The author considers a Kirchhoff type equation with strong damping written as , posed in . Here , , belongs to and is a nonlinear function which satisfies at least some -growth condition with respect to .
The first result of the paper proves an existence and uniqueness result for a solution of this problem, which is global in time and belongs to . Assuming some further hypotheses on the first-order partial derivatives of , the author proves that the underlying continuous semi-group of this problem has a global attractor in . The proof of the existence result follows the lines of the paper by M. Nakao and the author [J. Differ. Equations 227, No. 1, 204–229 (2006; Zbl 1096.35024)]. It is based on Banach’s fixed point theorem in an appropriate space-time functional space for .
Then the author proves a uniform bound on , which implies that the unique solution is global in time. The proof of the main result is based on a decomposition of the semi-group and on the construction of estimates. The paper ends with the proof of some further properties of the semi-group: it has finite fractal and Hausdorff dimension. The author here quotes a result from J. Hale [Asymptotic behavior of dissipative systems (Mathematical Surveys and Monographs, 25. Providence, RI: American Mathematical Society) (1988; Zbl 0642.58013)].