Qualitative Study of some Classes of Evolution Problems
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Date
2025
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20 August 1955 university of Skikda
Abstract
In this thesis, we present our results about the existence and exponential decay of certain
classes of evolution problems. The first problem focuses on a one-dimensional Lord-Schulman
thermoelastic system coupled with porous damping and time delay. The heat conduction in this
type of systems is described by the Lord–Shulman theory. The second problem focuses on the
swelling porous system with the Gurtin-Pipkin thermal effect as the only source of damping
with delay. In general, the study shows that the dissipation obtained from the Guertin-Pipkin
heat law is sufficient to stabilize the system exponentially, regardless of the system parameters.
The third problem focuses on a one-dimensional swelling porous-heat system with time-varying
delay in a bounded domain under Dirichlet boundary conditions, with thermodiffusion effects
and frictional damping. Overall, using the semi-group approach, the variable norm technique
of T. Kato, and the appropriate assumptions on the weight of delay, we establish the wellposedness of the considered systems. Then, we show that our systems are exponentially stable
by employing an appropriate Lyapunov functional. We point out that our results are achieved
without taking into account that we have the equal of speeds
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Keywords
Swelling porous systems