Mittag‐Leffler stability for a fractional viscoelastic telegraph problem

Abstract

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 In the fractional case, some basic rules are no longer valid, and therefore, the situation is more delicate.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.