Partial Differential Equations and Optimization
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Date
2022
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Université de M'sila
Abstract
In this thesis, we consider some viscoelastic problems for a strongly elliptic operator
of second order with variable coefficients in bounded domains. A review of the recent studies on
the generalized thermoelasticity theories and their associated modified models is also presented. In
this regard, we investigate several coupled systems with mixed Dirichlet-Neumann boundary
conditions and boundary interaction feedback. The coupling is via the acoustic boundary conditions
on a portion of the boundary. Using some interesting methods of mathematical analysis as,
semigroup theory, Schauder fixed point, Faedo-Galerkin approach, and compactness estimates, to
show the local and global existence of energy-associated solutions. In addition, taking into account
the Gearhart-Prüss theorem, the exponential stability of the corresponding semigroup is concluded.
Moreover, under a very wider class of generality of relaxation functions, we establish several
general decay results using the energy methods
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Keywords
Acoustic boundary conditions, Exponential stability, General decay, Global existence of solution, Thermoelastic effect, Viscoelastic damping