Global existence and energy decay for a transmission problem under a boundary fractional derivative type

The paper considers the effects of fractional derivative with a high degree of accuracy in the boundary conditions for the transmission problem. It is shown that the existence and uniqueness of the solutions for the transmission problem in a bounded domain with a boundary condition given by a fractional term in the second equation are guaranteed by using the semigroup theory. Under an appropriate assumptions on the transmission conditions and boundary conditions, we also discuss the exponential and strong stability of solution by also introducing the theory of semigroups.