Robustness of nonautonomous attractors for a family of nonlocal reaction–diffusion equations without uniqueness

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• 作者：Tomás Caraballo ; Marta Herrera-Cobos ; Pedro Marín-Rubio
• 关键词：Nonlocal diffusion ; Reaction–diffusion equations without uniqueness ; Pullback attractors ; Upper semicontinuity of attractors ; Multi ; valued dynamical systems
• 刊名：Nonlinear Dynamics
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：84
• 期：1
• 页码：35-50
• 全文大小：587 KB
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• 作者单位：Tomás Caraballo (1)
  Marta Herrera-Cobos (1)
  Pedro Marín-Rubio (1)
  
  1. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080, Seville, Spain
  
• 刊物类别：Engineering
• 刊物主题：Vibration, Dynamical Systems and Control
  Mechanics
  Mechanical Engineering
  Automotive and Aerospace Engineering and Traffic
  
• 出版者：Springer Netherlands
• ISSN：1573-269X

文摘

In this paper, we consider a nonautonomous nonlocal reaction–diffusion equation with a small perturbation in the nonlocal diffusion term and the nonautonomous force. Under the assumptions imposed on the viscosity function, the uniqueness of weak solutions cannot be guaranteed. In this multi-valued framework, the existence of weak solutions and minimal pullback attractors in the \(L^2\)-norm is analysed. In addition, some relationships between the attractors of the universe of fixed bounded sets and those associated to a universe given by a tempered condition are established. Finally, the upper semicontinuity property of pullback attractors w.r.t. the parameter is proved. Indeed, under suitable assumptions, we prove that the family of pullback attractors converges to the corresponding global compact attractor associated with the autonomous nonlocal limit problem when the parameter goes to zero.