ISSN:

1534-0392

eISSN:

1553-5258

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## Communications on Pure & Applied Analysis

April 2020 , Volume 19 , Issue 4

Special issue in honor of Prof. Tomás Caraballo on the occasion of his 60th birthday

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**Abstract:**

We provide a convenient Neimark-Sacker bifurcation result for time-periodic difference equations in arbitrary Banach spaces. It ensures the bifurcation of "discrete invariant tori" caused by a pair of complex-conjugated Floquet multipliers crossing the complex unit circle. This criterion is made explicit for integrodifference equations, which are infinite-dimensional discrete dynamical systems popular in theoretical ecology, and are used to describe the temporal evolution and spatial dispersal of populations with nonoverlapping generations. As an application, we combine analytical and numerical tools for a detailed bifurcation analysis of a spatial predator-prey model. Since such realistic models can frequently only be studied numerically, we formulate our assumptions in such a fashion as to allow for numerically stable verification.

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**Abstract:**

A nonautonomous mathematical model of chemotherapy cancer treatment with time-dependent infusion concentration of the chemotherapy agent is developed and studied. In particular, a mutual inhibition type model is adopted to describe the interactions between the chemotherapy agent and cells, in which the chemotherapy agent is modeled as the prey being consumed by both cancer and normal cells, thereby reducing the population of both. Properties of solutions and detailed dynamics of the nonautonomous system are investigated, and conditions under which the treatment is successful or unsuccessful are established. It can be shown both theoretically and numerically that with the same amount of chemotherapy agent infused during the same period of time, a treatment with variable infusion may over perform a treatment with constant infusion.

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**Abstract:**

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 domain shrinks to a fixed domain

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**Abstract:**

We introduce a non-instantaneous pulse vaccination model. Non-instantaneous impulsive nonlinear differential equations provide an adequate biomathematical model of some medical problems. In this paper we study some basic properties such as the attractiveness of the infection-free periodic solution and the permanence of some sub-population for a vaccine model where a constant fraction of the susceptible population is vaccinated in some periodic way. Our model is a system of nonlinear differential equations with impulses.

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**Abstract:**

This paper investigates the longtime behavior of delayed 3D Navier-Stokes equations in terms of attractors. The study will strongly rely on the investigation of the linearized Navier-Stokes system, and the relationship between the discrete dynamical flow for the linearized system and the continuous flow associated to the original system. Assuming the viscosity to be sufficiently large, there exists a unique attractor for the delayed 3D Navier-Stokes equations. Moreover, the attractor reduces to a singleton set.

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**Abstract:**

In this paper, we analyze the controllability properties under positivity constraints on the control or the state of a one-dimensional heat equation involving the fractional Laplacian

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**Abstract:**

In this paper, we study the theory of forward attractors for non-autonomous impulsive semidynamical systems. Moreover, we investigate some types of stability of the global attractor as orbital stability, asymptotic stability and stability in the sense of Lyapunov-Barbashin. We present an example to illustrate the theory.

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**Abstract:**

We investigate the forwards asymptotic dynamics of non-autonomous differential equations. Our approach is centred on those models for which the vector field is only defined for non-negative times, that is, the laws of evolution are not given, or simply not known, for times before a given time (say time

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**Abstract:**

In this work we consider two classical mathematical models of wine fermentation. The first model describes the wine-making process that is used to produce dry wine. The second model is obtained by introducing a term in the equation of the dynamics of the yeast. Thanks to this change it will be possible to inhibit the fermentation of the sugar and as a consequence a sweet wine will be obtained. We first prove the existence, uniqueness, positiveness and boundedness of solutions for both models. Then we pass to analyse the the long-time dynamics. For the second model we also provide estimates for the concentration of ethanol, nitrogen and sugar at the end of the process. Moreover, several numerical simulations are provided to support the theoretical results.

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**Abstract:**

We consider the nonclassical diffusion equation with hereditary memory

on a bounded three-dimensional domain. The main feature of the model is that the equation does not contain a term of the form

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**Abstract:**

We study stability of unidirectional flows for the linearized 2D

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**Abstract:**

Guo et al. [

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**Abstract:**

This article deals with the solution of some multi-objective optimal control problems for several PDEs: linear and semilinear elliptic equations and stationary Navier-Stokes systems. More precisely, we look for Pareto equilibria associated to standard cost functionals. First, we study the linear and semilinear cases. We prove the existence of equilibria, we deduce appropriate optimality systems, we present some iterative algorithms and we establish convergence results. Then, we analyze the existence and characterization of Pareto equilibria for the Navier-Stokes equations. Here, we use the formalism of Dubovitskii and Milyutin. In this framework, we also present a finite element approximation of the bi-objective problem and we illustrate the techniques with several numerical experiments.

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**Abstract:**

This paper treats the existence of pullback attractors for a 2D Navier–Stokes model with finite delay formulated in [Caraballo and Real, *J. Differential Equations* **205** (2004), 271–297]. Actually, we carry out our study under less restrictive assumptions than in the previous reference. More precisely, we remove a condition on square integrable control of the memory terms, which allows us to consider a bigger class of delay terms. Here we show that the asymptotic compactness of the corresponding processes required to establish the existence of pullback attractors, obtained in [García-Luengo, Marín-Rubio and Real, *Adv. Nonlinear Stud.* **13** (2013), 331–357] by using an energy method, can be also proved by verifying the flattening property – also known as "Condition (C)". We deal with dynamical systems in suitable phase spaces within two metrics, the

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**Abstract:**

In this paper we study several PDE problems where certain linear or nonlinear termsin the equation concentrate in the domain, typically (but not exclusively) near the boundary. We analyze some linear and nonlinear elliptic models, linear and nonlinear parabolic ones as well as some damped wave equations. We show that in all these singularly perturbed problems, the concentrating terms give rise in the limit to a modification in the original boundary condition of the problem. Hence we describe in each case which is the singular limit problem and analyze the convergence of solutions.

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**Abstract:**

We study properties of

First, we prove that under certain conditions the

Second, we give conditions ensuring that the

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**Abstract:**

This paper is concerned with long-time dynamics of semilinear wave equations defined on bounded domains of

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**Abstract:**

Linear skew-product semidynamical systems generated by random systems of delay differential equations are considered, both on a space of continuous functions as well as on a space of

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**Abstract:**

The main goal of this paper is to study the asymptotic behavior of a coupled Cahn-Hilliard/Allen-Cahn system with temperature. The work is divided into two parts: In the first part, the heat equation is based on the usual Fourier law. In the second one, it is based on the type Ⅲ heat conduction law. In both parts, we prove the existence of exponential attractors and, therefore, of finite-dimensional global attractors.

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**Abstract:**

In this work, we investigate the existence of positive (martingale and pathwise) solutions of stochastic partial differential equations (SPDEs) driven by a Lévy noise. The proof relies on the use of truncation, following the Stampacchia approach to maximum principle. Among the applications, the positivity and boundedness results for the solutions of some biological systems and reaction diffusion equations are provided under suitable hypotheses, as well as some comparison theorems. This article improves the results of [

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**Abstract:**

We study a mathematical model of anaerobic digestion with biomass recirculation, dedicated to landfill problems, and analyze its asymptotic behavior. We show that the global attractor is composed of an infinity of non-hyperbolic equilibria. For non-monotonic growth functions, this set is non connected, which impacts the performances of the bioprocess.

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**Abstract:**

In this work we consider a family of nonautonomous partial differential inclusions governed by

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In this paper, under a one-sided Lipschitz condition on the drift coefficient we adopt (via contraction principle) an exponential approximation argument to investigate large deviations for neutral stochastic functional differential equations.

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**Abstract:**

The approximation of Heaviside coefficient functions in delay neural lattice models with delays by sigmoidal functions is investigated. The solutions of the delay sigmoidal models are shown to converge to a solution of the delay differential inclusion as the sigmoidal parameter goes to zero. In addition, the existence of global attractors is established and compared for the various systems.

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**Abstract:**

In this work, we are concerned about the mean square exponential stability property for a class of stochastic neutral functional differential equations with small delay parameters. Both distributed and point delays under the neutral term are considered. Sufficient conditions are given to capture the exponential stability in mean square of the stochastic system under consideration. As an illustration, we present some practical systems to show their exponential stability which is not sensitive to small delays in the mean square sense.

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