Global attractors of impulsive parabolic inclusions
Sergey Dashkovskiy Oleksiy Kapustyan Iryna Romaniuk

In this work we consider an impulsive multi-valued dynamical system generated by a parabolic inclusion with upper semicontinuous right-hand side $\varepsilon F(y)$ 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 $\varepsilon>0$ this system has a global attractor.

keywords: Global attractor multi-valued dynamical system impulsive dynamical system parabolic inclusion
Regularity of global attractors for reaction-diffusion systems with no more than quadratic growth
Oleksiy V. Kapustyan Pavlo O. Kasyanov José Valero

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 $(H_{0}^{1} (Ω))^{N}$.

keywords: Reaction-diffusion system multivalued dynamical system global attractor
Regular solutions and global attractors for reaction-diffusion systems without uniqueness
Oleksiy V. Kapustyan Pavlo O. Kasyanov José Valero
In this paper we study the structural properties of global attractors of multi-valued semiflows generated by regular solutions of reaction-diffusion system without uniqueness of the Cauchy problem. Under additional gradient-like condition on the nonlinear term we prove that the global attractor coincides with the unstable manifold of the set of stationary points, and with the stable one when we consider only bounded complete trajectories. As an example we consider a generalized Fitz-Hugh-Nagumo system. We also suggest additional conditions, which provide that the global attractor is a bounded set in $(L^\infty(\Omega))^N$ and compact in $(H_0^1 (\Omega))^N$.
keywords: global attractor multi-valued dynamical system Reaction-diffusion system unstable manifold Fitz-Hugh-Nagumo system.
Preface to the special issue "Finite and infinite dimensional multivalued dynamical systems"
María J. Garrido-Atienza Oleksiy V. Kapustyan José Valero
Structure and regularity of the global attractor of a reaction-diffusion equation with non-smooth nonlinear term
Oleksiy V. Kapustyan Pavlo O. Kasyanov José Valero
In this paper we study the structure of the global attractor for a reaction-diffusion equation in which uniqueness of the Cauchy problem is not guarantied. We prove that the global attractor can be characterized using either the unstable manifold of the set of stationary points or the stable one but considering in this last case only solutions in the set of bounded complete trajectories.
keywords: unstable manifolds asymptotic behaviour. set-valued dynamical system Reaction-diffusion equations global attractor
Strong attractors for vanishing viscosity approximations of non-Newtonian suspension flows
Oleksiy V. Kapustyan Pavlo O. Kasyanov José Valero Michael Z. Zgurovsky

In this paper we prove the existence of global attractors in the strong topology of the phase space for semiflows generated by vanishing viscosity approximations of some class of complex fluids. We also show that the attractors tend to the set of all complete bounded trajectories of the original problem when the parameter of the approximations goes to zero.

keywords: Non-Newtonian fluids parabolic equations global attractors infinite-dimensional dynamical systems

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