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Longtime behavior of the Kirchhoff type equation with strong damping on N . (English) Zbl 1208.35147

The author considers a Kirchhoff type equation with strong damping written as u tt -M(u 2 )Δu-Δu t +u+u t +g(x,u)=f(x), posed in N × + . Here M(s)=1+s m/2 , m1, f belongs to L 2 ( N ) and g is a nonlinear function which satisfies at least some p-growth condition with respect to s.

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 C(0,;H 2 )C 1 (0,;H 1 )H 1 (0,;H 2 )H 2 (0,;L 2 ). Assuming some further hypotheses on the first-order partial derivatives of g, the author proves that the underlying continuous semi-group of this problem has a global attractor in H 2 H 1 . 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 T>0.

Then the author proves a uniform bound on u(t) H 2 2 +u t (t) H 1 2 , 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)].

MSC:
35Q74PDEs in connection with mechanics of deformable solids
35L70Nonlinear second-order hyperbolic equations