Stability of Solutions in a Fractured Rock Reservoir

Of concern is a problem which arises in oil recovery simulation in fractured rock reservoir. This problem models the fluid flow in a fissured media where the history of the flow must be taken into account. The model has been already derived and analyzed to some extent in few published papers. The well-posedness is now well understood. However, the asymptotic behavior and the stability of the system have been discussed by very few researchers and only for special classes of kernels. Namely, for kernels of exponential type, completely monotone kernels and for kernels of positive type. In this proposal we would like to contribute by filling part of this gap. It is our intention to consider wider classes of kernels so as to include more practical and theoretical situations. We would like to study the asymptotic behavior and stability of solutions of the following problem We will consider other classes of kernels than the ones already treated in the literature. In particular we will start by adapting some classes that have been discussed in the literature but in other contexts such as in viscoelasticity.