This paper studies a fractional differential inequality involving a new fractional derivative (Hilfer‐Hadamard type) with a polynomial source term. We obtain an exponent for which there does not exist any global solution for the problem. We also provide an example to show the existence of solutions in a wider space for some exponents.

We consider two identical beams on top of each other with an adhesive in between. A considerable natural slip occurs in the structure and will not be ignored as was done in the previous investigations. In this work we take into account this slip and prove that we can stabilize the system in an exponential manner using boundary controls. The model consists of three coupled equations. The first two are related to the well-known Timoshenko system, and the third one describes the dynamic of the slip. Our result improves the few existing similar works in the literature.

Two exponential stabilization results are proved for a vibrating structure subject to an interfacial slip. More precisely, the structure consists of two identical beams of Timoshenko type and clamped together but allowing for a longitudinal movement between the layers. We will stabilize the system through a transverse friction and also through a viscoelastic damping.