Long-time Dynamics and Observability in Nonlinear Swelling Porous Viscoelasticity

In this paper, we analyze the longtime properties of global and exponential attractors by observability inequalities for a swelling porous elastic beam with memory kernel <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>g</mml:mi> </mml:math> , the nonlinear frictional damping <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>h</mml:mi> </mml:math> and the nonlinear source terms. Further, we investigate the interaction between a viscoelastic damping <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>g</mml:mi> </mml:math> (without any decaying conditions) more general than the usual one and a nonlinear frictional damping. Under general assumptions on the relaxation function and the nonlinear feedback, we prove the well-posedness of the problem and the existence of an absorbing set. The existence of global and exponential attractors with finite fractal dimensions is achieved from observability inequalities and by the useful properties of asymptotic smoothness and quasi-stability (the latter needs some stronger condition on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>h</mml:mi> </mml:math> ). This paper is the first to study the dynamics of the nonlinear swelling porous viscoelastic beam from observability inequalities. The results obtained generalize and improve some previous results and can also be used in control theory.