obtain an estimate for large $|x|$ of the eigenfunctions of ${\mathcal A}$. Finally, we extend such estimates to a class of divergence type elliptic operators.

math.AP↗
arXiv2010

Compactness and invariance properties of evolution operators associated with Kolmogorov operators with unbounded coefficients

In this paper we consider nonautonomous elliptic operators ${\mathcal A}$ with nontrivial potential term defined in $I\times\mathbb R^d$, where $I$ is a right-halfline (possibly $I=\mathbb R$). We prove that we can associate an evolution operator $(G(t,s))$ with ${\mathcal A}$ in the space of all bounded and continuous functions on $\mathbb R^d$. We also study the compactness properties of the operator $G(t,s)$. Finally, we provide sufficient conditions guaranteeing that each operator $G(t,s)$ preserves the usual $L^p$-spaces and $C_0(\mathbb R^d)$.

math.AP↗
arXiv2009

Rigorous derivation of the Kuramoto-Sivashinsky equation in a 2D weakly nonlinear Stefan problem

In this paper we are interested in a rigorous derivation of the Kuramoto-Sivashinsky equation (K--S) in a Free Boundary Problem. As a paradigm, we consider a two-dimensional Stefan problem in a strip, a simplified version of a solid-liquid interface model. Near the instability threshold, we introduce a small parameter $\varepsilon$ and define rescaled variables accordingly. At fixed $\varepsilon$, our method is based on: definition of a suitable linear 1D operator, projection with respect to the longitudinal coordinate only, Lyapunov-Schmidt method. As a solvability condition, we derive a self-consistent parabolic equation for the front. We prove that, starting from the same configuration, the latter remains close to the solution of K--S on a fixed time interval, uniformly in $\varepsilon$ sufficiently small.

math.AP↗
arXiv2009

Heterogeneous Viral Environment in a HIV Spatial Model

We consider the basic model of virus dynamics in the modeling of Human Immunodeficiency Virus (HIV), in a 2D heterogenous environment. It consists of two ODEs for the non-infected and infected $CD_4^+$ $T$ lymphocytes, $T$ and $I$, and a parabolic PDE for the virus $V$. We define a new parameter $λ_0$ as an eigenvalue of some Sturm-Liouville problem, which takes the heterogenous reproductive ratio into account. For $λ_0<0$ the trivial non-infected solution is the only equilibrium. When $λ_0>0$, the former becomes unstable whereas there is only one positive infected equilibrium. Considering the model as a dynamical system, we prove the existence of a universal attractor. Finally, in the case of an alternating structure of viral sources, we define a homogenized limiting environment. The latter justifies the classical approach via ODE systems.

math.DS↗
arXiv2009

$L^p$--regularity for parabolic operators with unbounded time--dependent coefficients

We establish the maximal regularity for nonautonomous Ornstein-Uhlenbeck operators in $L^p$-spaces with respect to a family of invariant measures, where $p\in (1,+\infty)$. This result follows from the maximal $L^p$-regularity for a class of elliptic operators with unbounded, time-dependent drift coefficients and potentials acting on $L^p(\R^N)$ with Lebesgue measure.

math.AP↗