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- ItemSmooth attractors for strongly damped wave equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Pata, Vittorino; Zelik, SergeyThis 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.
- ItemGlobal and exponential attractors for 3-D wave equations with displacement dependent damping(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Pata, Vittorino; Zelik, SergeyA 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.
- ItemA remark on the weakly damped wave equation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Pata, Vittorino; Zelik, SergeyIn 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.
- ItemAttractors and their regularity for 2-D wave equations with nonlinear damping(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Pata, Vittorino; Zelik, SergeyWe 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.
- ItemGlobal attractors for semigroups of closed operators(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Pata, Vittorino; Zelik, SergeyIn 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.
- ItemAttractors for semilinear equations of viscoelasticity with very low disspation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Gatti, Stefania; Miranville, Alain; Pata, Vittorino; Zelik, SergeyWe 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.