Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 11, pp. 1-20.

Title: Long-time dynamics and upper-semicontinuity of attractors for a porous-elastic system with nonlinear localized damping

Authors: Mauro L. Santos (Federal Univ. of Para, Belem, PA, Brazil)
         Mirelson M. Freitas (Univ. of Brasilia, Brazil)
         Ronal Q. Caljaro (Federal Univ. of Para, Belem, PA, Brazil)
  
Abstract:
In this article we consider a one-dimensional porous-elastic system 
with nonlinear localized damping acting in an arbitrarily small region 
of the interval under consideration. We prove the existence of  a 
smooth global attractor with finite fractal dimension 
and the existence of exponential attractors  via 
quasi-stability theory recently proposed by Chueshov and Lasiecka. 
We also prove  the continuity of the attractors with respect to two 
parameters  in a residual dense set. Finally, we prove that the family 
of global attractors is upper-semicontinuous with respect to small
perturbations of external forces. These aspects were not previously 
considered for porous-elastic system with localized damping.


Submitted April 12, 2024. Published February 03, 2025.
Math Subject Classifications: 35B40, 35B41, 37L30, 35L75.
Key Words: Porous-elastic system; nonlinear localized damping;  quasi-stability;
  global attractor; upper-semicontinuity.