2006, Turaev, Dmitry, Vladimirov, Andrei, Zelik, Sergey

A new type of stable dynamic bound state of dissipative localized structures is found. It is characterized by chaotic oscillations of distance between the localized structures, their phase difference, and the center of mass velocity.

2006, Pata, Vittorino, Zelik, Sergey

A weakly damped wave equation in the three-dimensional (3-D) space with a damping coefficient depending on the displacement is studied. This equation is shown to generate a dissipative semigroup in the energy phase space, which possesses finite-dimensional global and exponential attractors in a slightly weaker topology.

2006, Pata, Vittorino, Zelik, Sergey

In this note, we establish a general result on the existence of global attractors for semigroups S(t) of operators acting on a Banach space X, where the strong continuity S(t) [Elemenz von] C(X,X) is replaced by the much weaker requirement that S(t) be a closed map.

2006, Zelik, Sergey, Mielke, Alexander

We study semilinear parabolic systems on the full space Rn that admit a family of exponentially decaying pulse-like steady states obtained via translations. The multi-pulse solutions under consideration look like the sum of infinitely many such pulses which are well separated. We prove a global center-manifold reduction theorem for the temporal evolution of such multi-pulse solutions and show that the dynamics of these solutions can be described by an infinite systems of ODEs for the positions of the pulses. As an application of the developed theory, we verify the existence of Sinai-Bunimovich space-time chaos in 1D space-time periodically forced Swift-Hohenberg equation.

2006, Pata, Vittorino, Zelik, Sergey

In this short note we present a direct method to establish the optimal regularity of the attractor for the semilinear damped wave equation with a nonlinearity of critical growth.

2011, Turaev, Dmitry, Vladimirov, Andrei G., Zelik, Sergey

We study interaction of well-separated oscillating localized structures (oscillons). We show that oscillons emit weakly decaying dispersive waves, which leads to formation of bound states due to subharmonic synchronization. We also show that in optical applications the Andronov-Hopf bifurcation of stationary localized structures leads to a drastic increase in their interaction strength.

2006, Pata, Vittorino, Zelik, Sergey

This paper is concerned with the semilinear strongly damped wave equation ptt u-Delta pt u-Delta u+varphi(u)=f. The existence of compact global attractors of optimal regularity is proved for nonlinearities phi of critical and supercritical growth.

2006, Pata, Vittorino, Zelik, Sergey

We address the study of a weakly damped wave equation in space-dimension two, with a damping coefficient depending on the displacement. The equation is shown to generate a semigroup possessing a compact global attractor of optimal regularity, as well as an exponential attractor.

2006, Gatti, Stefania, Miranville, Alain, Pata, Vittorino, Zelik, Sergey

We analyze a differential system arising in the theory of isothermal viscoelasticity. This system is equivalent to an integrodifferential equation of hyperbolic type with a cubic nonlinearity, where the dissipation mechanism is contained only in the convolution integral, accounting for the past history of the displacement. In particular, we consider here a convolution kernel which entails an extremely weak dissipation. In spite of that, we show that the related dynamical system possesses a global attractor of optimal regularity.