Mittag-Leffler stability and Lyapunov stability for a problem arising in porous media

Abstract

A fractional order problem arising in porous media is considered. Well-posedness as well as stability are discussed. Mittag-Leffler stability is proved in case of a strong fractional damping in the displacement component and a fractional frictional one in the volume fraction component. This extends an existing result from the integer-order (second-order) case to the non-integer case. In the absence of the fractional damping in the volume fraction component, it is shown a convergence to zero and a Lyapunov uniform stability.