Attractors and their properties for a class of Kirchhoff models with integro-differential damping

• Published in: Applicable analysis Vol. 101; no. 9; pp. 3284 - 3307
• Main Authors: Liu, Gongwei, Silva, Marcio A. Jorge
• Format: Journal Article
• Language: English
• Published: Abingdon Taylor & Francis 13-06-2022; Taylor & Francis Ltd
• Subjects: Attractors (mathematics); Damping; Fractal geometry; Fractals; global attractor; long-time dynamics; memory; nonlocal damping; p-Laplacian; Well posed problems
• ISSN: 0003-6811; 1563-504X
• DOI: 10.1080/00036811.2020.1846722

In this paper, we investigate a class of Kirchhoff models with integro-differential damping given by a possibly vanishing memory term in a past history framework and a nonlinear nonlocal strong dissipation
defined in a bounded Ω of
. Our main goal is to show the well-posedness and the long-time behavior through the corresponding autonomous dynamical system by regarding the relative past history. More precisely, under the assumptions that the exponent p and the growth of
are up to the critical range, the well-posedness and the existence of a global attractor with its geometrical structure are established. Furthermore, in the subcritical case, such a global attractor has finite fractal dimensions as well as regularity of trajectories. A result on generalized fractal exponential attractor is also proved. These results are presented for a wide class of nonlocal damping coefficient
and possibly degenerate memory term
, which deepen and extend earlier results on the subject.