Electron. J. Differential Equations, Vol. 2017 (2017), No. 303, pp. 1-16.

Stability of solutions for a heat equation with memory

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.

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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
Sebti Kerbal
Department of Mathematics and Statistics
Sultan Qaboos University, P.O. Box 36
Al-Khodh 123, Muscat, Oman
email: skerbal@squ.edu.om
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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