Mittag-Leffler stability for a fractional viscoelastic telegraph problem

We consider the well-known telegraph problem and replace the time-derivatives by non-integer derivatives. It is proved that the lower order fractional derivative provides enough damping to stabilize the system. Moreover, we show that the extremely weak dissipation produced by a viscelastic term is also able to drive the system to its equilibrium state. Both rates are of Mittag-Leffler type. We recall that this is the case in the integer-order case, that is, for the standard telegraph equation. In the fractional case, some basic rules are no longer valid, and therefore, the situation is more delicate. We prove Mittag-Leffler stability under certain smallness conditions on the relaxation function.