Artículos (Ecuaciones Diferenciales y Análisis Numérico)
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Artículo Analysis of the Darcy-Brinkman flow with viscous dissipation and non-homogeneous thermal boundary condition(Vilnius Gediminas Technical University , 2026-01-21) Pažanin, Igor; Suárez Grau, Francisco Javier; Ecuaciones Diferenciales y Análisis NuméricoThis study investigates the steady-state Darcy-Brinkman flow within a thin, saturated porous domain, focusing on the effects of viscous dissipation and non-homogeneous boundary condition for the temperature. Employing asymptotic techniques with respect to the domain’s thickness, we rigorously derive the simplified coupled model describing the fluid flow. The mathematical analysis is based on deriving the sharp a priori estimates and proving the compactness results of the rescaled functions. The resulting limit model incorporates contributions of viscous dissipation and thermal boundary conditions and thus could prove useful in the engineering applications involving porous media.Artículo A Review on the Analysis and Optimal Control of Chemotaxis-Consumption Models(Springer, 2025-07-19) Corrêa Vianna Filho, André Luiz; Guillén González, Francisco; Ecuaciones Diferenciales y Análisis Numérico; FQM131: Ecuaciones diferenciales, Simulación Num. y Desarrollo SoftwareIn the present review we focus on the chemotaxis-consumption model and in , for any fixed , endowed with isolated boundary conditions and nonnegative initial conditions, where (u, v) model cell density and chemical signal concentration. Our objective is to present an overview of the related literature and latest results on the aforementioned model concerning the following three distinct research lines we have obtained in Corrêa Vianna Filho and Guillén-González (Nonlinear Anal Real World Appl 70, 103795, 2023), Guillén-González and Corrêa Vianna Filho (SIAM J Numer Anal 61(5), 2509–2533, 2023), Guillén-González and Corrêa Vianna Filho (SIAM J Control Optim 61(5), 3156–3182, 2023), Corrêa Vianna Filho and Guillén-González (Appl Math Optim 89(2), 48, 2024): the mathematical analysis, the numerical analysis and the related optimal control theory with a bilinear control acting on the chemical equation.Artículo Remarks on control and inverse problems for PDEs(Springer, 2025-07-11) Fernández Cara, Enrique; Ecuaciones Diferenciales y Análisis Numérico; FQM131: Ecuaciones diferenciales, Simulación Num. y Desarrollo SoftwareThis paper deals with recent results and open questions on the control and parameter identification of systems governed by PDEs. Among them, we find a few parabolic and hyperbolic equations, sometimes in the framework of a free-boundary problem. In the considered control problems, we try to govern the behavior of the solution(s) with a good (judicious) choice of the data. On the other hand, in the other (inverse) problems, the goal is to identify the value(s) of unknown data from particular observations of the solutions. We will recall a collection of optimal control, controllability and inverse problem assertions and will explain the arguments of proof. We will also present the results of some numerical experiments. Finally, we will state several open problems that can motivate future research on the subject.Artículo On a chemotaxis model with competitive terms arising in angiogenesis(AIMS, 2018-02-01) Delgado, Manuel; Gayte Delgado, Inmaculada; Morales Rodrigo, Cristian; Suárez Fernández, Antonio; Ecuaciones Diferenciales y Análisis NuméricoIn this paper we study an anti-angiogenic therapy model that deactivates the tumor angiogenic factors. The model consists of four parabolic equations and considers the chemotaxis and a logistic law for the endothelial cells and several boundary conditions, some of them are non homogeneous. We study the parabolic problem, proving the existence of a unique global positive solution for positive initial conditions, and the stationary problem, justifying the existence of one real number, an eigenvalue of a certain problem, which determines if the semi-trivial solutions are stable or unstable and the existence of a coexistence state.Artículo Some superlinear problems with nonlocal diffusion coefficient(Elsevier, 2019-09-20) Figueiredo Sousa, Tarcyana S.; Morales Rodrigo, Cristian; Suárez Fernández, Antonio; Ecuaciones Diferenciales y Análisis NuméricoWe study a superlinear elliptic problem with a non-local diffusion coefficient. We show that there exists a drastic change on the structure of the set of positive solutions when the non-local coefficient grows fast enough to infinity. We combine mainly sub-super and bifurcation methods to obtain our results.Artículo Non-local Degenerate Diffusion Coefficients Break Down the Components of Positive Solutions(De Gruyter, 2019-05-08) Delgado Delgado, Manuel; Morales Rodrigo, Cristian; Santos Júnior, J. R.; Suárez Fernández, Antonio; Ecuaciones Diferenciales y Análisis NuméricoThis paper deals with nonlinear elliptic problems where the diffusion coefficient is a degenerate non-local term. We show that this degeneration implies the growth of the complexity of the structure of the set of positive solutions of the equation. Specifically, when the reaction term is of logistic type, the continuum of positive solutions breaks into two disjoint pieces. Our approach uses mainly fixed point arguments.Artículo The Lotka-Volterra models with nonlocal cross-diffusivity terms(Elsevier, 2025-01-31) Costa, Marcos Antonio Viana; Morales Rodrigo, Cristian; Suárez Fernández, Antonio; Ecuaciones Diferenciales y Análisis NuméricoWe consider the Lotka-Volterra systems in their three classic forms: competition, prey-predator, and cooperation. These systems include nonlocal cross-diffusivity terms, meaning that the diffusion velocity rate of one species depends on the total population of the other species. The inclusion of these nonlocal diffusivity terms causes a significant change in the structure of coexistence states compared to the classical Lotka-Volterra systems. To obtain these results, we employ mainly the fixed point index in cones.Artículo Lotka-Voterra competition model with nonlocal coefficient diffusion(Elsevier, 2025-05-09) Costa, Marcos Antonio Viana; Morales Rodrigo, Cristian; Suárez Fernández, Antonio; Ecuaciones Diferenciales y Análisis NuméricoWe consider the classical Lotka-Volterra competition system with non-local diffusion, specifically, the diffusion coefficients depend on the total population in a nonlinear way. This kind of diffusion models that the species tends to leave crowded areas or is attracted to regions with higher population density, depending on whether the nonlinear function increases or decreases, respectively. The inclusion of these non-local terms in the diffusion coefficients entails significant technical difficulties. We show results of the existence and non-existence of coexistence states of the models depending on the coefficients of the model.Artículo Machine learning techniques to identify synchronization patterns in multiple timescale dynamical systems networks(Elsevier, 2025-12-17) Bandera, A.; Fernández García, Soledad; Gómez Mármol, Macarena; Vida, A.; Ecuaciones Diferenciales y Análisis Numérico; FQM120: Modelado Matemático y Simulación de Sistemas MedioambientalesWe present a novel methodology that combines machine learning techniques with dynamical analysis to classify and interpret the behavior distribution of network models of coupled dynamical systems. Our methodology determines the optimal number of distinct behaviors and classifies them based on time-series features, allowing for an interpretable and automated partition of the parameter space. Applying this approach to a homogeneous two-clusters model of intracellular calcium concentration dynamics, we identify nine different long-term behaviors, including complex and chaotic regimes, mapping experimental data available in the literature. The results highlight the complementarity between data-driven classification and classical dynamical analysis in capturing rich synchronization patterns and detecting subtle transitions in multiple timescale biological systems.Artículo Nonautonomous perturbations of morse–smale semigroups: stability of the phase diagram(Springer, 2021-08-05) Bortolan, M. C.; Carvalho, A. N.; Langa Rosado, José Antonio; Raugel, G.; Ecuaciones Diferenciales y Análisis Numérico; FQM314: Análisis Estocástico de Sistemas DiferencialesIn this work we study Morse–Smale semigroups under nonautonomous perturbations, which leads us to introduce the concept of Morse–Smale evolution processes of hyperbolic type, associated to nonautonomous evolutionary equations. They are amongst the dynamically gradient evolution processes (in the sense of Carvalho et al., in: Applied Mathematical Sciences, vol 182, Springer, New York, 2013) with a finite number of hyperbolic global solutions, for which the stable and unstable manifolds intersect transversally. We prove the stability of the phase diagram of the attractors for a small continuously differentiable nonautonomous perturbation of a Morse–Smale semigroup with a finite number of hyperbolic equilibria. We present the complete proofs of the local and global -lemmas in the infinite-dimensional case. Such results are due to D. Henry, presented in his handwritten notes Henry (in: Manuscript, IME-USP), and are included here for completeness.Artículo Forward Attraction of Nonautonomous Dynamical Systems and Applications to Navier-Stokes Equations(SIAM, 2024) Cui, Hongyong; Figueroa López, Rodiak N.; Langa Rosado, José Antonio; Nascimento, Marcelo José Dias; Ecuaciones Diferenciales y Análisis Numérico; FQM314: Análisis Estocástico de Sistemas DiferencialesIn this paper we studied the forward dynamics of nonautonomous dynamical systems in terms of forward attractors. We first reviewed the well-known uniform attractor theory, and then by weakening the uniformity of attraction we introduced semiuniform forward attractors and minimal (nonuniform) forward attractors. With these semiuniform attractors, a characterization of the structure of uniform attractors was given: a uniform attractor is composed of two semiuniform attractors and bounded complete trajectories connecting them. As a consequence, the nature of the forward attraction of a dissipative nonautonomous dynamical system was then revealed: the vector field in the distant future of the system determines the (nonuniform) forward asymptotic behavior. A criterion for certain semiuniform attractors to have finite fractal dimension was given and the finite dimensionality of uniform attractors was discussed. Forward attracting time-dependent sets were studied also. A sufficient condition and a necessary condition for a time-dependent set to be forward attracting were given with illustrative counterexamples. Forward attractors of a Navier–Stokes equation with asymptotically vanishing viscosity (with an Euler equation as the limit equation) and with time-dependent forcing were studied as applications.Artículo Smoothing and finite-dimensionality of uniform attractors in Banach spaces(Elsevier, 2021-06-05) Cui, Hongyong; Carvalho, Alexandre N.; Cunha, Arthur C.; Langa Rosado, José Antonio; Ecuaciones Diferenciales y Análisis Numérico; FQM314: Análisis Estocástico de Sistemas DiferencialesThe aim of this paper is to find an upper bound for the fractal dimension of uniform attractors in Banach spaces. The main technique we employ is essentially based on a compact embedding of some auxiliary Banach space into the phase space and a corresponding smoothing effect between these spaces. Our bounds on the fractal dimension of uniform attractors are given in terms of the dimension of the symbol space and the Kolmogorov entropy number of the embedding. In addition, a dynamical analysis on the symbol space is also given, showing that the finite-dimensionality of the hull of a time-dependent function is fully determined by the tails of the function, which allows us to consider more general non-autonomous terms than quasi-periodic functions. As applications, we show that the uniform attractor of the 2D Navier-Stokes equation is finite-dimensional in H and in V, and that of a reaction-diffusion equation is finite-dimensional in L2 and in Lp, with p > 2.Artículo A three population Lotka-Volterra competition model with two populations interacting through an interface(2026-02-01) Álvarez Caudevilla, Pablo; Brändle Cerqueira, Cristina; Molina Becerra, Mónica; Suárez Fernández, Antonio; Matemática Aplicada II; Ecuaciones Diferenciales y Análisis Numérico; Ministerio de Ciencia, Innovación y Universidades (MICIU). España; TIC130: Investigación en Sistemas Dinámicos en Ingeniería; FQM131: Ecuaciones diferenciales, Simulación Num. y Desarrollo SoftwareIn this work we consider three species competing with each other in the same habitat. One of the species lives in the entire habitat, competing with the other two species, while the other two inhabit two disjoint regions of the habitat. These two populations just interact on a region/interface which acts as a geographical barrier. This barrier condition causes a drastic change in species behaviour compared to the classical Lotka-Volterra competitive model, showing very rich and new different situations depending on the several parameters involved in the system.Artículo Remarks on exponential attractors for a non-autonomous PDE with H^(-1)-valued forces(AIMS, 2025-01) Aguilar Reyes, Álvaro; Marín Rubio, Pedro; Ecuaciones Diferenciales y Análisis Numérico; FQM314: Análisis Estocástico de Sistemas DiferencialesA reaction-diffusion problem with time-dependent force with values in is considered. Some results ensuring the existence of pullback exponential attractors are established, combining some classical and more recent developments on the topic. Two different sets of assumptions are considered. One involves translation bounded forces. The second one allows time-exponential growth in the past.Artículo Weak global attractor for the 3D-Navier–Stokes equations via the globally modified Navier–Stokes equations(Springer, 2024-11-26) Bortolan, Matheus Cheque; Carvalho, Alexandre Nolasco; Marín Rubio, Pedro; Valero, José; Ecuaciones Diferenciales y Análisis Numérico; FQM314: Análisis Estocástico de Sistemas DiferencialesIn this paper, we obtain the existence of a weak global attractor for the three-dimensional Navier–Stokes equations, that is, a weakly compact set with an invariance property, that uniformly attracts solutions, with respect to the weak topology, for initial data in bounded sets. To that end, we define this weak global attractor in terms of limits of solutions of the globally modified Navier–Stokes equations in the weak topology. We use the theory of semilinear parabolic equations and -regularity to obtain the local well-posedness for the globally modified Navier–Stokes equations, the existence of the global attractor and its regularity.Artículo Remarks on non-homogeneous Cauchy problems with time-dependent generators(Springer, 2025-03-24) Marín Rubio, Pedro; Seminario Huertas, Paulo Nicanor; Ecuaciones Diferenciales y Análisis Numérico; FQM314: Análisis Estocástico de Sistemas DiferencialesIn this paper it is studied the well-posedness in several senses for non-homogeneous Cauchy problems where the infinitesimal generator depends on the time parameter. More specifically, we analyze the existence of classical, mild and weak solutions and their relationships. Thanks to uniqueness arguments, mild solutions are proved to satisfy a classical variational formulation. Finally, these results are applied to a thermoelastic plate model where the thermal part is of Cattaneo type and all the physical coefficients depend on time.Artículo Mathematical Modeling of Neuroblast Migration Toward the Olfactory Bulb(Elsevier, 2025) Acosta Soba, Daniel; Castro, Carmen; Geribaldi Doldán, Noelia; Guillén González, Francisco Manuel; Núñez Abades, Pedro Antonio; Ortega Román, Noelia; Pérez García, Patricia; Rodríguez Galván, J. Rafael; Fisiología; Ecuaciones Diferenciales y Análisis Numérico; Ministerio de Ciencia, Innovación y Universidades (MICIU). España; European Commission (EC). Fondo Europeo de Desarrollo Regional (FEDER); Universidad de Cádiz; University of Tennessee; Junta de AndalucíaThis article is devoted to the mathematical modeling of the migration of neuroblasts, precursor cells of neurons, along the Rostral Migratory Stream (RMS), the pathway they usually follow before maturing. According to our model, this way is determined mainly by attraction forces to the olfactory bulb, and also by the heterogeneous mobility of neuroblasts in different regions of the brain. Carefully identifying them as solutions to partial differential equations allows us to determine the movement of neuroblasts along the RMS in a realistic fashion. For solving the equations we develop numerical schemes where the application of novel discontinuous Galerkin methods allows to maintain the properties of the continuous model such as the maximum principle. We present some successful computer tests including parameter adjustment to fit real data from rodent brains.Artículo On the null controllability of a one-dimensional fluid–solid interaction model(Académie des sciences, 2003-11-05) Doubova Krasotchenko, Anna; Fernández Cara, Enrique; Ecuaciones Diferenciales y Análisis Numérico; FQM131: Ecuaciones diferenciales, Simulación Num. y Desarrollo SoftwareWe analyze the null controllability of a one-dimensional nonlinear system which models the interaction of a fluid and a particle. The fluid is governed by the Burgers equation and the control is exerted on the boundary points. We present two main results: the global null controllability of a linearized system and the local null controllability of the nonlinear original model. The proofs rely on appropriate global Carleman inequalities and fixed point arguments.Artículo A non-overlapping domain decomposition method for the Stokes equations via a penalti term on the interface(Académie des sciences, 2002-02-01) Chacón Rebollo, Tomás; Chacón Vera, Eliseo; Ecuaciones Diferenciales y Análisis NuméricoThe purpose of this Note is to perform a theoretical analysis of the domain decomposition method introduced in [2]. We motivate and introduce an improvement of this method and carry out the analysis when it is applied to solving the Stokes equations. Our method is based on a penalty term on the interface between subdomains that enforces the appropriate transmission conditions and may be seen as variation of the Robin method. We obtain strong convergence results for velocity and pressure in the standard H1 and L2 norms and precise rates of convergence, together with error estimates. These error estimates are of optimal order with respect to the precision of the interpolation. We conclude with some numerical tests.Artículo Derivation of the k-epsilon model for locally homogeneous turbulence by homogenization techniques(Académie des sciences, 2003-09-15) Chacón Rebollo, Tomás; Franco Coronil, Daniel; Ecuaciones Diferenciales y Análisis Numérico; FQM120: Modelado Matemático y Simulación de Sistemas MedioambientalesWe derive the incompressible and compressible k–ε model for locally homogeneous turbulence. The model is rigorously derived on formal mathematical grounds using the MPP modelling technique. This lets us calculate by either analytical or numerical means the closure constants of the model.