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Marcone C. Pereira

Publications and source records attributed to Marcone C. Pereira.

At least 19 recordsLinked to original sources
arXiv2026

Periodic Homogenization of Local/Nonlocal Systems

In this paper, we study the homogenization of elliptic equations that combine a local part, given by the Laplacian with Neumann boundary conditions, and its nonlocal version, defined through an integral operator with a smooth kernel. These two components are coupled through an additional nonlocal operator also given by a smooth kernel. We consider a sequence of partitions of a fixed spatial domain into two regions - local and nonlocal - which are periodically distributed in space (with one of the regions consisting of small, periodically arranged holes). Depending on the relative location of the local and nonlocal regions, we obtain qualitatively different limit behaviors. When the local part of the equation is confined to the small periodic holes, the sequence of solutions converges to the unique solution of a limit system in which the local component vanishes, while the nonlocal part persists and splits into two distinct components. On the other hand, when the local part of the problem lies outside the holes, the limit system exhibits a homogenized local diffusion operator coupled with a nonlocal equation. Finally, we analyze an intermediate regime in which only part of the local diffusion survives in the limit, by considering configurations consisting of parallel thin strips instead of holes.

math.AP↗
arXiv2026

Attractor Continuity for Semilinear Parabolic Equations on Thin Domains with Degenerating Outward Peaks

In this work, we analyze the asymptotic behavior of the attractors associated with a semilinear parabolic equation subject to homogeneous Neumann boundary conditions and defined on a thin domain $R^\varepsilon \subset \mathbb{R}^{1+n}$. We assume that the thin domain exhibits a cusp, known as an outward peak, whose geometry is characterized by a nonnegative function that vanishes at a point on the boundary. Our objective is to rigorously establish the continuity of the attractors as $\varepsilon \to 0$ and to determine their rate of convergence.

math.AP↗
arXiv2023

The Hadamard formula and the Rayleigh-Faber-Krahn inequality for nonlocal eigenvalue problems

In this paper we obtain a Hadamard type formula for simple eigenvalues and an analog to the Rayleigh-Faber-Krahn inequality for a class of nonlocal eigenvalue problems. Such class of equations include among others, the classical nonlocal problems with Dirichlet and Neumann conditions. The Hadamard formula is computed allowing domain perturbations given by embeddings of $n$-dimensional Riemannian manifolds (possibly with boundary) of finite volume while the Rayleigh-Faber-Krahn inequality is shown by rearrangement techniques.

math.AP↗
arXiv2020

Homogenization for nonlocal evolution problems with three different smooth kernels

In this paper we consider the homogenization of the evolution problem associated with a jump process that involves three different smooth kernels that govern the jumps to/from different parts of the domain. We assume that the spacial domain is divided into a sequence of two subdomains $A_n \cup B_n$ and we have three different smooth kernels, one that controls the jumps from $A_n$ to $A_n$, a second one that controls the jumps from $B_n$ to $B_n$ and the third one that governs the interactions between $A_n$ and $B_n$.Assuming that $χ_{A_n} (x) \to X(x)$ weakly in $L^\infty$ (and then $χ_{B_n} (x) \to 1-X(x)$ weakly in $L^\infty$) as $n \to \infty$ and that the initial condition is given by a density $u_0$ in $L^2$ we show that there is an homogenized limit system in which the three kernels and the limit function $X$ appear. When the initial condition is a delta at one point, $δ_{\bar{x}}$ (this corresponds to the process that starts at $\bar{x}$) we show that there is convergence along subsequences such that $\bar{x} \in A_{n_j}$ or $\bar{x} \in B_{n_j}$ for every $n_j$ large enough. We also provide a probabilistic interpretation of this evolution equation in terms of a stochastic process that describes the movement of a particle that jumps in $Ω$ according to the three different kernels and show that the underlying process converges in distribution to a limit process associated with the limit equation. We focus our analysis in Neumann type boundary conditions and briefly describe at the end how to deal with Dirichlet boundary conditions.

math.AP↗
arXiv2020

Homogenization for nonlocal problems with smooth kernels

In this paper we consider the homogenization problem for a nonlocal equation that involve different smooth kernels. We assume that the spacial domain is divided into a sequence of two subdomains $A_n \cup B_n$ and we have three different smooth kernels, one that controls the jumps from $A_n$ to $A_n$, a second one that controls the jumps from $B_n$ to $B_n$ and the third one that governs the interactions between $A_n$ and $B_n$. Assuming that $χ_{A_n} (x) \to X(x)$ weakly-* in $L^\infty$ (and then $χ_{B_n} (x) \to (1-X)(x)$ weakly-* in $L^\infty$) as $n \to \infty$ we show that there is an homogenized limit system in which the three kernels and the limit function $X$ appear. We deal with both Neumann and Dirichlet boundary conditions. Moreover, we also provide a probabilistic interpretation of our results.

math.AP↗
arXiv2020

Time-scale analysis non-local diffusion systems, applied to disease models

The objective of the present paper is to use the well known Ross-Macdonald models as a prototype, incorporating spatial movements, identifying different times scales and proving a singular perturbation result using a system of local and non-local diffusion. This results can be applied to the prototype model, where the vector has a fast dynamics, local in space, and the host has a slow dynamics, non-local in space.

math.AP↗
arXiv2020

Nonlocal and nonlinear evolution equations in perforated domains

In this work we analyze the behavior of the solutions to nonlocal evolution equations of the form $u_t(x,t) = \int J(x-y) u(y,t) \, dy - h_ε(x) u(x,t) + f(x,u(x,t))$ with $x$ in a perturbed domain $Ω^ε\subset Ω$ which is thought as a fixed set $Ω$ from where we remove a subset $A^ε$ called the holes. We choose an appropriated families of functions $h_ε\in L^\infty$ in order to deal with both Neumann and Dirichlet conditions in the holes setting a Dirichlet condition outside $Ω$. Moreover, we take $J$ as a non-singular kernel and $f$ as a nonlocal nonlinearity. % Under the assumption that the characteristic functions of $Ω^ε$ have a weak limit, we study the limit of the solutions providing a nonlocal homogenized equation.

math.AP↗
arXiv2019

Remarks on the spectrum of a nonlocal Dirichlet problem

In this paper we analyse the spectrum of nonlocal Dirichlet problems with non-singular kernels in bounded open sets. The novelty is the continuity of eigenvalues with respect to domain perturbation via Lebesgue measure. Also, under additional smooth condition on the kernel and domain, we prove differentiability of simple eigenvalues computing their first derivative.

math.AP↗
arXiv2018

Nonlocal problems in perforated domains

In this paper we analyze nonlocal equations in perforated domains. We consider nonlocal problems of the form $f(x) = \int_{B} J(x-y) (u(y) - u(x)) dy$ with $x$ in a perforated domain $Ω^ε\subset Ω$. Here $J$ is a non-singular kernel. We think about $Ω^ε$ as a fixed set $Ω$ from where we have removed a subset that we call the holes. We deal both with the Neumann and Dirichlet conditions in the holes and assume a Dirichlet condition outside $Ω$. In the later case we impose that $u$ vanishes in the holes but integrate in the whole $\mathbb{R}^N$ ($B=\mathbb{R}^N$) and in the former we just consider integrals in $\mathbb{R}^N$ minus the holes ($B=\mathbb{R}^N \setminus (Ω\setminus Ω^ε)$). Assuming weak convergence of the holes, specifically, under the assumption that the characteristic function of $Ω^ε$ has a weak limit, $χ_ε \rightharpoonup \mathcal{X}$ weakly$^*$ in $L^\infty(Ω)$, we analyze the limit as $ε\to 0$ of the solutions to the nonlocal problems proving that there is a nonlocal limit problem. In the case in which the holes are periodically removed balls we obtain that the critical radius is of order of the size of the typical cell (that gives the period). In addition, in this periodic case, we also study the behavior of these nonlocal problems when we rescale the kernel in order to approximate local PDE problems.

math.AP↗
arXiv2018

Semilinear elliptic equations in thin regions with terms concentrating on oscillatory boundaries

In this work we study the behavior of a family of solutions of a semilinear elliptic equation, with homogeneous Neumann boundary condition, posed in a two-dimensional oscillating thin region with reaction terms concentrated in a neighborhood of the oscillatory boundary. Our main result is concerned with the upper and lower semicontinuity of the set of solutions. We show that the solutions of our perturbed equation can be approximated with ones of a one-dimensional equation, which also captures the effects of all relevant physical processes that take place in the original problem.

math.AP↗
arXiv2017

Dynamical boundary conditions in a non-cylindrical domain for the Laplace equation

In this paper, we study existence, uniqueness and asymptotic behavior of the Laplace equation with dynamical boundary conditions on regular non-cylindrical domains. We write the problem as a non-autonomous Dirichlet-to-Neumann operator and use form methods in a more general framework to accomplish our goal. A class of non-autonomous elliptic problems with dynamical boundary conditions on Lipschitz domains is also considered in this same context.

math.AP↗
arXiv2016

Continuity of attractors for a family of $C^1$ perturbations of the square

We consider here the family of semilinear parabolic problems \begin{equation*} \begin{array}{rcl} \left\{ \begin{array}{rcl} u_t(x,t)&=&Δu(x,t) -au(x,t) + f(u(x,t)) ,\,\,\ x \in Ω_ε\,\,\,\mbox{and}\,\,\,\,\,\,t>0\,, \\ \displaystyle\frac{\partial u}{\partial N}(x,t)&=&g(u(x,t)), \,\, x \in \partialΩ_ε\,\,\,\mbox{and}\,\,\,\,\,\,t>0\,, \end{array} \right. \end{array} \end{equation*} where $ Ω $ is the unit square, $Ω_ε=h_ε(Ω)$ and $h_ε$ is a family of diffeomorphisms converging to the identity in the $C^1$-norm. We show that the problem is well posed for $ε>0$ sufficiently small in a suitable phase space, the associated semigroup has a global attractor $\mathcal{A}_ε$ and the family $\{\mathcal{A}_ε\}$ is continuous at $ε= 0$.

math.DS↗
arXiv2015

The Neumann problem in thin domains with very highly oscillatory boundaries

In this paper we analyze the behavior of solutions of the Neumann problem posed in a thin domain of the type $R^ε= \{(x_1,x_2) \in \R^2 \; | \; x_1 \in (0,1), \, - \, ε\, b(x_1) < x_2 < ε\, G(x_1, x_1/ε^α) \}$ with $α>1$ and $ε> 0$, defined by smooth functions $b(x)$ and $G(x,y)$, where the function $G$ is supposed to be $l(x)$-periodic in the second variable $y$. The condition $α> 1$ implies that the upper boundary of this thin domain presents a very high oscillatory behavior. Indeed, we have that the order of its oscillations is larger than the order of the amplitude and height of $R^ε$ given by the small parameter $ε$. We also consider more general and complicated geometries for thin domains which are not given as the graph of certain smooth functions, but rather more comb-like domains.

math.AP↗
arXiv2013

Parabolic problems in highly oscillating thin domains

In this work we consider the asymptotic behavior of the nonlinear semigroup defined by a semilinear parabolic problem with homogeneous Neumann boundary conditions posed in a bounded region of the plane that degenerates into a line segment when a positive parameter $ε$ goes to zero (a thin domain). Here we also allow that its boundary presents highly oscillatory behavior with different orders and variable profile. We take thin domains possessing the same order $ε$ to the thickness and amplitude of the oscillations but assuming different order to the period of oscillations on the top and the bottom of the boundary. We combine methods from linear homogenization theory and the theory on nonlinear dynamics of dissipative systems to obtain the limit problem establishing convergence properties for the solutions. At the end we show the upper semicontinuity of the attractors and stationary states.

math.AP↗
arXiv2013

Correctors for the Neumann problem in thin domains with locally periodic oscillatory structure

In this paper we are concerned with convergence of solutions of the Poisson equation with Neumann boundary conditions in a two-dimensional thin domain exhibiting highly oscillatory behavior in part of its boundary. We deal with the resonant case in which the height, amplitude and period of the oscillations are all of the same order which is given by a small parameter $ε> 0$. Applying an appropriate corrector approach we get strong convergence when we replace the original solutions by a kind of first-order expansion through the Multiple-Scale Method.

math.AP↗
arXiv2012

Continuity of attractors for a nonlinear parabolic problem with terms concentrating in the boundary

We analyze the dynamics of the flow generated by a nonlinear parabolic problem when some reaction and potential terms are concentrated in a neighborhood of the boundary. We assume that this neighborhood shrinks to the boundary as a parameter εgoes to zero. Also, we suppose that the "inner boundary" of this neighborhood presents a highly oscillatory behavior. Our main goal here is to show the continuity of the family of attractors with respect to ε. Indeed, we prove upper semicontinuity under the usual properties of regularity and dissipativeness and, assuming hyperbolicity of the equilibria, we also show the lower semicontinuity of the attractors at ε=0.

math.AP↗