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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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Preface to the special issue in honor of Prof. Tomás Caraballo on occasion of his 60th birthday
María J. Garrido-Atienza, José A. Langa, Pedro Marín Rubio and José Valero
2020, 19(4): ⅰ-ⅵ doi: 10.3934/cpaa.2020080 +[Abstract](776) +[HTML](147) +[PDF](667.39KB)
Abstract:
Bifurcations in periodic integrodifference equations in $ C(\Omega) $ Ⅱ: Discrete torus bifurcations
Christian Aarset and Christian Pötzsche
2020, 19(4): 1847-1874 doi: 10.3934/cpaa.2020081 +[Abstract](560) +[HTML](109) +[PDF](1636.03KB)
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.

A mathematical model of chemotherapy with variable infusion
Ismail Abdulrashid and Xiaoying Han
2020, 19(4): 1875-1890 doi: 10.3934/cpaa.2020082 +[Abstract](713) +[HTML](149) +[PDF](514.32KB)
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.

Elliptic and parabolic problems in thin domains with doubly weak oscillatory boundary
José M. Arrieta and Manuel Villanueva-Pesqueira
2020, 19(4): 1891-1914 doi: 10.3934/cpaa.2020083 +[Abstract](608) +[HTML](111) +[PDF](424.75KB)
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 \begin{document}$ \omega\subset \mathbb{R}^n $\end{document}. 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.

On a delayed epidemic model with non-instantaneous impulses
Liang Bai, Juan J. Nieto and José M. Uzal
2020, 19(4): 1915-1930 doi: 10.3934/cpaa.2020084 +[Abstract](751) +[HTML](157) +[PDF](356.02KB)
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.

Longtime behavior for 3D Navier-Stokes equations with constant delays
Hakima Bessaih and María J. Garrido-Atienza
2020, 19(4): 1931-1948 doi: 10.3934/cpaa.2020085 +[Abstract](423) +[HTML](99) +[PDF](346.83KB)
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.

Controllability of the one-dimensional fractional heat equation under positivity constraints
Umberto Biccari, Mahamadi Warma and Enrique Zuazua
2020, 19(4): 1949-1978 doi: 10.3934/cpaa.2020086 +[Abstract](805) +[HTML](100) +[PDF](3734.8KB)
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 \begin{document}$ (-d_x^{\,2})^{s}{} $\end{document} (\begin{document}$ 0<s<1 $\end{document}) on the interval \begin{document}$ (-1,1) $\end{document}. We prove the existence of a minimal (strictly positive) time \begin{document}$ T_{\rm min} $\end{document} such that the fractional heat dynamics can be controlled from any initial datum in \begin{document}$ L^2(-1,1) $\end{document} to a positive trajectory through the action of a positive control, when \begin{document}$ s>1/2 $\end{document}. Moreover, we show that in this minimal time constrained controllability is achieved by means of a control that belongs to a certain space of Radon measures. We also give some numerical simulations that confirm our theoretical results.

Stability and forward attractors for non-autonomous impulsive semidynamical systems
Everaldo de Mello Bonotto and Daniela Paula Demuner
2020, 19(4): 1979-1996 doi: 10.3934/cpaa.2020087 +[Abstract](584) +[HTML](110) +[PDF](353.65KB)
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.

Forwards dynamics of non-autonomous dynamical systems: Driving semigroups without backwards uniqueness and structure of the attractor
Alexandre N. Carvalho, José A. Langa and James C. Robinson
2020, 19(4): 1997-2013 doi: 10.3934/cpaa.2020088 +[Abstract](588) +[HTML](123) +[PDF](182.73KB)
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 \begin{document}$ t = 0 $\end{document}). We will be interested in the cases for which the 'driving' (time shift) semigroup has a global attractor in which backwards solutions are not necessarily unique. Considering vector fields in the global attractor of the driving semigroup allows for a natural way to extend vector fields, defined only for non-negative times, to the whole real line. These objects play a crucial role in the description of the asymptotic dynamics of our non-autonomous differential equation. We will study, in some particular cases, the isolated invariant sets of the associated skew-product semigroup with the aim of characterising the global attractor. We develop an example for which we derive decomposition for the global attractor of skew-product semigroup from the characterisation of the attractor of the associated driving semigroup.

Dynamics of fermentation models for the production of dry and sweet wine
Renato Colucci and Javier López-de-la-Cruz
2020, 19(4): 2015-2034 doi: 10.3934/cpaa.2020089 +[Abstract](558) +[HTML](122) +[PDF](2821.1KB)
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.

Nonclassical diffusion with memory lacking instantaneous damping
Monica Conti, Filippo Dell'Oro and Vittorino Pata
2020, 19(4): 2035-2050 doi: 10.3934/cpaa.2020090 +[Abstract](523) +[HTML](120) +[PDF](370.48KB)
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 \begin{document}$ -\Delta u $\end{document}, contributing as an instantaneous damping. Setting the problem in the past history framework, we prove that the related solution semigroup possesses a global attractor of optimal regularity.

Instability of unidirectional flows for the 2D α-Euler equations
Holger Dullin, Yuri Latushkin, Robert Marangell, Shibi Vasudevan and Joachim Worthington
2020, 19(4): 2051-2079 doi: 10.3934/cpaa.2020091 +[Abstract](439) +[HTML](94) +[PDF](449.03KB)
Abstract:

We study stability of unidirectional flows for the linearized 2D \begin{document}$ \alpha $\end{document}-Euler equations on the torus. The unidirectional flows are steady states whose vorticity is given by Fourier modes corresponding to a vector \begin{document}$ \mathbf p \in \mathbb Z^{2} $\end{document}. We linearize the \begin{document}$ \alpha $\end{document}-Euler equation and write the linearized operator \begin{document}$ L_{B} $\end{document} in \begin{document}$ \ell^{2}(\mathbb Z^{2}) $\end{document} as a direct sum of one-dimensional difference operators \begin{document}$ L_{B,\mathbf q} $\end{document} in \begin{document}$ \ell^{2}(\mathbb Z) $\end{document} parametrized by some vectors \begin{document}$ \mathbf q\in\mathbb Z^2 $\end{document} such that the set \begin{document}$ \{\mathbf q +n \mathbf p:n \in \mathbb Z\} $\end{document} covers the entire grid \begin{document}$ \mathbb Z^{2} $\end{document}. The set \begin{document}$ \{\mathbf q +n \mathbf p:n \in \mathbb Z\} $\end{document} can have zero, one, or two points inside the disk of radius \begin{document}$ \|\mathbf p\| $\end{document}. We consider the case where the set \begin{document}$ \{\mathbf q +n \mathbf p:n \in \mathbb Z\} $\end{document} has exactly one point in the open disc of radius \begin{document}$ \mathbf p $\end{document}. We show that unidirectional flows that satisfy this condition are linearly unstable. Our main result is an instability theorem that provides a necessary and sufficient condition for the existence of a positive eigenvalue to the operator \begin{document}$ L_{B, {\mathbf q}} $\end{document} in terms of equations involving certain continued fractions. Moreover, we are also able to provide a complete characterization of the corresponding eigenvector. The proof is based on the use of continued fractions techniques expanding upon the ideas of Friedlander and Howard.

Advances in the truncated Euler–Maruyama method for stochastic differential delay equations
Weiyin Fei, Liangjian Hu, Xuerong Mao and Dengfeng Xia
2020, 19(4): 2081-2100 doi: 10.3934/cpaa.2020092 +[Abstract](592) +[HTML](118) +[PDF](151.93KB)
Abstract:

Guo et al. [8] are the first to study the strong convergence of the explicit numerical method for the highly nonlinear stochastic differential delay equations (SDDEs) under the generalised Khasminskii-type condition. The method used there is the truncated Euler–Maruyama (EM) method. In this paper we will point out that a main condition imposed in [8] is somehow restrictive in the sense that the condition could force the step size to be so small that the truncated EM method would be inapplicable. The key aim of this paper is then to establish the convergence rate without this restriction.

Theoretical and numerical results for some bi-objective optimal control problems
Enrique Fernández-Cara and Irene Marín-Gayte
2020, 19(4): 2101-2126 doi: 10.3934/cpaa.2020093 +[Abstract](574) +[HTML](123) +[PDF](1092.94KB)
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.

Pullback attractors for 2D Navier–Stokes equations with delays and the flattening property
Julia García-Luengo and Pedro Marín-Rubio
2020, 19(4): 2127-2146 doi: 10.3934/cpaa.2020094 +[Abstract](537) +[HTML](101) +[PDF](382.66KB)
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 \begin{document}$ L^2 $\end{document} norm and the \begin{document}$ H^1 $\end{document} norm. Moreover, we provide results on the existence of pullback attractors for two possible choices of the attracted universes, namely, the standard one of fixed bounded sets, and secondly, one given by a tempered condition.

PDE problems with concentrating terms near the boundary
Ángela Jiménez-Casas and Aníbal Rodríguez-Bernal
2020, 19(4): 2147-2195 doi: 10.3934/cpaa.2020095 +[Abstract](588) +[HTML](122) +[PDF](533.33KB)
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.

Chain recurrence and structure of $ \omega $-limit sets of multivalued semiflows
Olexiy V. Kapustyan, Pavlo O. Kasyanov and José Valero
2020, 19(4): 2197-2217 doi: 10.3934/cpaa.2020096 +[Abstract](540) +[HTML](93) +[PDF](369.34KB)
Abstract:

We study properties of \begin{document}$ \omega $\end{document}-limit sets of multivalued semiflows like chain recurrence or the existence of cyclic chains.

First, we prove that under certain conditions the \begin{document}$ \omega $\end{document}-limit set of a trajectory is chain recurrent, applying this result to an evolution differential inclusion with upper semicontinous right-hand side.

Second, we give conditions ensuring that the \begin{document}$ \omega $\end{document}-limit set of a trajectory contains a cyclic chain. Using this result we are able to check that the \begin{document}$ \omega $\end{document}-limit set of every trajectory of a reaction-diffusion equation without uniqueness of solutions is an equilibrium.

Attractors for semilinear wave equations with localized damping and external forces
To Fu Ma and Paulo Nicanor Seminario-Huertas
2020, 19(4): 2219-2233 doi: 10.3934/cpaa.2020097 +[Abstract](676) +[HTML](130) +[PDF](352.32KB)
Abstract:

This paper is concerned with long-time dynamics of semilinear wave equations defined on bounded domains of \begin{document}$ \mathbb{R}^3 $\end{document} with cubic nonlinear terms and locally distributed damping. The existence of regular finite-dimensional global attractors established by Chueshov, Lasiecka and Toundykov (2008) reflects a good deal of the current state of the art on this matter. Our contribution is threefold. First, we prove uniform boundedness of attractors with respect to a forcing parameter. Then, we study the continuity of attractors with respect to the parameter in a residual dense set. Finally, we show the existence of generalized exponential attractors. These aspects were not previously considered for wave equations with localized damping.

Lyapunov exponents and Oseledets decomposition in random dynamical systems generated by systems of delay differential equations
Janusz Mierczyński, Sylvia Novo and Rafael Obaya
2020, 19(4): 2235-2255 doi: 10.3934/cpaa.2020098 +[Abstract](615) +[HTML](89) +[PDF](365.83KB)
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 \begin{document}$ p $\end{document}-summable functions. The main result states that in both cases, the Lyapunov exponents are identical, and that the Oseledets decompositions are related by natural embeddings.

Asymptotic behavior of a Cahn-Hilliard/Allen-Cahn system with temperature
Alain Miranville, Ramon Quintanilla and Wafa Saoud
2020, 19(4): 2257-2288 doi: 10.3934/cpaa.2020099 +[Abstract](694) +[HTML](102) +[PDF](392.05KB)
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.

The stampacchia maximum principle for stochastic partial differential equations forced by lévy noise
Phuong Nguyen and Roger Temam
2020, 19(4): 2289-2331 doi: 10.3934/cpaa.2020100 +[Abstract](590) +[HTML](103) +[PDF](656.5KB)
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 [15] where the authors only considered the case of the Wiener noise; even in this case we improve on [15] because the coefficients of the principal differential operator are now allowed to depend upon \begin{document}$ t $\end{document}.

Analysis of an anaerobic digestion model in landfill with mortality term
S. Ouchtout, Z. Mghazli, J. Harmand, A. Rapaport and Z. Belhachmi
2020, 19(4): 2333-2346 doi: 10.3934/cpaa.2020101 +[Abstract](589) +[HTML](97) +[PDF](448.21KB)
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.

Convergence of nonautonomous multivalued problems with large diffusion to ordinary differential inclusions
Jacson Simsen, Mariza Stefanello Simsen and José Valero
2020, 19(4): 2347-2368 doi: 10.3934/cpaa.2020102 +[Abstract](514) +[HTML](96) +[PDF](379.1KB)
Abstract:

In this work we consider a family of nonautonomous partial differential inclusions governed by \begin{document}$ p $\end{document}-laplacian operators with variable exponents and large diffusion and driven by a forcing nonlinear term of Heaviside type. We prove first that this problem generates a sequence of multivalued nonautonomous dynamical systems possessing a pullback attractor. The main result of this paper states that the solutions of the family of partial differential inclusions converge to the solutions of a limit ordinary differential inclusion for large diffusion and when the exponents go to \begin{document}$ 2 $\end{document}. After that we prove the upper semicontinuity of the pullback attractors.

Large deviations for neutral stochastic functional differential equations
Yongqiang Suo and Chenggui Yuan
2020, 19(4): 2369-2384 doi: 10.3934/cpaa.2020103 +[Abstract](543) +[HTML](97) +[PDF](345.71KB)
Abstract:

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.

Sigmoidal approximations of a delay neural lattice model with Heaviside functions
Xiaoli Wang, Meihua Yang and Peter E. Kloeden
2020, 19(4): 2385-2402 doi: 10.3934/cpaa.2020104 +[Abstract](454) +[HTML](96) +[PDF](374.27KB)
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.

Sensitivity to small delays of mean square stability for stochastic neutral evolution equations
Wei Wang, Kai Liu and Xiulian Wang
2020, 19(4): 2403-2418 doi: 10.3934/cpaa.2020105 +[Abstract](558) +[HTML](97) +[PDF](330.99KB)
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.

2019  Impact Factor: 1.105

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