In this paper we investigate the long-term behavior of the solutions of the one-dimensional porous-elasticity problem with porous dissipation and nonlinear feedback force. We prove that the porous-elasticity problem converges to a quasi-static problem for the microvoids motion as a suitable parameter J tends to zero. Finite dimensional global attractor with additional regularity in J is obtained using the recent quasi-stability theory. Finally, we compare the porous-elasticity problem with quasi-static problem, in the sense of the upper-semicontinuity of their attractors as .
I.Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Springer, Berlin, 2015.
2.
I.Chueshov and I.Lasiecka, Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping, Memoirs of AMS, Vol. 912, AMS, Providence, 2008.
3.
I.Chueshov and I.Lasiecka, Von Karman Evolution Equations, Well-Posedness and Long Time Dynamics, Springer Monographs in Mathematics, Springer, New York, 2010.
4.
I.C.M.Eller and I.Lasiecka, On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Commun. Partial Differ. Equ.27 (2002), 1901–1951. doi:10.1081/PDE-120016132.
5.
P.G.Geredeli and I.Lasiecka, Asymptotic analysis and upper semicontinuity with respect to rotational inertia of attractors to von Karman plates with geometrically localized dissipation and critical nonlinearity, Nonlinear Anal.91 (2013), 72–92. doi:10.1016/j.na.2013.06.008.
6.
J.K.Hale and G.Raugel, Upper semicontinuity of the attractor for a singulary perturbed hyperbolic equation, J. Differ. Equ.73 (1988), 197–214. doi:10.1016/0022-0396(88)90104-0.
7.
I.Lasiecka and D.Tataru, Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping, Differential and Integral Equations6 (1993), 507–533.
8.
Z.Liu and S.Zheng, Semigroups Associated with Dissipative Systems, Chapman and Hall/CRC, Boca Raton, 1999.
9.
A.Magaña and R.Quintanilla, On the time decay of solutions in one-dimensional theories of porous materials, International Journal of Solids and Structures43 (2006), 3414–3427. doi:10.1016/j.ijsolstr.2005.06.077.
10.
A.Magaña and R.Quintanilla, On the time decay of solutions in porous-elasticity with quasi-static microvoids, J. Math. Anal. Appl.331 (2007), 617–630. doi:10.1016/j.jmaa.2006.08.086.
11.
J.E.Muñoz Rivera and R.Quintanilla, On the time polynomial decay in elastic solids with voids, J. Math. Anal. Appl.338 (2008), 1296–1309. doi:10.1016/j.jmaa.2007.06.005.
12.
J.Muñoz-Rivera and R.Quintanilla, On the time polynomial decay in elastic solids with voids, J. Math. Anal. Appl.338 (2008), 296–1309.
13.
J.W.Nunziato and S.C.Cowin, A nonlinear theory of elastic materials with voids, Arch. Ration. Mech. Anal.72 (1979), 175–201. doi:10.1007/BF00249363.
14.
J.Prüss, On the spectrum of -semigroups, Trans. AMS28 (1984), 847–857.
15.
M.L.Santos, A.D.S.Campelo and D.S.Almeida Júnior, On the decay rates of porous elastic systems, J. Elast.127 (2017), 79–101. doi:10.1007/s10659-016-9597-y.
16.
J.Simon, Compact sets in the space , Ann. Math. Pura Appl.148 (1987), 5–96.
