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Discrete & Continuous Dynamical Systems - B

2017 , Volume 22 , Issue 5

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Preface to the special issue "Finite and infinite dimensional multivalued dynamical systems"
María J. Garrido-Atienza , Oleksiy V. Kapustyan and  José Valero
2017, 22(5): ⅰ-ⅳ doi: 10.3934/dcdsb.201705i +[Abstract](71) +[HTML](8) +[PDF](112.5KB)
Stability results for discontinuous nonlinear elliptic and parabolic problems with a S-shaped bifurcation branch of stationary solutions
Sabri Bensid and  Jesús Ildefonso Díaz
2017, 22(5): 1757-1778 doi: 10.3934/dcdsb.2017105 +[Abstract](36) +[HTML](2) +[PDF](532.0KB)

We study stability of the nonnegative solutions of a discontinuous elliptic eigenvalue problem relevant in several applications as for instance in climate modeling. After giving the explicit expresion of the S-shaped bifurcation diagram \begin{document} $\left( \lambda ,{{\left\| {{\mu }_{\lambda }} \right\|}_{\infty }} \right)$ \end{document} we show the instability of the decreasing part of the bifurcation curve and the stability of the increasing part. This extends to the case of non-smooth nonlinear terms the well known 1971 result by M.G. Crandall and P.H. Rabinowitz concerning differentiable nonlinear terms. We point out that, in general, there is a lacking of uniquenees of solutions for the associated parabolic problem. Nevertheless, for nondegenerate solutions (crossing the discontinuity value of u in a transversal way) the comparison principle and the uniqueness of solutions hold. The instability is obtained trough a linearization process leading to an eigenvalue problem in which a Dirac delta distribution appears as a coefficient of the differential operator. The stability proof uses a suitable change of variables, the continuuity of the bifurcation branch and the comparison principle for nondegenerate solutions of the parabolic problem.

Attractors for a random evolution equation with infinite memory: Theoretical results
Tomás Caraballo , María J. Garrido-Atienza , Björn Schmalfuss and  José Valero
2017, 22(5): 1779-1800 doi: 10.3934/dcdsb.2017106 +[Abstract](45) +[HTML](3) +[PDF](516.5KB)

The long-time behavior of solutions (more precisely, the existence of random pullback attractors) for an integro-differential parabolic equation of diffusion type with memory terms, more particularly with terms containing both finite and infinite delays, as well as some kind of randomness, is analyzed in this paper. We impose general assumptions not ensuring uniqueness of solutions, which implies that the theory of multivalued dynamical system has to be used. Furthermore, the emphasis is put on the existence of random pullback attractors by exploiting the techniques of the theory of multivalued nonautonomous/random dynamical systems.

Global attractor for a nonlocal p-Laplacian equation without uniqueness of solution
Tomás Caraballo , Marta Herrera-Cobos and  Pedro Marín-Rubio
2017, 22(5): 1801-1816 doi: 10.3934/dcdsb.2017107 +[Abstract](49) +[HTML](8) +[PDF](412.2KB)

In this paper, the existence of solution for a \begin{document} $p$ \end{document}-Laplacian parabolic equation with nonlocal diffusion is established. To do this, we make use of a change of variable which transforms the original problem into a nonlocal one but with local diffusion. Since the uniqueness of solution is unknown, the asymptotic behaviour of the solutions is analysed in a multi-valued framework. Namely, the existence of the compact global attractor in \begin{document} $L^2(Ω)$ \end{document} is ensured.

Asymptotic behaviour of a non-classical and non-autonomous diffusion equation containing some hereditary characteristic
Tomás Caraballo , Antonio M. Márquez-Durán and  Rivero Felipe
2017, 22(5): 1817-1833 doi: 10.3934/dcdsb.2017108 +[Abstract](40) +[HTML](1) +[PDF](471.2KB)

Our aim in this work is the study of the existence and uniqueness of solutions for a non-classical and non-autonomous diffusion equation containing infinite delay terms. We also analyze the asymptotic behaviour of the system in the pullback sense and, under suitable additional conditions, we obtain global exponential decay of the solutions of the evolutionary problem to stationary solutions.

Strong trajectory and global $\mathbf{W^{1,p}}$-attractors for the damped-driven Euler system in $\mathbb R^2$
Vladimir Chepyzhov , Alexei Ilyin and  Sergey Zelik
2017, 22(5): 1835-1855 doi: 10.3934/dcdsb.2017109 +[Abstract](38) +[HTML](6) +[PDF](505.8KB)

We consider the damped and driven two-dimensional Euler equations in the plane with weak solutions having finite energy and enstrophy. We show that these (possibly non-unique) solutions satisfy the energy and enstrophy equality. It is shown that this system has a strong global and a strong trajectory attractor in the Sobolev space \begin{document} $H^1$ \end{document}. A similar result on the strong attraction holds in the spaces \begin{document} $H^1\cap\{u:\ \|\text{curl}\, u\|_{L^p}<∞\}$ \end{document} for \begin{document} $p≥2$ \end{document}.

Smooth attractors for weak solutions of the SQG equation with critical dissipation
Michele Coti Zelati and  Piotr Kalita
2017, 22(5): 1857-1873 doi: 10.3934/dcdsb.2017110 +[Abstract](36) +[HTML](5) +[PDF](457.5KB)

We consider the evolution of weak vanishing viscosity solutions to the critically dissipative surface quasi-geostrophic equation. Due to the possible non-uniqueness of solutions, we rephrase the problem as a set-valued dynamical system and prove the existence of a global attractor of optimal Sobolev regularity. To achieve this, we derive a new Sobolev estimate involving Hölder norms, which complement the existing estimates based on commutator analysis.

Global attractors of impulsive parabolic inclusions
Sergey Dashkovskiy , Oleksiy Kapustyan and  Iryna Romaniuk
2017, 22(5): 1875-1886 doi: 10.3934/dcdsb.2017111 +[Abstract](57) +[HTML](1) +[PDF](389.5KB)

In this work we consider an impulsive multi-valued dynamical system generated by a parabolic inclusion with upper semicontinuous right-hand side \begin{document}$\varepsilon F(y)$\end{document} and with impulsive multi-valued perturbations. Moments of impulses are not fixed and defined by moments of intersection of solutions with some subset of the phase space. We prove that for sufficiently small value of the parameter \begin{document}$\varepsilon>0$\end{document} this system has a global attractor.

Long-time behavior of state functions for climate energy balance model
Nataliia V. Gorban , Olha V. Khomenko , Liliia S. Paliichuk and  Alla M. Tkachuk
2017, 22(5): 1887-1897 doi: 10.3934/dcdsb.2017112 +[Abstract](28) +[HTML](2) +[PDF](420.3KB)

We study the long time behavior of state functions for a climate energy balance model (so-called Budyko Model). Existence and properties of weak solutions, and existence of Lyapunov function are obtained. Existence, structure and regularity properties for global and trajectory attractors are justified.

Regularity of global attractors for reaction-diffusion systems with no more than quadratic growth
Oleksiy V. Kapustyan , Pavlo O. Kasyanov and  José Valero
2017, 22(5): 1899-1908 doi: 10.3934/dcdsb.2017113 +[Abstract](30) +[HTML](1) +[PDF](394.0KB)

We consider reaction-diffusion systems in a three-dimensional bounded domain under standard dissipativity conditions and quadratic growth conditions. No smoothness or monotonicity conditions are assumed. We prove that every weak solution is regular and use this fact to show that the global attractor of the corresponding multi-valued semiflow is compact in the space \begin{document}$(H_{0}^{1} (Ω))^{N}$\end{document}.

Pullback attractors of reaction-diffusion inclusions with space-dependent delay
Peter E. Kloeden and  Thomas Lorenz
2017, 22(5): 1909-1964 doi: 10.3934/dcdsb.2017114 +[Abstract](53) +[HTML](2) +[PDF](823.6KB)

Inspired by biological phenomena with effects of switching off (maybe just for a while), we investigate non-autonomous reaction-diffusion inclusions whose multi-valued reaction term may depend on the essential supremum over a time interval in the recent past (but) pointwise in space. The focus is on sufficient conditions for the existence of pullback attractors. If the multi-valued reaction term satisfies a form of inclusion principle standard tools for non-autonomous dynamical systems in metric spaces can be applied and provide new results (even) for infinite time intervals of delay. More challenging is the case without assuming such a monotonicity assumption. Then we consider the parabolic differential inclusion with the time interval of delay depending on space and extend the approaches of norm-to-weak semigroups to a purely metric setting. This provides completely new tools for proving pullback attractors of non-autonomous dynamical systems in metric spaces.

Topological stability in set-valued dynamics
Roger Metzger , Carlos Arnoldo Morales Rojas and  Phillipe Thieullen
2017, 22(5): 1965-1975 doi: 10.3934/dcdsb.2017115 +[Abstract](55) +[HTML](10) +[PDF](361.3KB)

We propose a definition of topological stability for set-valued maps. We prove that a single-valued map which is topologically stable in the set-valued sense is topologically stable in the classical sense [14]. Next, we prove that every upper semicontinuous closed-valued map which is positively expansive [15] and satisfies the positive pseudo-orbit tracing property [9] is topologically stable. Finally, we prove that every topologically stable set-valued map of a compact metric space has the positive pseudo-orbit tracing property and the periodic points are dense in the nonwandering set. These results extend the classical single-valued ones in [1] and [14].

On practical stability of differential inclusions using Lyapunov functions
Volodymyr Pichkur
2017, 22(5): 1977-1986 doi: 10.3934/dcdsb.2017116 +[Abstract](28) +[HTML](1) +[PDF](331.8KB)

In this paper we consider the problem of practical stability for differential inclusions. We prove the necessary and sufficient conditions using Lyapunov functions. Then we solve the practical stability problem of linear differential inclusion with ellipsoidal righthand part and ellipsoidal initial data set. In the last section we apply the main result of this paper to the problem of practical stabilization.

The averaging of fuzzy hyperbolic differential inclusions
Andrej V. Plotnikov , Tatyana A. Komleva and  Liliya I. Plotnikova
2017, 22(5): 1987-1998 doi: 10.3934/dcdsb.2017117 +[Abstract](44) +[HTML](2) +[PDF](443.1KB)

In this paper the substantiation of a possibility of application of partial averaging method for hyperbolic differential inclusions with the fuzzy right-hand side with the small parameters is considered.

Averaging of fuzzy integral equations
Natalia Skripnik
2017, 22(5): 1999-2010 doi: 10.3934/dcdsb.2017118 +[Abstract](36) +[HTML](3) +[PDF](338.3KB)

The integral equations are encountered in various fields of science and in numerous applications, including elasticity, plasticity, heat and mass transfer, oscillation theory, fluid dynamics, filtration theory, electrostatics, electrodynamics, biomechanics, game theory, control, queuing theory, electrical engineering, economics, and medicine. In this paper the fuzzy integral equation is considered and the existence and uniqueness theorem, the theorem of continuous dependence on the right-hand side and initial fuzzy set are proved. Also the possibility of using the scheme of full averaging for fuzzy integral equation with a small parameter is considered.

Multivalued non-autonomous random dynamical systems for wave equations without uniqueness
Bixiang Wang
2017, 22(5): 2011-2051 doi: 10.3934/dcdsb.2017119 +[Abstract](33) +[HTML](1) +[PDF](717.4KB)

This paper deals with the multivalued non-autonomous random dynamical system generated by the non-autonomous stochastic wave equations on unbounded domains, which has a non-Lipschitz nonlinearity with critical exponent in the three dimensional case. We introduce the concept of weak upper semicontinuity of multivalued functions and use such continuity to prove the measurability of multivalued functions from a metric space to a separable Banach space. By this approach, we show the measurability of pullback attractors of the multivalued random dynamical system of the wave equations regardless of the completeness of the underlying probability space. The asymptotic compactness of solutions is proved by the method of energy equations, and the difficulty caused by the non-compactness of Sobolev embeddings on $\mathbb{R}^n$ is overcome by the uniform estimates on the tails of solutions.

Uniform global attractors for non-autonomous dissipative dynamical systems
Michael Zgurovsky , Mark Gluzman , Nataliia Gorban , Pavlo Kasyanov , Liliia Paliichuk and  Olha Khomenko
2017, 22(5): 2053-2065 doi: 10.3934/dcdsb.2017120 +[Abstract](35) +[HTML](1) +[PDF](417.9KB)

In this paper we consider sufficient conditions for the existence of uniform compact global attractor for non-autonomous dynamical systems in special classes of infinite-dimensional phase spaces. The obtained generalizations allow us to avoid the restrictive compactness assumptions on the space of shifts of non-autonomous terms in particular evolution problems. The results are applied to several evolution inclusions.

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