The nonlinear damped wave equation
is considered on a bounded domain imposing Dirichlet boundary conditions. For the nonlinearity it is assumed that , with the first eigenvalue of . In addition, the growth condition is supposed if , whereas may grow exponentially for .
The main result of the paper asserts that the equation has a connected global attractor in , identifying with . It is further shown that for each global orbit in the attractor the - resp. -limit set is a connected subset of the critical points of the Lyapunov functional , where . If the set of critical points is totally disconnected, then the solutions do not only approach the attractor as a set, but they converge to an individual critical point as . The proofs rely on the application of suitable abstract results concerning the existence of attractors.