Nasser-eddine Tatar, Sebti Kerbal, Asma Al-Ghassani
Abstract:
This article concerns the heat equation with a memory term in the form of a
time-convolution of a kernel with the time-derivative of the state.
This problem appears in oil recovery simulation in fractured rock reservoir.
It models the fluid flow in a fissured media where the history of the flow
must be taken into account. Most of the existing papers on related works
treat only (in addition to the well-posedness which is by now well understood
in various spaces) the convergence of solutions to the equilibrium state
without establishing any decay rate. In the present work we shall improve
and extend the existing results. In addition to weakening the conditions
on the kernel leading to exponential decay, we extend the decay rate to
a general one.
Submitted September 30, 2017. Published December 11, 2017.
Math Subject Classifications: 93D20, 35K20, 35K05.
Key Words: Heat equation; memory term; exponential stability;
fractured reservoir; fissure media.
Show me the PDF file (268 KB), TEX file for this article.
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Nasser-eddine Tatar Department of Mathematics and Statistics King Fahd University of Petroleum and Minerals Dhahran 31261, Saudi Arabia email: tatarn@kfupm.edu.sa |
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Sebti Kerbal Department of Mathematics and Statistics Sultan Qaboos University, P.O. Box 36 Al-Khodh 123, Muscat, Oman email: skerbal@squ.edu.om |
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Asma Al-Ghassani Department of Mathematics and Statistics Sultan Qaboos University, P.O. Box 36 Al-Khodh 123, Muscat, Oman email: ghassani@squ.edu.om |
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