Stability results of some distributed systems involving Mindlin‐Timoshenko plates in the plane

Abstract

In , Belkacem and Kasimov studied the stability of an one‐dimensional Timoshenko system in with one distributed temperature or Cattaneo dissipation damping. They proved that the heat dissipation alone is sufficient to stabilize the system. But there is a difference between the Timoshenko system in and its analogous system in . For this reason, the stability results are no longer the same and of intrinsic difference. In this paper, we consider the stability of some distributed systems involving Mindlin‐Timoshenko plate in the plane. If the plate is subject to two internal distributed damping then, using a direct approach based on the Fourier transform, we establish a polynomial energy decay rate for initial data in . In the case of indirect internal stability, when only one among the two equations is effectively damped while the second is indirectly damped through the coupling, we have two different situations. To be more precise, if the equation of the displacement in the vertical direction of the plate is only damped then, the system is unstable. Next, when the control is acting on the equation of the angles of rotation of a filament of the plate, no decay can be proved but our conjecture is a polynomial stability. Finally, unlike the one‐dimensional case, we show that, under a heat conduction (by Fourier or Cattaneo law), the plate is unstable.