arXiv subjects

Tomás Caraballo

Publications and source records attributed to Tomás Caraballo.

At least 19 recordsLinked to original sources

arXiv2026

A Stochastic Model of Intracellular Calcium Concentrations

In this paper, we study a slow-fast model of Intracellular Calcium Concentrations (ICC) within a stochastic framework, where stochastic perturbations are introduced to capture the intrinsic randomness of such biological systems. We first analyse the three-dimensional single-cell model, based on the FitzHugh-Nagumo structure, and subsequently extend the analysis to a coupled two-cell six-dimensional setting. We prove existence and uniqueness of solutions and establish positivity of the calcium variable. We obtain explicit mean-square stability estimates around deterministic equilibria and quantify how they depend on the noise intensity. Numerical simulations for both the single-cell and coupled systems illustrate relevant qualitative behaviours and reveal additional dynamical features arising in the stochastic framework.

math.DS↗

arXiv2026

SIR model with random diffusion, reinfection, and random transmission: exponential attractors and spread of the disease

We introduce a stochastic SIR-type partial differential equation model incorporating random diffusion, reinfection, vital dynamics, and a randomly varying transmission rate. For the associated random dynamical system, we prove the existence of both random and exponential attractors. We construct a non-stationary, random disease-free global solution, which serves to localize the random attractor. Furthermore, we analyze the mean value of the random transmission coefficient to establish conditions under which the disease may either be eradicated or persist in an endemic state, depending on the system's parameters.

math.DS↗

arXiv2026

The Stochastic TR-BDF2 Scheme of Order 2

Our main objective in this paper is to develop a second-order stochastic numerical method which generalizes the well-known deterministic TR-BDF2 scheme. Since most stochastic techniques used for approximating the solution of a stochastic differential equation may have lower order compared to the deterministic case, we have elaborated a scheme which not only preserves the second-order accuracy of the original scheme in the stochastic framework, but also its $A$-stability. Once we obtain the scheme and prove its second-order accuracy and $A$-stability, which is not a trivial task, we also state a result concerning its $MS$-stability. This concept is also analyzed for different parameter ranges in our scheme and the It{ô}--Taylor approximation of order 2, revealing scenarios where, for certain time step sizes, the developed method is $MS$-stable while the It{ô}--Taylor one is not. This concept is really useful to tackle slow-fast problems such as stiff ones, which we aim to explore further in future work. Finally, we validate the theoretical results with some academic test cases.

math.NA↗

arXiv2025

On a retarded stochastic system with discrete diffusion modeling life tables

This work proposes a method for modeling and forecasting mortality rates. It constitutes an improvement over previous studies by incorporating both the historical evolution of the mortality phenomenon and its random behavior. In the first part, we introduce the model and analyze mathematical properties such as the existence of solutions and their asymptotic behavior. In the second part, we apply this model to forecast mortality rates in Spain, showing that it yields better results than classical methods.

math.ST↗

arXiv2025

Dynamics and large deviations for fractional stochastic partial differential equations with Lévy noise

This paper is mainly concerned with a kind of fractional stochastic evolution equations driven by Lévy noise in a bounded domain. We first state the well-posedness of the problem via iterative approximations and energy estimates. Then, the existence and uniqueness of weak pullback mean random attractors for the equations {are} established by defining a mean random dynamical system. Next, we prove the existence of invariant measures when the problem is autonomous by means of the fact that $H^γ(\mathcal{O})$ is compactly embedded in $L^2(\mathcal{O})$ with $γ\in (0,1)$. Moreover, the uniqueness of this invariant measure is presented which ensures the ergodicity of the problem. Finally, a large deviation principle result for solutions of SPDEs perturbed by small Lévy noise and Brownian motion is obtained by a variational formula for positive functionals of a Poisson random measure and Brownian motion. Additionally, the results are illustrated by the fractional stochastic Chafee-Infante equations

math.PR↗

arXiv2025

Dynamics and Wong-Zakai approximations of stochastic nonlocal PDEs with long time memory

In this paper, a combination of Galerkin's method and Dafermos' transformation is first used to prove the existence and uniqueness of solutions for a class of stochastic nonlocal PDEs with long time memory driven by additive noise. Next, the existence of tempered random attractors for such equations is established in an appropriate space for the analysis of problems with delay and memory. Eventually, the convergence of solutions of Wong-Zakai approximations and upper semicontinuity of random attractors of the approximate random system, as the step sizes of approximations approach zero, are analyzed in a detailed way.

math.DS↗

arXiv2024

Asymptotic behavior of a semilinear problem in heat conduction with long time memory and non-local diffusion

In this paper, the asymptotic behavior of a semilinear heat equation with long time memory and non-local diffusion is analyzed in the usual set-up for dynamical systems generated by differential equations with delay terms. This approach is different from the previous published literature on the long time behavior of heat equations with memory which is carried out by the Dafermos transformation. As a consequence, the obtained results provide complete information about the attracting sets for the original problem, instead of the transformed one. In particular, the proved results also generalize and complete previous literature in the local case.

math.AP↗

arXiv2024

A delay nonlocal quasilinear Chafee-Infante problem: An approach via semigroup theory

In this work we study a dissipative one dimensional scalar parabolic problem with non-local nonlinear diffusion with delay. We consider the general situation in which the functions involved are only continuous and solutions may not be unique. We establish conditions for global existence and prove the existence of global attractors. All results are presented only in the autonomous since the non-autonomous case follows in the same way, including the existence of pullback attractors. A particularly interesting feature is that there is a semilinear problem (nonlocal in space and in time) from which one can obtain all solutions of the associated quasilinear problem and that for this semilinear problem the delay depends on the initial function making its study more involved.

math.AP↗

arXiv2024

Existence, regularity and asymptotic behavior of solutions for a nonlocal Chafee-Infante Problem via semigroup theory

This article deals with the study of a non-local one-dimensional quasilinear problem with continuous forcing. We use a time-reparameterization to obtain a semilinear problem and study a more general equation using semigroup theory. The existence of mild solutions is established without uniqueness with the aid of the formula of variation of constants and asking only a suitable modulus of continuity on the nonlinearity this mild solution is shown to be strong. Comparison results are also established with the aid of the formula of variation of constants and using these comparison results, global existence is obtained with the additional requirement that the nonlinearity satisfy a structural condition. The existence of pullback attractor is also established for the associated multivalued process along with the uniform bounds given by the comparison results with the additional requirement that the nonlinearity be dissipative. As much as possible the results are abstract so that they can be also applied to other models.

math.AP↗

arXiv2024

Exponential attractors for a nonlocal delayed reaction-diffusion equation on an unbounded domain

The main objective of this paper is to investigate exponential attractors for a nonlocal delayed reaction-diffusion equation on an unbounded domain. We first obtain the existence of a globally attractive absorbing set for the dynamical system generated by the equation under the assumption that the nonlinear term is bounded. Then, we construct exponential attractors of the equation directly in its natural phase space, i.e., a Banach space with explicit fractal dimension by combining squeezing properties of the system as well as a covering lemma of finite subspace of Banach spaces. Our result generalizes the methods established in Hilbert spaces and weighted spaces, and the fractal dimension of the obtained exponential attractor does not depend on the entropy number but only depends on some inner characteristic of the studied equation.

math.AP↗

arXiv2024

A way to model stochastic perturbations in population dynamics models with bounded realizations

In this paper, we analyze the use of the Ornstein-Uhlenbeck process to model dynamical systems subjected to bounded noisy perturbations. In order to discuss the main characteristics of this new approach we consider some basic models in population dynamics such as the logistic equations and competitive Lotka-Volterra systems. The key is the fact that these perturbations can be ensured to keep inside some interval that can be previously fixed, for instance, by practitioners, even though the resulting model does not generate a random dynamical system. However, one can still analyze the forwards asymptotic behavior of these random differential systems. Moreover, to illustrate the advantages of this type of modeling, we exhibit an example testing the theoretical results with real data, and consequently one can see this method as a realistic one, which can be very useful and helpful for scientists.

math.DS↗

arXiv2024

Dynamics and numerical simulations to predict empirical antibiotic treatment of multi-resistant Pseudomonas aeruginosa infection

This work discloses an epidemiological mathematical model to predict an empirical treatment for dogs infected by Pseudomonas aeruginosa. This dangerous pathogen is one of the leading causes of multi-resistant infections and can be transmitted from dogs to humans. Numerical simulations and appropriated codes were developed using Matlab software to gather information concerning long-time dynamics of the susceptible, infected and recovered individuals. All data compiled from the mathematical model was used to provide an appropriated antibiotic sensitivity panel for this specific infection. In this study, several variables have been included in this model to predict which treatment should be prescribed in emergency cases, when there is no time to perform an antibiogram or the cost of it could not be assumed. In particular, we highlight the use of this model aiming to become part of the convenient toolbox of Public Health research and decision-making in the design of the mitigation strategy of bacterial pathogens.

q-bio.PE↗

arXiv2023

Topological dimensions of attractors for partial functional differential equations in Banach spaces

The main objective of this paper is to obtain estimations of Hausdorff dimension as well as fractal dimension of global attractors and pullback attractors for both autonomous and nonautonomous functional differential equations (FDEs) in Banach spaces. New criterions for the finite Hausdorff dimension and fractal dimension of attractors in Banach spaces are firslty proposed by combining the squeezing property and the covering of finite subspace of Banach spaces, which generalize the method established in Hilbert spaces. In order to surmount the barrier caused by the lack of orthogonal projectors with finite rank, which is the key tool for proving the squeezing property of partial differential equations in Hilbert spaces, we adopt the state decomposition of phase space based on the exponential dichotomy of the studied FDEs to obtain similar squeezing property. The theoretical results are applied to a retarded nonlinear reaction-diffusion equation and a non-autonomous retarded functional differential equation in the natural phase space, for which explicit bounds of dimensions that do not depend on the entropy number but only depend on the spectrum of the linear parts and Lipschitz constants of the nonlinear parts are obtained.

math.DS↗

arXiv2023

Existence and dimensions of global attractors for a delayed reaction-diffusion equation on an unbounded domain

The purpose of this paper is to investigate the existence and Hausdorff dimension as well as fractal dimension of global attractors for a delayed reaction-diffusion equation on an unbounded domain. The noncompactness of the domain causes the Laplace operator has a continuous spectrum, the semigroup generated by the linear part and the Sobolev embeddings are no longer compact, making the problem more difficult compared with the equations on bounded domains. We first obtain the existence of an absorbing set for the infinite dimensional dynamical system generated by the equation by a priori estimate of the solutions. Then, we show the asymptotic compactness of the solution semiflow by an uniform a priori estimates for far-field values of solutions together with the Arzelà-Ascoli theorem, which facilitates us to show the existence of global attractors. By decomposing the solution into three parts and establishing a squeezing property of each part, we obtain the explicit upper estimation of both Hausdorff and fractal dimension of the global attractors, which only depend on the inner characteristic of the equation, while not related to the entropy number compared with the existing literature.

math.AP↗

arXiv2023

Topological dimensions of global attractors for a delayed reaction-diffusion equation on an unbounded domain

The purpose of this paper is to investigate the existence and estimation of Hausdorff and fractal dimension of global attractors for a delayed reaction-diffusion equation on an unbounded domain. The noncompactness of the domain cause the Laplace operator has a continuous spectrum, the semigroup generated by the linear part and the Sobolev embeddings no longer compact, making the problem more difficult compared with the equation on bounded domain. In order to adopt Hausdorff and fractal dimension dimension estimation tools established for dynamical systems in Hilbert space, we recast the equation in an auxiliary Hilbert space. We first obtain the existence of global solutions on unbounded domain by a perturbation semigroup approach which generates an infinite dimensional dynamical system. Then, we show the existence of global attractors by firstly showing the existence of an absorbing set and then give a an uniform a priori estimates for far-field values of solutions which facilitate us to prove the asymptotic compactness of the generated dynamical system. After establishing the variational equation in the auxiliary Hilbert space and the differentiable properties of the generated dynamical system, the upper estimate of both Hausdorff and fractal dimensions of the global attractors is obtained.

math.DS↗

arXiv2023

Lyapunov exponents and invariant manifolds for stochastic linear partial functional differential equations

The main purpose of this work is to characterize the almost sure local structure stability of solutions to a class of linear stochastic partial functional differential equations (SPFDEs) by investigating the Lyapunov exponents and invariant manifolds near the stationary point. It is firstly proved that the trajectory field of the stochastic delayed stochastic partial functional differential equation admits an almost sure continuous version which is compact for $t>τ$ by a delicate construction based on the random semiflow generated by the diffusion term. Then it is proved that the version generates a random dynamical system(RDS) by the Wong-Zakai approximation of the stochastic partial differential equation constructed by the diffusion term. Subsequently, it is shown that the constructed linear cocycle admits fixed (at most) countable set of Lyapunov exponents and the associate Oseledets random filtration of the Banach space is obtained by adopting the infinite-dimensional multiplicative ergodic theorem in Banach spaces established by Lian and Lu [\textit{Mem Amer Math Soc, 2010, 206: 967}]. As a by product, the stable-manifolds theorem for the linear SPFDE in the hyperbolic case is also established.

math.DS↗

arXiv2023

Invariant manifolds for stochastic delayed partial differential equations of parabolic type

The aim of this paper is to prove the existence and smoothness of stable and unstable invariant manifolds for a stochastic delayed partial differential equation of parabolic type. The stochastic delayed partial differential equation is firstly transformed into a random delayed partial differential equation by a conjugation, which is then recast into a Hilbert space. For the auxiliary equation, the variation of constants formula holds and we show the existence of Lipschitz continuous stable and unstable manifolds by the Lyapunov-Perron method. Subsequently, we prove the smoothness of these invariant manifolds under appropriate spectral gap condition by carefully investigating the smoothness of auxiliary equation, after which, we obtain the invariant manifolds of the original equation by projection and inverse transformation. Eventually, we illustrate the obtained theoretical results by their application to a stochastic single-species population model.

math.DS↗

arXiv2023

Convergence of solutions for a reaction-diffusion problem with fractional Laplacian

A kind of nonlocal reaction-diffusion equations on an unbounded domain containing fractional Laplacian operator is analyzed. To be precise, we prove the convergence of solutions of the equation governed by the fractional Laplacian to the solutions of the classical equation governed by the standard Laplacian, when the fractional parameter grows to 1. The existence of global attractors is investigated as well. The novelty of this paper is concerned with the convergence of solutions when the fractional parameter varies, which, as far as the authors are aware, seems to be the first result of this kind of problems in the literature.

math.AP↗