Global attractor for damped abstract nonlinear hyperbolic systems
In this work, the author considers the following nonlinear second order hyperbolic systems with damping: \[ w_{tt}+A_1 w+A_2w_t +N^\ast g(Nw)= 0, \quad w(0)=\phi_0, \quad w_t(0)=\phi_1, \] where \(A_1, A_2\) and \(N\) are unbounded linear operators, \(N^*\) is the dual of \(N\) and \(g\) is a continuous nonlinear operator in a complex separable Hilbert space \(H\). Using an analysis method based partly on the results of H. T. Banks, D. S. Gilliam and V. I. Shubov on the existence and uniqueness of the weak solutions, the author proves that the problem generates a strong dynamical system which is continuous in the sense that the system continuously depends on the initial data and time. The main results is about the existence of a compact global attractor. The author shows that trajectories of the system are asymptotically compact and the dynamical system is bounded and point dissipative. At last, using the existence of a global Lyapunov function, the author shows that the global attractor is the unstable set of the set of its fixed points.
scientific article; zbMATH DE number 1903496
scientific article