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Longtime behavior of the Kirchhoff type equation with strong damping on ${ℝ}^{N}$. (English) Zbl 1208.35147

The author considers a Kirchhoff type equation with strong damping written as ${u}_{tt}-M\left({∥\nabla u∥}^{2}\right){\Delta }u-{\Delta }{u}_{t}+u+{u}_{t}+g\left(x,u\right)=f\left(x\right)$, posed in ${ℝ}^{N}×{ℝ}^{+}$. Here $M\left(s\right)=1+{s}^{m/2}$, $m\ge 1$, $f$ belongs to ${L}^{2}\left({ℝ}^{N}\right)$ 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 $C\left(\left[0,\infty \right);{H}^{2}\right)\cap {C}^{1}\left(\left[0,\infty \right);{H}^{1}\right)\cap {H}^{1}\left(\left[0,\infty \right);{H}^{2}\right)\cap {H}^{2}\left(\left[0,\infty \right);{L}^{2}\right)$. 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 ${H}^{2}\cap {H}^{1}$. 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 ${∥u\left(t\right)∥}_{{H}^{2}}^{2}+{∥{u}_{t}\left(t\right)∥}_{{H}^{1}}^{2}$, 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:
 35Q74 PDEs in connection with mechanics of deformable solids 35L70 Nonlinear second-order hyperbolic equations