Finite-dimensional attractors for the Kirchhoff equation with a strong dissipation

Abstract

The paper studies the existence of the finite-dimensional global attractors and exponential attractors for the dynamical system associated with the Kirchhoff type equation with a strong dissipation u t t − M ( ‖ ∇ u ‖ 2 ) Δ u − Δ u t + h ( u t ) + g ( u ) = f ( x ) . It proves that the above mentioned dynamical system possesses a global attractor which has finite fractal dimension and an exponential attractor. For application, the fact shows that for the concerned viscoelastic flow the permanent regime (global attractor) can be observed when the excitation starts from any bounded set in phase space, and the dimension of the attractor, that is, the number of degree of freedom of the turbulent phenomenon and thus the level of complexity concerning the flow, is finite.