arXiv subjects

José M. Arrieta

Publications and source records attributed to José M. Arrieta.

At least 19 recordsLinked to original sources

arXiv2026

A brush problem. Homogenization involving thin domains and PDEs in graphs

This work analyses the homogenization of a linear elliptic equation with Neumann boundary conditions in a comb/brush domain, composed of a fixed base and a family of thin teeth. The teeth are defined as rescalings of order less than or equal to $\varepsilon$ of a model tooth of arbitrary shape. Periodicity in their distribution is not assumed; instead, the existence of an asymptotic limit density $θ$, which may vanish in certain regions, is assumed. The convergence analysis is performed using an adaptation of the unfolding operator method to a non-periodic framework. Finally, it is shown that, under certain conditions on the geometry of the teeth, the resulting limit problem can be interpreted as a differential equation on a graph.

math.AP↗

arXiv2025

Quenching phenomena in a system of non-local diffusion equations

In this paper we study the quenching phenomena occurring in a non-local diffusion system of two equations with intertwined singular absorption terms of the type $u^{-p}$. We prove that there exists a range of multiplicative parameters for which every solution presents quenching, while outside this range there are both global and quenching solutions. We also characterize in terms of the exponents of the absorption terms when the quenching is simultaneous or non-simultaneous and obtain the quenching rates.

math.AP↗

arXiv2025

Shadowing for Infinite Dimensional Dynamical Systems

In this paper we extend to an infinite dimensional setting some results on the shadowing property that are known on finite dimensional compact manifolds without border and in $\mathbb{R}^n$. In fact, we show that if $\{\T(t):t\ge 0\}$ is a Morse-Smale semigroup defined in a Hilbert space with a global attractor $\mathcal{A}$ and non-wandering set given only by its equilibria, then $\T(1)|_{\mathcal{A}}:\mathcal{A}\to \mathcal{A} $ admits the Lipschitz Shadowing property. Moreover, for any positively invariant bounded neighborhood $\U\supset\mathcal{A}$ of the global attractor, the map $\T(1)|_{\U}:\U\to \U$ has the Hölder-Shadowing property. We obtain results related to the structural stability of Morse-Smale semigroups, that were only known on finite dimension and continuity of global attractors.

math.DS↗

arXiv2025

On bifurcation from infinity: a compactification approach

We consider a scalar parabolic partial differential equation on the interval with nonlinear boundary conditions that are asymptotically sublinear. As the parameter crosses critical values (e.g. the Steklov eigenvalues), it is known that there are large equilibria that arise through a bifurcation from infinity (i.e., such equilibria converge, after rescaling, to the Steklov eigenfunctions). We provide a compactification approach to the study of such unbounded bifurcation curves of equilibria, their stability, and heteroclinic orbits. In particular, we construct an induced semiflow at infinity such that the Steklov eigenfunctions are equilibria. Moreover, we prove the existence of infinite-time blow-up solutions that converge, after rescaling, to certain eigenfunctions that are equilibria of the induced semiflow at infinity.

math.AP↗

arXiv2024

Continuity of attractors of parabolic equations with nonlinear boundary conditions and rapidly varying boundaries. The case of a Lipschitz deformation

In this paper we obtain the continuity of attractors for nonlinear parabolic equations with nonlinear boundary conditions when the boundary of the domain varies very rapidly as a parameter $ε$ goes to zero. We consider the case where the boundary of the domain presents a highly oscillatory behavior as the parameter $ε$ goes to zero. For the case where we have a Lipschitz deformation of the boundary with the Lipschitz constant uniformly bounded in $ε$ but the boundaries do not approach in a Lipschitz sense, the solutions of these equations converge in certain sense to the solution of a limit parabolic equation of the same type but where the boundary condition has a factor that captures the oscillations of the boundary. To address this problem, it is necessary to consider the notion of convergence of functions defined in varying domains and the convergence of a family of operators defined in different Banach spaces. Moreover, since we consider problems with nonlinear boundary conditions, it is necessary to extend these concepts to the case of spaces with negative exponents and to operators defined between these spaces.

math.AP↗

arXiv2024

Homogenization in 3D thin domains with oscillating boundaries of different orders

This paper presents an extension of the unfolding operator technique, initially applied to two-dimensional domains, to the realm of three-dimensional thin domains. The advancement of this methodology is pivotal, as it enhances our understanding and analysis of three-dimensional geometries, which are crucial in various practical fields such as engineering and physics. Our work delves into the asymptotic behavior of solutions to a reaction-diffusion equation with Neumann boundary conditions set within such a oscillatory 3-dimensional thin domain. The method introduced enables the deduction of effective problems across all scenarios, tackling the intrinsic complexity of these domains. This complexity is especially pronounced due to the possibility of diverse types of oscillations occurring along their boundaries.

math.AP↗

arXiv2024

Elliptic and parabolic problems in thin domains with doubly weak oscillatory boundary

In this work we consider higher dimensional thin domains with the property that both boundaries, bottom and top, present oscillations of weak type. We consider the Laplace operator with Neumann boundary conditions and analyze the behavior of the solutions as the thin domains shrinks to a fixed domain $ω\subset \R^n$. We obtain the convergence of the resolvent of the elliptic operators in the sense of compact convergence of operators, which in particular implies the convergence of the spectra. This convergence of the resolvent operators will allow us to conclude the global dynamics, in terms of the global attractors of a reaction diffusion equation in the thin domains. In particular, we show the upper semicontinuity of the attractors and stationary states. An important case treated is the case of a quasiperiodic situation, where the bottom and top oscillations are periodic but with period rationally independent.

math.AP↗

arXiv2024

Boundedness of solutions of nonautonomous degenerate logistic equations

In this work we analyze the boundedness properties of the solutions of a nonautonomous parabolic degenerate logistic equation in a bounded domain. The equation is degenerate in the sense that the logistic nonlinearity vanishes in a moving region, $K(t)$, inside the domain. The boundedness character of the solutions depends not only on, roughly speaking, the first eigenvalue of the Laplace operator in $K(t)$ but also on the way this moving set evolves inside the domain and in particular on the speed at which it moves.

math.AP↗

arXiv2023

Bifurcation and hyperbolicity for a nonlocal quasilinear parabolic problem

In this article, we study a one-dimensional nonlocal quasilinear problem of the form $u_t=a(\Vert u_x\Vert^2)u_{xx}+νf(u)$, with Dirichlet boundary conditions on the interval $[0,π]$, where $0<m\leq a(s)\leq M$ for all $s\in \mathbb{R}^+$ and $f$ satisfies suitable conditions. We give a complete characterization of the bifurcations and of the hyperbolicity of the corresponding equilibria. With respect to the bifurcations we extend the existing result when the function $a(\cdot)$ is non-decreasing to the case of general smooth nonlocal diffusion functions showing that bifurcations may be pitchfork or saddle-node, subcritical or supercritical. We also give a complete characterization of hyperbolicity specifying necessary and sufficient conditions for its presence or absence. We also explore some examples to exhibit the variety of possibilities that may occur, depending of the function $a$, as the parameter $ν$ varies.

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

Spectral analysis of the biharmonic operator subject to Neumann boundary conditions on dumbbell domains

We consider the biharmonic operator subject to homogeneous boundary conditions of Neumann type on a planar dumbbell domain which consists of two disjoint domains connected by a thin channel. We analyse the spectral behaviour of the operator, characterizing the limit of the eigenvalues and of the eigenprojections as the thickness of the channel goes to zero. In applications to linear elasticity, the fourth order operator under consideration is related to the deformation of a free elastic plate, a part of which shrinks to a segment. In contrast to what happens with the classical second order case, it turns out that the limiting equation is here distorted by a strange factor depending on a parameter which plays the role of the Poisson coefficient of the represented plate.

math.SP↗

arXiv2017

Distance of attractors for thin domains

In this work we consider a dissipative reaction-diffusion equation in a $d$-dimensional thin domain shrinking to a one dimensional segment and obtain good rates for the convergence of the attractors. To accomplish this, we use estimates on the convergence of inertial manifolds as developed previously in \cite{Arrieta-Santamaria-C0} and Shadowing theory.

math.AP↗

arXiv2017

$C^{1,θ}$-Estimates on the distance of Inertial Manifolds

In this paper we obtain $C^{1,θ}$-estimates on the distance of inertial manifolds for dynamical systems generated by evolutionary parabolic type equations. We consider the situation where the systems are defined in different phase spaces and we estimate the distance in terms of the distance of the resolvent operators of the corresponding elliptic operators and the distance of the nonlinearities of the equations.

math.AP↗

arXiv2015

Unfolding operator method for thin domains with a locally periodic highly oscillatory boundary

We analyze the behavior of solutions of the Poisson equation with homogeneous Neumann boundary conditions in a two-dimensional thin domain which presents locally periodic oscillations at the boundary. The oscillations are such that both the amplitude and period of the oscillations may vary in space. We obtain the homogenized limit problem and a corrector result by extending the unfolding operator method to the case of locally periodic media. We emphasize the fact that the techniques developed in this paper can be adapted to other locally periodic cases like reticulated or perforated domains where the period may be space-dependent.

math.AP↗

arXiv2015

Higher order elliptic operators on variable domains. Stability results and boundary oscillations for intermediate problems

We study the spectral behavior of higher order elliptic operators upon domain perturbation. We prove general spectral stability results for Dirichlet, Neumann and intermediate boundary conditions. Moreover, we consider the case of the bi-harmonic operator with those intermediate boundary conditions which appears in study of hinged plates. In this case, we analyze the spectral behavior when the boundary of the domain is subject to a periodic oscillatory perturbation. We will show that there is a critical oscillatory behavior and the limit problem depends on whether we are above, below or just sitting on this critical value. In particular, in the critical case we identify the strange term appearing in the limiting boundary conditions by using the unfolding method from homogenization theory.

math.AP↗

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

Estimates on the distance of inertial manifolds

In this paper we obtain estimates on the distance of inertial manifolds for dynamical systems generated by evolutionary parabolic type equations. We consider the situation where the systems are defined in different phase spaces and we estimate the distance in terms of the distance of the resolvent operators of the corresponding elliptic operators and the distance of the nonlinearities of the equations.

math.AP↗

arXiv2013

Thin domains with doubly oscillatory boundary

We consider a 2-dimensional thin domain with order of thickness ε which presents oscillations of amplitude also ε on both boundaries, top and bottom, but the period of the oscillations are of different order at the top and at the bottom. We study the behavior of the Laplace operator with Neumann boundary condition and obtain its asymptotic homogenized limit as the parameter ε goes to 0. We are interested in understanding how this different oscillatory behavior at the boundary, influences the limit problem.

math.AP↗