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 $\begin{array}{rl}& u_{tt}+{\alpha }_{\mu }{\mathrm{△}}^{2}u-{\mathrm{△}}_{p}u-{\int }_{-\mathrm{\infty }}^t\mu (t-s){\mathrm{△}}^{2}u\left(s\right)\mathrm{d}s\\ & -N\left({\int }_{\mathrm{\Omega }}|\mathrm{\nabla }u\left(t\right){|}^2\mathrm{d}x\right)\mathrm{△}{u}_t+f\left(u\right)=h,\end{array}$defined in a bounded Ω of ${\mathbb{R}}^N$. 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 $f\left(u\right)$ 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 $N(\cdot )$ and possibly degenerate memory term $(\mu \equiv 0)$, which deepen and extend earlier results on the subject.

摘要

在本文中，我们研究了一类具有积分微分阻尼的基尔霍夫模型，该模型由过去历史框架中可能消失的记忆项和非线性非局部强耗散给出$\begin{array}{rl}& {你}_{吨吨}+{\alpha }_{\mu }{\mathrm{△}}^2你-{\mathrm{△}}_p你-{\int }_{-\mathrm{\infty }}^{吨}\mu (吨-s){\mathrm{△}}^2你\left(s\right)\mathrm{d}s\\ & -ñ\left({\int }_{\mathrm{\Omega }}|\mathrm{\nabla }你\left(吨\right){|}^2\mathrm{d}X\right)\mathrm{△}{你}_{吨}+F\left(你\right)=H,\end{array}$在有界 Ω 中定义${\mathbb{R}}^{ñ}$. 我们的主要目标是通过相对过去的历史，通过相应的自主动力系统来展示适定性和长期行为。更准确地说，假设指数p和增长$F\left(你\right)$达到临界范围，建立了具有其几何结构的全局吸引子的适定性和存在性。此外，在亚临界情况下，这种全局吸引子具有有限的分形维数以及轨迹的规律性。还证明了广义分形指数吸引子的一个结果。这些结果适用于广泛的非局部阻尼系数$ñ(\cdot )$和可能退化的记忆词$(\mu \equiv 0)$，这加深和扩展了有关该主题的早期结果。