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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 dissipationdefined 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 ofare 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 coefficientand possibly degenerate memory term, which deepen and extend earlier results on the subject.

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