The weekly Wednesday seminar on 03 December 2014, 10.00 a.m., will feature Professor Varga Kalantarov (Department of Mathematics, Koç University, Turkey) who will talk about «Finite-dimensional Asymptotic Behavior of Dynamical Systems Generated by Strongly Damped Nonlinear Wave Equations».

In this talk the problem of finite-dimensional asymptotic behavior (in time) of dynamical systems generated by initial boundary value problems for nonlinear strongly damped wave equation of the form

u_tt + f(u)u_t – bΔu_t – Δu + g(u) = h,      x ∈Ω,     t > 0   (1)

will be discussed. Here   Ω ⊂ R³ is a bounded domain with smooth boundary, b > 0 is a given parameter, h, f (•), g (•) are given functions.

It will be show that under the standard smoothness and dissipativity restrictions on the nonlinear terms f(u) and g(u) the initial boundary value problem under the homogeneous Dirichlet boundary condition for the equation (1) is globally well-posed in the class of sufficiently regular solutions. We will show that the semigroup generated by the initial boundary value problem for the equation (1) has a global attractor in the corresponding phase space.

In the case when  f(u) = 0 we show that the asymptotic behavior (in time) of solutions of the initial boundary value problem for (1) is determined by finitely many Fourier modes.

These results are obtained for the nonlinearities  f(u) and g(u) of an arbitrary polynomial growth and without the assumption that the considered problem has a global Lyapunov function.