Attractors and their properties for a class of Kirchhoff models with integro-differential damping. Issue 9 (13th June 2022) - Liu, Gongwei; Silva, Marcio A. Jorge
ABSTRACT: 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 1 u t t + α μ △ 2 u − △ p u − ∫ − ∞ t μ ( t − s ) △ 2 u ( s ) d s − N ∫ Ω | ∇ u ( t ) | 2 d x △ u t + f ( u ) = h, defined in a bounded Ω of R N . 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 f ( u ) 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 N ( ⋅ ) and possibly degenerate memory term ( μ ≡ 0 ), which deepen and extend earlier results on the subject.