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Quasi‐stability and upper semicontinuity for coupled parabolic equations with memory
Author(s) -
Aouadi Moncef
Publication year - 2020
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/sapm.12331
Subject(s) - attractor , mathematics , parabolic partial differential equation , nonlinear system , property (philosophy) , stability (learning theory) , metric (unit) , mathematical analysis , set (abstract data type) , upper and lower bounds , metric space , heat equation , partial differential equation , physics , computer science , philosophy , operations management , epistemology , quantum mechanics , machine learning , economics , programming language
This current study deals with the long‐time dynamics of a nonlinear system of coupled parabolic equations with memory. The system describes the thermodiffusion phenomenon where the fluxes of mass diffusion and heat depend on the past history of the chemical potential and the temperature gradients, respectively, according to Gurtin‐Pipkin's law. Inspired by the works of Chueshov and Lasiecka on the property of quasi‐stability of dynamic systems, we prove this property for the problem considered in this study. This property allows us to analyze certain properties of global and exponential attractors in a more efficient and practical way. This approach is applied for the first time for coupled parabolic equations. We analyze the continuity of global attractors with respect to a pair of parameters in a residual dense set and their upper semicontinuity in a complete metric space. Finally, we analyze the upper semicontinuity of global attractors with respect to small perturbations of the damping terms.