Micro-inertia effects on existence of attractors for Form II Mindlin’s strain gradient viscoelastic plate

In this paper, we study the long-time dynamics of solutions to strain gradient viscoelastic plates equations taking into account micro-inertia effects and subjected to three different types of external nonlinear terms. First we derive briefly the equations of strain gradient viscoelastic plate corresponding to anti-plane shear deformations which is assumed to be consistent with the Mindlin Form II. Based on semigroup theory, we prove the existence and uniqueness of global solution. Then, we show that the existence of finite dimensional global attractors depends on the value of the micro-inertia parameter (whether or not zero) and on the assumptions on the external non-linearities. Sufficient conditions on existence of exponential and global minimal attractors can be deduced. Finally, we show the upper-semicontinuity of global attractors with respect to the micro-inertia parameter.