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.

2008, Sprekels, Jürgen, Wu, Hao

We consider a semilinear parabolic equation subject to a nonlinear dynamical boundary condition that is related to the so-called Wentzell boundary condition. First, we prove the existence and uniqueness of global solutions as well as the existence of a global attractor. Then we derive a suitable Lojasiewicz-Simon type inequality to show the convergence of global solutions to single steady states as time tends to infinity under the assumption that the nonlinear terms f, g are real analytic. Moreover, we provide an estimate for the convergence rate.

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.