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Discrete & Continuous Dynamical Systems - B

March 2019 , Volume 24 , Issue 3

Special issue on dynamics and control in distributed systems: Dedicated to the memory of Valery S. Melnik (1952-2007)

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Preface to the special issue "Dynamics and control in distributed systems: Dedicated to the memory of Valery S. Melnik (1952-2007)"
Tomás Caraballo Garrido, Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero and Michael Zgurovsky
2019, 24(3): ⅰ-ⅴ doi: 10.3934/dcdsb.20193i +[Abstract](1247) +[HTML](115) +[PDF](112.13KB)
Abstract:
Stability analysis of a chemotherapy model with delays
Ismail Abdulrashid, Abdallah A. M. Alsammani and Xiaoying Han
2019, 24(3): 989-1005 doi: 10.3934/dcdsb.2019002 +[Abstract](1482) +[HTML](132) +[PDF](543.61KB)
Abstract:

A chemotherapy model for cancer treatment is studied, where the chemotherapy agent and cells are assumed to follow a predator-prey type relation. The time delays from the instant that the chemotherapy agent is injected to the instant that the treatment is effective are taken into account and dynamics of systems with or without delays are compared. First basic properties of solutions including existence and uniqueness, boundedness and positiveness are discussed. Then conditions on model parameters are established for different outcomes of the treatment. Numerical simulations are provided to illustrate theoretical results.

Periodic orbits for the perturbed planar circular restricted 3–body problem
Elbaz I. Abouelmagd, Juan Luis García Guirao and Jaume Llibre
2019, 24(3): 1007-1020 doi: 10.3934/dcdsb.2019003 +[Abstract](1238) +[HTML](104) +[PDF](374.87KB)
Abstract:

We characterize when the classical first and second kind of periodic orbits of the planar circular restricted \begin{document}$ 3 $\end{document}–body problem obtained by Poincaré, can be extended to perturbed planar circular restricted \begin{document}$ 3 $\end{document}–body problems. We put special emphasis when the perturbed forces are due to zonal harmonic or to a solar sail.

Pursuit differential-difference games with pure time-lag
Lesia V. Baranovska
2019, 24(3): 1021-1031 doi: 10.3934/dcdsb.2019004 +[Abstract](1075) +[HTML](137) +[PDF](380.23KB)
Abstract:

The analytical approach for solution of pursuit differential-difference games with pure time-lag is considered. For the pursuit local problem with the fixed time the scheme of the method of resolving functions and Pontryagin's first direct method are developed. The integral presentation of game solution based on the time-delay exponential is proposed at first time. The guaranteed times of the game termination are found, and corresponding control laws are constructed. Comparison of the times of approach by the method of resolving functions and Pontryagin's first direct method for the initial problem are made.

On the exact number of monotone solutions of a simplified Budyko climate model and their different stability
Sabri Bensid and Jesús Ildefonso Díaz
2019, 24(3): 1033-1047 doi: 10.3934/dcdsb.2019005 +[Abstract](1053) +[HTML](69) +[PDF](1986.81KB)
Abstract:

We consider a simplified version of the Budyko diffusive energy balance climate model. We obtain the exact number of monotone stationary solutions of the associated discontinuous nonlinear elliptic with absorption. We show that the bifurcation curve, in terms of the solar constant parameter, is S-shaped. We prove the instability of the decreasing part and the stability of the increasing part of the bifurcation curve. In terms of the Budyko climate problem the above results lead to an important qualitative information which is far to be evident and which seems to be new in the mathematical literature on climate models. We prove that if the solar constant is represented by \begin{document}$ \lambda \in (\lambda _{1}, \lambda _{2}), $\end{document} for suitable \begin{document}$ \lambda _{1}<\lambda _{2}, $\end{document} then there are exactly two stationary solutions giving rise to a free boundary (i.e. generating two symmetric polar ice caps: North and South ones) and a third solution corresponding to a totally ice covered Earth. Moreover, we prove that the solution with smaller polar ice caps is stable and the one with bigger ice caps is unstable.

Robustness of dynamically gradient multivalued dynamical systems
Rubén Caballero, Alexandre N. Carvalho, Pedro Marín-Rubio and José Valero
2019, 24(3): 1049-1077 doi: 10.3934/dcdsb.2019006 +[Abstract](918) +[HTML](104) +[PDF](498.13KB)
Abstract:

In this paper we study the robustness of dynamically gradient multivalued semiflows. As an application, we describe the dynamical properties of a family of Chafee-Infante problems approximating a differential inclusion studied in [3], proving that the weak solutions of these problems generate a dynamically gradient multivalued semiflow with respect to suitable Morse sets.

Some remarks on an environmental defensive expenditures model
Tomás Caraballo, Renato Colucci and Luca Guerrini
2019, 24(3): 1079-1093 doi: 10.3934/dcdsb.2019007 +[Abstract](955) +[HTML](99) +[PDF](1112.39KB)
Abstract:

In this paper, we consider the environmental defensive expenditures model with delay proposed by Russu in [16] and obtain different results about stability of equilibria in the case of absence of delay. Moreover we provide a more detailed analysis of the stability for equilibria and Hopf bifurcation in the case with delay. Finally, we discuss possible modifications of the model in order to make it more accurate and realistic.

I. U. Bronshtein's conjecture for monotone nonautonomous dynamical systems
David Cheban
2019, 24(3): 1095-1113 doi: 10.3934/dcdsb.2019008 +[Abstract](1057) +[HTML](82) +[PDF](446.64KB)
Abstract:

In this paper we study the problem of Levitan/Bohr almost periodicity of solutions for dissipative differential equations (Bronshtein's conjecture for Bohr almost periodic case). We give a positive answer to this conjecture for monotone Levitan/Bohr almost periodic systems of differential/difference equations.

Inertial manifolds for the hyperbolic relaxation of semilinear parabolic equations
Vladimir V. Chepyzhov, Anna Kostianko and Sergey Zelik
2019, 24(3): 1115-1142 doi: 10.3934/dcdsb.2019009 +[Abstract](1001) +[HTML](83) +[PDF](517.13KB)
Abstract:

The paper gives a comprehensive study of Inertial Manifolds for hyperbolic relaxations of an abstract semilinear parabolic equation in a Hilbert space. A new scheme of constructing Inertial Manifolds for such type of problems is suggested and optimal spectral gap conditions which guarantee their existence are established. Moreover, the dependence of the constructed manifolds on the relaxation parameter in the case of the parabolic singular limit is also studied.

Bibliography: 38 titles.

A topological characterization of the $\omega$-limit sets of analytic vector fields on open subsets of the sphere
José Ginés Espín Buendía and Víctor Jiménez Lopéz
2019, 24(3): 1143-1173 doi: 10.3934/dcdsb.2019010 +[Abstract](583) +[HTML](69) +[PDF](570.29KB)
Abstract:

In [15], V. Jiménez López and J. Llibre characterized, up to homeomorphism, the \begin{document}$ \omega $\end{document}-limit sets of analytic vector fields on the sphere and the projective plane. The authors also studied the same problem for open subsets of these surfaces.

Unfortunately, an essential lemma in their programme for general surfaces has a gap. Although the proof of this lemma can be amended in the case of the sphere, the plane, the projective plane and the projective plane minus one point (and therefore the characterizations for these surfaces in [15] are correct), the lemma is not generally true, see [6].

Consequently, the topological characterization for analytic vector fields on open subsets of the sphere and the projective plane is still pending. In this paper, we close this problem in the case of open subsets of the sphere.

Forward attraction of pullback attractors and synchronizing behavior of gradient-like systems with nonautonomous perturbations
Xuewei Ju, Desheng Li and Jinqiao Duan
2019, 24(3): 1175-1197 doi: 10.3934/dcdsb.2019011 +[Abstract](1226) +[HTML](80) +[PDF](489.66KB)
Abstract:

We consider the nonautonomous perturbation \begin{document}$ x_t+Ax = f(x)+\varepsilon h(t) $\end{document} of a gradient-like system \begin{document}$ x_t+Ax = f(x) $\end{document} in a Banach space \begin{document}$ X $\end{document}, where \begin{document}$ A $\end{document} is a sectorial operator with compact resolvent. Assume the non-perturbed system \begin{document}$ x_t+Ax = f(x) $\end{document} has an attractor \begin{document}$ {\mathscr A} $\end{document}. Then it can be shown that the perturbed one has a pullback attractor \begin{document}$ {\mathscr A} _\varepsilon $\end{document} near \begin{document}$ {\mathscr A} $\end{document}. If all the equilibria of the non-perturbed system in \begin{document}$ {\mathscr A} $\end{document} are hyperbolic, we also infer from [4,6] that \begin{document}$ {\mathscr A} _\varepsilon $\end{document} inherits the natural Morse structure of \begin{document}$ {\mathscr A} $\end{document}. In this present work, we introduce the notion of nonautonomous equilibria and give a more precise description on the Morse structure of \begin{document}$ {\mathscr A} _\varepsilon $\end{document} and the asymptotically synchronizing behavior of the perturbed system. Based on the above result we further prove that the sections of \begin{document}$ {\mathscr A} _\varepsilon $\end{document} depend on time symbol continuously in the sense of Hausdorff distance. Consequently, one concludes that \begin{document}$ {\mathscr A} _\varepsilon $\end{document} is a forward attractor of the perturbed nonautonomous system. It will also be shown that the perturbed system exhibits completely a global forward synchronizing behavior with the external force.

On relation between attractors for single and multivalued semiflows for a certain class of PDEs
Piotr Kalita, Grzegorz Łukaszewicz and Jakub Siemianowski
2019, 24(3): 1199-1227 doi: 10.3934/dcdsb.2019012 +[Abstract](1031) +[HTML](84) +[PDF](534.4KB)
Abstract:

Sometimes it is not possible to prove the uniqueness of the weak solutions for problems of mathematical physics, but it is possible to bootstrap their regularity to the regularity of strong solutions which are unique. In this paper we formulate an abstract setting for such class of problems and we provide the conditions under which the global attractors for both strong and weak solutions coincide and the fractal dimension of the common attractor is finite. We present two problems belonging to this class: planar Rayleigh–Bénard flow of thermomicropolar fluid and surface quasigeostrophic equation on torus.

Attractors of multivalued semi-flows generated by solutions of optimal control problems
Olexiy V. Kapustyan, Pavlo O. Kasyanov, José Valero and Mikhail Z. Zgurovsky
2019, 24(3): 1229-1242 doi: 10.3934/dcdsb.2019013 +[Abstract](1166) +[HTML](67) +[PDF](392.05KB)
Abstract:

In this paper we study the dynamical system generated by the solutions of optimal control problems. We obtain suitable conditions under which such systems generate multivalued semiprocesses. We prove the existence of uniform attractors for the multivalued semiprocess generated by the solutions of controlled reaction-diffusion equations and study its properties.

Quasi-optimal control with a general quadratic criterion in a special norm for systems described by parabolic-hyperbolic equations with non-local boundary conditions
Volodymyr O. Kapustyan, Ivan O. Pyshnograiev and Olena A. Kapustian
2019, 24(3): 1243-1258 doi: 10.3934/dcdsb.2019014 +[Abstract](1155) +[HTML](68) +[PDF](353.96KB)
Abstract:

In this work, we consider a dynamical system generated by a parabolic-hyperbolic equation with non-local boundary conditions. The optimal control problem for this system is studied using a notion of quasi-optimal solution. Existence and uniqueness of quasi-optimal control are proved.

Forward attracting sets of reaction-diffusion equations on variable domains
Peter E. Kloeden and Meihua Yang
2019, 24(3): 1259-1271 doi: 10.3934/dcdsb.2019015 +[Abstract](1085) +[HTML](92) +[PDF](365.28KB)
Abstract:

Reaction-diffusion equations on time-variable domains are instrinsically nonautonomous even if the coefficients in the equation do not depend explicitly on time. Thus the appropriate asymptotic concepts, such as attractors, are nonautonomous. Forward attracting sets based on omega-limit sets are considered in this paper. These are related to the Vishik uniform attractor but are not as restrictive since they depend only on the dynamics in the distant future. They are usually not invariant. Here it is shown that they are asymptotically positively invariant, in general, and, if the future dynamics is appropriately uniform, also asymptotically negatively invariant as well as upper semi continuous dependence in a parameter will be established. These results also apply to reaction-diffusion equations on a fixed domain.

On optimal control problem for an ill-posed strongly nonlinear elliptic equation with $p$-Laplace operator and $L^1$-type of nonlinearity
Peter I. Kogut and Olha P. Kupenko
2019, 24(3): 1273-1295 doi: 10.3934/dcdsb.2019016 +[Abstract](1135) +[HTML](69) +[PDF](557.11KB)
Abstract:

We study an optimal control problem for one class of non-linear elliptic equations with \begin{document}$p$\end{document}-Laplace operator and \begin{document}$L^1$\end{document}-nonlinearity. We deal with such case of nonlinearity when we cannot expect to have a solution of the state equation for any given control. After defining a suitable functional class in which we look for solutions, we reformulate the original problem and prove the existence of optimal pairs. In order to ensure the validity of such reformulation, we provide its substantiation using a special family of fictitious optimal control problems. The idea to involve the fictitious optimization problems was mainly inspired by the brilliant book of V.S. Mel'nik and V.I. Ivanenko "Variational Methods in Control Problems for the Systems with Distributed Parameters", Kyiv, 1998.

Existence of solutions for space-fractional parabolic hemivariational inequalities
Yongjian Liu, Zhenhai Liu and Ching-Feng Wen
2019, 24(3): 1297-1307 doi: 10.3934/dcdsb.2019017 +[Abstract](1075) +[HTML](63) +[PDF](380.94KB)
Abstract:

This paper is devoted to the existence of solutions for space-fractional parabolic hemivariational inequalities by means of the well-known surjectivity result for multivalued ($S_+$) type mappings.

Partial differential inclusions of transport type with state constraints
Thomas Lorenz
2019, 24(3): 1309-1340 doi: 10.3934/dcdsb.2019018 +[Abstract](1272) +[HTML](116) +[PDF](685.64KB)
Abstract:

The focus is on the existence of weak solutions to the quasilinear first-order partial differential inclusion

with values in \begin{document}$L^p({{\mathbb{R}}^{N}})$\end{document} for \begin{document}$p ∈ (1, ∞)$\end{document}. The solution is to satisfy state constraints in addition, i.e., all its values belong to a given set \begin{document}$\mathcal{V} \subset L^p({{\mathbb{R}}^{N}})$\end{document} of constraints. We specify sufficient conditions such that every function in \begin{document}$\mathcal{V}$\end{document} initializes at least one weak solution with all its values in \begin{document}$\mathcal{V}$\end{document}(so-called weak invariance a.k.a. viability of \begin{document}$\mathcal{V}$\end{document}). Due to the regularity assumptions about the set-valued coefficient mappings, these solutions prove to be renormalized (in the sense of Di Perna and Lions).

On the nonconserved Caginalp phase-field system based on the Maxwell-Cattaneo law with two temperatures and logarithmic potentials
Ahmad Makki, Alain Miranville and Georges Sadaka
2019, 24(3): 1341-1365 doi: 10.3934/dcdsb.2019019 +[Abstract](1113) +[HTML](67) +[PDF](756.26KB)
Abstract:

Our aim in this article is to study generalizations of the nonconserved Caginalp phase-field system based on the Maxwell-Cattaneo law with two temperatures for heat conduction and with logarithmic nonlinear terms. We obtain well-posedness results and study the asymptotic behavior of the system. In particular, we prove the existence of the global attractor. Furthermore, we give some numerical simulations, obtained with the $\mathtt{FreeFem++}$ software [24], comparing the nonconserved Caginalp phase-field model with regular and logarithmic nonlinear terms.

Asymptotic behavior of the stochastic Keller-Segel equations
Yadong Shang, Jianjun Paul Tian and Bixiang Wang
2019, 24(3): 1367-1391 doi: 10.3934/dcdsb.2019020 +[Abstract](1388) +[HTML](87) +[PDF](566.44KB)
Abstract:

This paper deals with the asymptotic behavior of the solutions of the non-autonomous one-dimensional stochastic Keller-Segel equations defined in a bounded interval with Neumann boundary conditions. We prove the existence and uniqueness of tempered pullback random attractors under certain conditions. We also establish the convergence of the solutions as well as the pullback random attractors of the stochastic equations as the intensity of noise approaches zero.

An optimal control problem for some nonlinear elliptic equations with unbounded coefficients
Gabriella Zecca
2019, 24(3): 1393-1409 doi: 10.3934/dcdsb.2019021 +[Abstract](1218) +[HTML](100) +[PDF](482.53KB)
Abstract:

We study an optimal control problem associated to a Dirichlet boundary value problem of the type

\begin{document}$ 1<p\leqslant 2, $\end{document} where \begin{document}$ \Omega $\end{document} is a bounded regular domain of \begin{document}$ \mathbb{R}^N $\end{document}, \begin{document}$ 0\in \Omega , $\end{document} \begin{document}$ \beta: \Omega \rightarrow {\mathbb R} $\end{document} is an unbounded function satisfying \begin{document}$ \beta(x)\geqslant\lambda_0>0 $\end{document} a.e., \begin{document}$ A $\end{document} is a suitably small constant, and \begin{document}$ g\in L^\infty( \Omega ; \mathbb{R}^N ) $\end{document}.

We consider the vector field \begin{document}$ \mathcal F $\end{document} as the control and the corresponding weak solution \begin{document}$ u $\end{document} to (BVP) as the state. Our aim is to find the optimal vector field \begin{document}$ \mathcal F\in L^p( \Omega ) $\end{document} so that the corresponding state \begin{document}$ u\in W^{1,p}_0( \Omega ) $\end{document} is close to the desired profile in \begin{document}$ L^p( \Omega ) $\end{document} while the norm of \begin{document}$ u $\end{document} in \begin{document}$ W^{1,p}( \Omega ) $\end{document} is not too large.

We prove that, for every \begin{document}$ p $\end{document} less than \begin{document}$ 2 $\end{document} and suitably close to \begin{document}$ 2 $\end{document}, (BVP) admits an unique weak solution and for such values of \begin{document}$ p $\end{document}, we prove the existence of optimal pairs.

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