Unboundedness for the Euler–Bernoulli beam equation with a fractional boundary dissipation

Abstract

We consider the Euler–Bernoulli beam problem with some boundary controls involving a fractional derivative. The fractional derivative here represents a fractional dissipation of lower order than one. We prove that the classical energy associated to the system is unbounded in presence of a polynomial nonlinearity. In fact, it will be proved that the energy will grow up as an exponential function as time goes to infinity.