# American Institute of Mathematical Sciences

November  2016, 15(6): 2301-2328. doi: 10.3934/cpaa.2016038

## Finite dimensional smooth attractor for the Berger plate with dissipation acting on a portion of the boundary

 1 Department of Mathematics, University of Nebraska-Lincoln, Lincoln, Nebraska 68588 2 Department of Mathematics, Faculty of Science, Hacettepe University, Beytepe 06800, Ankara 3 Haceteppe University, Ankara , Turkey

Received  February 2016 Revised  July 2016 Published  September 2016

We consider a (nonlinear) Berger plate in the absence of rotational inertia acted upon by nonlinear boundary dissipation. We take the boundary to have two disjoint components: a clamped (inactive) portion and a controlled portion where the feedback is active via a hinged-type condition. We emphasize the damping acts only in one boundary condition on a portion of the boundary. In [24] this type of boundary damping was considered for a Berger plate on the whole boundary and shown to yield the existence of a compact global attractor. In this work we address the issues arising from damping active only on a portion of the boundary, including deriving a necessary trace estimate for $(\Delta u)\big|_{\Gamma_0}$ and eliminating a geometric condition in [24] which was utilized on the damped portion of the boundary.
Additionally, we use recent techniques in the asymptotic behavior of hyperbolic-like dynamical systems [11, 18] involving a stabilizability" estimate to show that the compact global attractor has finite fractal dimension and exhibits additional regularity beyond that of the state space (for finite energy solutions).
Citation: George Avalos, Pelin G. Geredeli, Justin T. Webster. Finite dimensional smooth attractor for the Berger plate with dissipation acting on a portion of the boundary. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2301-2328. doi: 10.3934/cpaa.2016038
##### References:
 [1] J. P. Aubin, Une théorè de compacité, C.R. Acad. Sci. Paris, 256 (1963), 5042-5044. [2] G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system without mechanical dissipation, Rend. Istit. Mat. Univ. Trieste, 28 (1997), 1-28. [3] G. Avalos and I. Lasiecka, Boundary controllability of thermoelastic plates via the free boundary conditions, SIAM J. Control. Optim., 38 (2000), 337-383. doi: 10.1137/S0363012998339836. [4] A. Babin and M. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992. [5] J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Cont. Dyn. Sys, 10 (2004), 31-52. doi: 10.3934/dcds.2004.10.31. [6] H. M. Berger, A new approach to the analysis of large deflections of plates, J. Appl. Mech., 22 (1955), 465-472. [7] V. V. Bolotin, Nonconservative Problems of Elastic Stability, Pergamon Press, Oxford, 1963. [8] S. C. Brenner and R. Scott, The Mathematical Theory of Finite Element Methods, 15, Springer Science & Business Media, 2008. doi: 10.1007/978-0-387-75934-0. [9] F. Bucci, I. Chueshov and I. Lasiecka, Global attractor for a composite system of nonlinear wave and plate equations, Comm. Pure and Appl. Anal., 6 (2007), 113-140. [10] F. Bucci and I. Chueshov, Long-time dynamics of a coupled system of nonlinear wave and thermoelastic plate equations, Dynam. Sys., 22 (2008), 557-586. doi: 10.3934/dcds.2008.22.557. [11] I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Springer, 2015. doi: 10.1007/978-3-319-22903-4. [12] I. Chueshov, Long-time dynamics of Kirchhoff wave models with strong nonlinear damping, J. Diff. Equs., 252 (2012), 1229-1262. doi: 10.1016/j.jde.2011.08.022. [13] I. Chueshov, Introduction to the Theory of Infinite Dimensional Dissipative Systems, Acta, Kharkov, 1999, in Russian; English translation: Acta, Kharkov, 2002; http://www.emis.de/monographs/Chueshov/ [14] I. Chueshov, M. Eller and I. Lasiecka, Finite dimensionality of the attractor for a semilinear wave equation with nonlinear boundary dissipation, Comm. PDE, 29 (2004), 1847-1976. doi: 10.1081/PDE-200040203. [15] I. Chueshov and I. Lasiecka, Global attractors for von Karman evolutions with a nonlinear boundary dissipation, J. Differ. Equs., 198 (2004), 196-231. doi: 10.1016/j.jde.2003.08.008. [16] I. Chueshov and I. Lasiecka, Long-time behavior of second-order evolutions with nonlinear damping, Memoires of AMS, 195, 2008. doi: 10.1090/memo/0912. [17] I. Chueshov and I. Lasiecka, Long-time dynamics of von Karman semi-flows with non-linear boundary/interior damping, J. Differ. Equs., 233 (2008), 42-86. doi: 10.1016/j.jde.2006.09.019. [18] I. Chueshov and I. Lasiecka, Von Karman Evolution Equations, Springer-Verlag, 2010. doi: 10.1007/978-0-387-87712-9. [19] I. Chueshov, I. Lasiecka and D. Toundykov, Global attractor for a wave equation with nonlinear localized boundary damping and a source term of critical exponent, J. Dyn. Diff. Equs., 21 (2009), 269-314. doi: 10.1007/s10884-009-9132-y. [20] I. Chueshov, I. Lasiecka and J. T. Webster, Attractors for delayed, non-rotational von Karman plates with applications to flow-structure interactions without any damping, Comm. in PDE, 39 (2014), 1965-1997. doi: 10.1080/03605302.2014.930484. [21] P. Ciarlet and P. Rabier, Les Equations de Von Karman, Springer, 1980. [22] A. Eden and A. J. Milani, Exponential attractors for extensible beam equations, Nonlinearity, 6 (1993), 457-479. [23] P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation, Discrete Cont. Dyn. Sys, 10 (2004), 211-238. doi: 10.3934/dcds.2004.10.211. [24] P. G. Geredeli and J. T. Webster, Qualitative results on the dynamics of a Berger plate with nonlinear boundary damping, Nonlin. Anal: Real World Applications, 31 (2016), 227-256. doi: 10.1016/j.nonrwa.2016.02.002. [25] P. G. Geredeli, I. Lasiecka and J. T. Webster, Smooth attractors of finite dimension for von Karman evolutions with nonlinear frictional damping localized in a boundary layer, J. Diff. Eqs., 254 (2013), 1193-1229. doi: 10.1016/j.jde.2012.10.016. [26] P. G. Geredeli and J. T. Webster, Decay rates to eqilibrium for nonlinear plate equations with geometrically constrained, degenerate dissipation, Appl. Math. and Optim., 68 (2013), 361-390. Erratum, Appl. Math. and Optim., 70 (2014), 565-566. [27] J. K. Hale and G. Raugel, Attractors for dissipative evolutionary equations, In International Conference on Differential Equations (Vol. 1, p. 2), 1993, World Scientific River Edge, NJ. [28] G. Ji and I. Lasiecka, Nonlinear boundary feedback stabilization for a semilinear Kirchhoff plate with dissipation acting only via moments-limiting behavior, JMAA, 229 (1999), 452-479. doi: 10.1006/jmaa.1998.6170. [29] A. Kh. Khanmamedov, Global attractors for von Karman equations with non-linear dissipation, J. Math. Anal. Appl, 318 (2006), 92-101. doi: 10.1016/j.jmaa.2005.05.031. [30] J. Lagnese, Boundary Stabilization of Thin Plates, SIAM, 1989. doi: 10.1137/1.9781611970821. [31] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations, Cambridge University Press, Cambridge, 2000. [32] I. Lasiecka and R. Triggiani, Sharp trace estimates of solutions to Kirchhoff and Euler-Bernoulli equations, Appl. Math Optim, 28 (1993), 277-306. doi: 10.1007/BF01200382. [33] V. Kalantarov and S. Zelik, Finite-dimensional attractors for the quasi-linear strongly-damped wave equation, J. Diff. Equs., 247 (2009), 1120-1155. doi: 10.1016/j.jde.2009.04.010. [34] J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, 1971. [35] J. L. Lions, Contrôlabilité exacte, perturbations et stabilization de systèmes distribués, Vol. I, Masson, Paris, 1989. [36] J. Málek and D. Pražak, Large time behavior via the method of $l$-trajectories, J. Diff. Eqs., 181 (2002), 243-279. doi: 10.1006/jdeq.2001.4087. [37] A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, In Handbook of Differential Equations: Evolutionary Equations (M. C. Dafermos and M. Pokorny eds.) doi: 10.1016/S1874-5717(08)00003-0. [38] V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations, Nonlinearity, 19 (2006), 1495-1506. doi: 10.1088/0951-7715/19/7/001. [39] J.-P. Puel and M. Tucsnak, Boundary stabilization for the von Karman equations, SIAM J. Control and Optim., 33 (1995), 255-273. doi: 10.1137/S0363012992228350. [40] D. Pražak, On finite fractal dimension of the global attractor for the wave equation with nonlinear damping, J. Dyn. Diff. Eqs., 14 (2002), 764-776. doi: 10.1023/A:1020756426088. [41] G. Raugel, Global attractors in partial differential equations, In Handbook of Dynamical Systems (B. Fiedler ed.), v. 2, Elsevier Sciences, Amsterdam, 2002. doi: 10.1016/S1874-575X(02)80038-8. [42] J. Simon, Compact sets in the space $L^p(0,T;B)$, Annali di Matematica pura ed applicata IV, CXLVI (1987), 65-96. doi: 10.1007/BF01762360. [43] C. P. Vendhan, A study of Berger equations applied to nonlinear vibrations of elastic plates, Int. J. Mech. Sci, 17 (1975), 461-468.

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##### References:
 [1] J. P. Aubin, Une théorè de compacité, C.R. Acad. Sci. Paris, 256 (1963), 5042-5044. [2] G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system without mechanical dissipation, Rend. Istit. Mat. Univ. Trieste, 28 (1997), 1-28. [3] G. Avalos and I. Lasiecka, Boundary controllability of thermoelastic plates via the free boundary conditions, SIAM J. Control. Optim., 38 (2000), 337-383. doi: 10.1137/S0363012998339836. [4] A. Babin and M. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992. [5] J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Cont. Dyn. Sys, 10 (2004), 31-52. doi: 10.3934/dcds.2004.10.31. [6] H. M. Berger, A new approach to the analysis of large deflections of plates, J. Appl. Mech., 22 (1955), 465-472. [7] V. V. Bolotin, Nonconservative Problems of Elastic Stability, Pergamon Press, Oxford, 1963. [8] S. C. Brenner and R. Scott, The Mathematical Theory of Finite Element Methods, 15, Springer Science & Business Media, 2008. doi: 10.1007/978-0-387-75934-0. [9] F. Bucci, I. Chueshov and I. Lasiecka, Global attractor for a composite system of nonlinear wave and plate equations, Comm. Pure and Appl. Anal., 6 (2007), 113-140. [10] F. Bucci and I. Chueshov, Long-time dynamics of a coupled system of nonlinear wave and thermoelastic plate equations, Dynam. Sys., 22 (2008), 557-586. doi: 10.3934/dcds.2008.22.557. [11] I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Springer, 2015. doi: 10.1007/978-3-319-22903-4. [12] I. Chueshov, Long-time dynamics of Kirchhoff wave models with strong nonlinear damping, J. Diff. Equs., 252 (2012), 1229-1262. doi: 10.1016/j.jde.2011.08.022. [13] I. Chueshov, Introduction to the Theory of Infinite Dimensional Dissipative Systems, Acta, Kharkov, 1999, in Russian; English translation: Acta, Kharkov, 2002; http://www.emis.de/monographs/Chueshov/ [14] I. Chueshov, M. Eller and I. Lasiecka, Finite dimensionality of the attractor for a semilinear wave equation with nonlinear boundary dissipation, Comm. PDE, 29 (2004), 1847-1976. doi: 10.1081/PDE-200040203. [15] I. Chueshov and I. Lasiecka, Global attractors for von Karman evolutions with a nonlinear boundary dissipation, J. Differ. Equs., 198 (2004), 196-231. doi: 10.1016/j.jde.2003.08.008. [16] I. Chueshov and I. Lasiecka, Long-time behavior of second-order evolutions with nonlinear damping, Memoires of AMS, 195, 2008. doi: 10.1090/memo/0912. [17] I. Chueshov and I. Lasiecka, Long-time dynamics of von Karman semi-flows with non-linear boundary/interior damping, J. Differ. Equs., 233 (2008), 42-86. doi: 10.1016/j.jde.2006.09.019. [18] I. Chueshov and I. Lasiecka, Von Karman Evolution Equations, Springer-Verlag, 2010. doi: 10.1007/978-0-387-87712-9. [19] I. Chueshov, I. Lasiecka and D. Toundykov, Global attractor for a wave equation with nonlinear localized boundary damping and a source term of critical exponent, J. Dyn. Diff. Equs., 21 (2009), 269-314. doi: 10.1007/s10884-009-9132-y. [20] I. Chueshov, I. Lasiecka and J. T. Webster, Attractors for delayed, non-rotational von Karman plates with applications to flow-structure interactions without any damping, Comm. in PDE, 39 (2014), 1965-1997. doi: 10.1080/03605302.2014.930484. [21] P. Ciarlet and P. Rabier, Les Equations de Von Karman, Springer, 1980. [22] A. Eden and A. J. Milani, Exponential attractors for extensible beam equations, Nonlinearity, 6 (1993), 457-479. [23] P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation, Discrete Cont. Dyn. Sys, 10 (2004), 211-238. doi: 10.3934/dcds.2004.10.211. [24] P. G. Geredeli and J. T. Webster, Qualitative results on the dynamics of a Berger plate with nonlinear boundary damping, Nonlin. Anal: Real World Applications, 31 (2016), 227-256. doi: 10.1016/j.nonrwa.2016.02.002. [25] P. G. Geredeli, I. Lasiecka and J. T. Webster, Smooth attractors of finite dimension for von Karman evolutions with nonlinear frictional damping localized in a boundary layer, J. Diff. Eqs., 254 (2013), 1193-1229. doi: 10.1016/j.jde.2012.10.016. [26] P. G. Geredeli and J. T. Webster, Decay rates to eqilibrium for nonlinear plate equations with geometrically constrained, degenerate dissipation, Appl. Math. and Optim., 68 (2013), 361-390. Erratum, Appl. Math. and Optim., 70 (2014), 565-566. [27] J. K. Hale and G. Raugel, Attractors for dissipative evolutionary equations, In International Conference on Differential Equations (Vol. 1, p. 2), 1993, World Scientific River Edge, NJ. [28] G. Ji and I. Lasiecka, Nonlinear boundary feedback stabilization for a semilinear Kirchhoff plate with dissipation acting only via moments-limiting behavior, JMAA, 229 (1999), 452-479. doi: 10.1006/jmaa.1998.6170. [29] A. Kh. Khanmamedov, Global attractors for von Karman equations with non-linear dissipation, J. Math. Anal. Appl, 318 (2006), 92-101. doi: 10.1016/j.jmaa.2005.05.031. [30] J. Lagnese, Boundary Stabilization of Thin Plates, SIAM, 1989. doi: 10.1137/1.9781611970821. [31] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations, Cambridge University Press, Cambridge, 2000. [32] I. Lasiecka and R. Triggiani, Sharp trace estimates of solutions to Kirchhoff and Euler-Bernoulli equations, Appl. Math Optim, 28 (1993), 277-306. doi: 10.1007/BF01200382. [33] V. Kalantarov and S. Zelik, Finite-dimensional attractors for the quasi-linear strongly-damped wave equation, J. Diff. Equs., 247 (2009), 1120-1155. doi: 10.1016/j.jde.2009.04.010. [34] J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, 1971. [35] J. L. Lions, Contrôlabilité exacte, perturbations et stabilization de systèmes distribués, Vol. I, Masson, Paris, 1989. [36] J. Málek and D. Pražak, Large time behavior via the method of $l$-trajectories, J. Diff. Eqs., 181 (2002), 243-279. doi: 10.1006/jdeq.2001.4087. [37] A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, In Handbook of Differential Equations: Evolutionary Equations (M. C. Dafermos and M. Pokorny eds.) doi: 10.1016/S1874-5717(08)00003-0. [38] V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations, Nonlinearity, 19 (2006), 1495-1506. doi: 10.1088/0951-7715/19/7/001. [39] J.-P. Puel and M. Tucsnak, Boundary stabilization for the von Karman equations, SIAM J. Control and Optim., 33 (1995), 255-273. doi: 10.1137/S0363012992228350. [40] D. Pražak, On finite fractal dimension of the global attractor for the wave equation with nonlinear damping, J. Dyn. Diff. Eqs., 14 (2002), 764-776. doi: 10.1023/A:1020756426088. [41] G. Raugel, Global attractors in partial differential equations, In Handbook of Dynamical Systems (B. Fiedler ed.), v. 2, Elsevier Sciences, Amsterdam, 2002. doi: 10.1016/S1874-575X(02)80038-8. [42] J. Simon, Compact sets in the space $L^p(0,T;B)$, Annali di Matematica pura ed applicata IV, CXLVI (1987), 65-96. doi: 10.1007/BF01762360. [43] C. P. Vendhan, A study of Berger equations applied to nonlinear vibrations of elastic plates, Int. J. Mech. Sci, 17 (1975), 461-468.
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