arXiv subjects

Daniele Andreucci

Publications and source records attributed to Daniele Andreucci.

16 recordsLinked to original sources

arXiv2025

A supersolution approach to doubly degenerate parabolic equations with weights

We consider the Cauchy problem in the Euclidean space for a doubly degenerate parabolic equation with a space-dependent exponential weight, where the exponent satisfies the doubling condition. In particular, both the so called logconvex and logconcave cases may be considered. Under the additional natural assumptions we construct supersolutions and subsolutions allowing us to control the precise sharp temporal decay bounds. We apply our results also to an equation with inhomogeneous density, via a suitable variable transformation.

math.AP↗

arXiv2024

The Cauchy problem for doubly degenerate parabolic equations with weights

We consider the Cauchy problem in the Euclidean space for a doubly degenerate parabolic equation with a space-dependent exponential weight, roughly speaking of the type of the exponential of a power of the distance from the origin. We assume here the solutions of the Cauchy problem to be globally integrable in space (in the appropriate weighted sense) and non-negative. Under suitable assumptions, we prove for the solutions sup estimates, i.e., the decay rate at infinity, the property of finite speed of propagation and support estimates. All our estimates are given explicitly in terms of the weight appearing in the equation.

math.AP↗

arXiv2021

Some remarks on the Sobolev inequality in Riemannian manifolds

We investigate Sobolev and Hardy inequalities, specifically weighted Minerbe's type estimates, in noncompact complete connected Riemannian manifolds whose geometry is described by an isoperimetric profile. In particular, we assume that the manifold satisfies the $p$-hyperbolicity property, stated in terms of a necessary integral Dini condition on the isoperimetric profile. Our method seems to us to combine sharply the knowledge of the isoperimetric profile and the optimal Bliss type Hardy inequality depending on the geometry of the manifold. We recover the well known best Sobolev constant in the Euclidean case.

math.FA↗

arXiv2021

Homogenization of a modified bidomain model involving imperfect transmission

We study, by means of the periodic unfolding technique, the homogenization of a modified bidomain model, which describes the propagation of the action potential in the cardiac electrophysiology. Such a model, allowing the presence of pathological zones in the heart, involves various geometries and non-standard transmission conditions on the interface between the healthy and the damaged part of the cardiac muscle.

math.AP↗

arXiv2021

Well-posedness for a modified bidomain model describing bioelectric activity in damaged heart tissues

We prove the existence and the uniqueness of a solution for a modified bidomain model, describing the electrical behaviour of the cardiac tissue in pathological situations. The leading idea is to reduce the problem to an abstract parabolic setting, which requires to introduce several auxiliary differential systems and a non-standard bilinear form. The main difficulties are due to the degeneracy of the bidomain system and to its non-standard coupling with a diffusion equation, accounting for the presence of the pathological zone in the heart tissue.

math.AP↗

arXiv2020

Asymptotic properties of solutions to the Cauchy problem for degenerate parabolic equations with inhomogeneous density on manifolds

We consider the Cauchy problem for doubly nonlinear degenerate parabolic equations with inhomogeneous density on noncompact Riemannian manifolds. We give a qualitative classification of the behavior of the solutions of the problem depending on the behavior of the density function at infinity and the geometry of the manifold, which is described in terms of its isoperimetric function. We establish for the solutions properties as: stabilization of the solution to zero for large times, finite speed of propagation, universal bounds of the solution, blow up of the interface. Each one of these behaviors of course takes place in a suitable range of parameters, whose definition involves a universal geometrical characteristic function, depending both on the geometry of the manifold and on the asymptotics of the density at infinity.

math.AP↗

arXiv2020

Diffusion in inhomogeneous media with periodic microstructures

Diffusion in inhomogeneous materials can be described by both the Fick and Fokker--Planck diffusion equations. Here, we study a mixed Fick and Fokker-Planck diffusion problem with coefficients rapidly oscillating both in space and time. We obtain macroscopic models performing the homogenization limit by means of the unfolding technique.

math.AP↗

arXiv2020

Well-posedness of two pseudo-parabolic problems for electrical conduction in heterogeneous media

We prove a well-posedness result for two pseudo-parabolic problems, which can be seen as two models for the same electrical conduction phenomenon in heterogeneous media, neglecting the magnetic field. One of the problems is the concentration limit of the other one, when the thickness of the dielectric inclusions goes to zero. The concentrated problem involves a transmission condition through interfaces, which is mediated by a suitable Laplace-Beltrami type equation.

math.AP↗

arXiv2020

Concentration and homogenization in electrical conduction in heterogeneous media involving the Laplace-Beltrami operator

We study a concentration and homogenization problem modelling electrical conduction in a composite material. The novelty of the problem is due to the specific scaling of the physical quantities characterizing the dielectric component of the composite. This leads to the appearance of a peculiar displacement current governed by a Laplace-Beltrami pseudo-parabolic equation. This pseudo-parabolic character is present also in the homogenized equation, which is obtained by the unfolding technique.

math.AP↗

arXiv2019

Long time behavior of solutions of degenerate parabolic equations with inhomogeneous density on manifolds

We consider the Cauchy problem for doubly non-linear degenerate parabolic equations on Riemannian manifolds of infinite volume, or in $\R^N$. The equation contains a weight function as a capacitary coefficient which we assume to decay at infinity. We connect the behavior of non-negative solutions to the interplay between such coefficient and the geometry of the manifold, obtaining, in a suitable subcritical range, estimates of the vanishing rate for long times and of the finite speed of propagation. In supercritical ranges we obtain universal bounds and prove blow up in a finite time of the (initially bounded) support of solutions.

math.AP↗

arXiv2019

Alternating and variable controls for the wave equation

The present article discusses the exact observability of the wave equation when the observation subset of the boundary is variable in time. In the one-dimensional case, we prove an equivalent condition for the exact observability, which takes into account only the location in time of the observation. To this end we use Fourier series. Then we investigate the two specific cases of single exchange of the control position, and of exchange at a constant rate. In the multi-dimensional case, we analyse sufficient conditions for the exact observability relying on the multiplier method. In the last section, the multi-dimensional results are applied to specific settings and some connections between the one and multi-dimensional case are discussed; furthermore some open problems are presented.

math.OC↗

arXiv2018

Fick and Fokker--Planck diffusion law in inhomogeneous media

We discuss diffusion of particles in a spatially inhomogeneous medium. From the microscopic viewpoint we consider independent particles randomly evolving on a lattice. We show that the reversibility condition has a discrete geometric interpretation in terms of weights associated to un--oriented edges and vertices. We consider the hydrodynamic diffusive scaling that gives, as a macroscopic evolution equation, the Fokker--Planck equation corresponding to the evolution of the probability distribution of a reversible spatially inhomogeneous diffusion process. The geometric macroscopic counterpart of reversibility is encoded into a tensor metrics and a positive function. The Fick's law with inhomogeneous diffusion matrix is obtained in the case when the spatial inhomogeneity is associated exclusively with the edge weights. We discuss also some related properties of the systems like a non-homogeneous Einstein relation and the possibility of uphill diffusion.

cond-mat.stat-mech↗

arXiv2018

Asymptotic estimates for the p-Laplacian on infinite graphs with decaying initial data

We consider the Cauchy problem for the evolutive discrete p-Laplacian in infinite graphs, with initial data decaying at infinity. We prove optimal sup and gradient bounds for nonnegative solutions, when the initial data has finite mass, and also sharp evaluation for the confinement of mass, i.e., the effective speed of propagation. We provide estimates for some moments of the solution, defined using the distance from a given vertex. Our technique relies on suitable inequalities of Faber-Krahn type, and looks at the local theory of continuous nonlinear partial differential equations. As it is known, however, not all of this approach can have a direct counterpart in graphs. A basic tool here is a result connecting the supremum of the solution at a given positive time with the measure of its level sets at previous times. We also consider the case of slowly decaying initial data, where the total mass is infinite.

math.AP↗

arXiv2018

Variable Support Control for the Wave Equation: A Multiplier Approach

We study the controllability of the multidimensional wave equation in a bounded domain with Dirichlet boundary condition, in which the support of the control is allowed to change over time. The exact controllability is reduced to the proof of the observability inequality, which is proven by a multiplier method. Besides our main results, we present some applications.

math.OC↗

arXiv2015

Asymptotic decay under nonlinear and noncoercive dissipative effects for electrical conduction in biological tissues

We consider a nonlinear model for electrical conduction in biological tissues. The nonlinearity appears in the interface condition prescribed on the cell membrane. The purpose of this paper is proving asymptotic convergence for large times to a periodic solution when time-periodic boundary data are assigned. The novelty here is that we allow the nonlinearity to be noncoercive. We consider both the homogenized and the non-homogenized version of the problem.

math.AP↗

arXiv2013

Flux through a time-periodic gate: Monte Carlo test of a homogenization result

We investigate via Monte Carlo numerical simulations and theoretical considerations the outflux of random walkers moving in an interval bounded by an interface exhibiting channels (pores, doors) which undergo an open/close cycle according to a periodic schedule. We examine the onset of a limiting boundary behavior characterized by a constant ratio between the outflux and the local density, in the thermodynamic limit. We compare such a limit with the predictions of a theoretical model already obtained in the literature as the homogenization limit of a suitable diffusion problem.

math-ph↗