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 & Applied Analysis, 2016, 15 (6) : 2301-2328. doi: 10.3934/cpaa.2016038
References:
[1]

J. P. Aubin, Une théorè de compacité,, \emph{C.R. Acad. Sci. Paris}, 256 (1963), 5042.

[2]

G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system without mechanical dissipation,, \emph{Rend. Istit. Mat. Univ. Trieste}, 28 (1997), 1.

[3]

G. Avalos and I. Lasiecka, Boundary controllability of thermoelastic plates via the free boundary conditions,, \emph{SIAM J. Control. Optim.}, 38 (2000), 337. doi: 10.1137/S0363012998339836.

[4]

A. Babin and M. Vishik, Attractors of Evolution Equations,, North-Holland, (1992).

[5]

J. M. Ball, Global attractors for damped semilinear wave equations,, \emph{Discrete Cont. Dyn. Sys}, 10 (2004), 31. doi: 10.3934/dcds.2004.10.31.

[6]

H. M. Berger, A new approach to the analysis of large deflections of plates,, \emph{J. Appl. Mech.}, 22 (1955), 465.

[7]

V. V. Bolotin, Nonconservative Problems of Elastic Stability,, Pergamon Press, (1963).

[8]

S. C. Brenner and R. Scott, The Mathematical Theory of Finite Element Methods,, \textbf{15}, 15 (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,, \emph{Comm. Pure and Appl. Anal.}, 6 (2007), 113.

[10]

F. Bucci and I. Chueshov, Long-time dynamics of a coupled system of nonlinear wave and thermoelastic plate equations,, \emph{Dynam. Sys.}, 22 (2008), 557. 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,, \emph{J. Diff. Equs.}, 252 (2012), 1229. 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: \emph{Acta}, (2002).

[14]

I. Chueshov, M. Eller and I. Lasiecka, Finite dimensionality of the attractor for a semilinear wave equation with nonlinear boundary dissipation,, \emph{Comm. PDE}, 29 (2004), 1847. doi: 10.1081/PDE-200040203.

[15]

I. Chueshov and I. Lasiecka, Global attractors for von Karman evolutions with a nonlinear boundary dissipation,, \emph{J. Differ. Equs.}, 198 (2004), 196. doi: 10.1016/j.jde.2003.08.008.

[16]

I. Chueshov and I. Lasiecka, Long-time behavior of second-order evolutions with nonlinear damping,, \emph{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,, \emph{J. Differ. Equs.}, 233 (2008), 42. 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,, \emph{J. Dyn. Diff. Equs.}, 21 (2009), 269. 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,, \emph{Comm. in PDE}, 39 (2014), 1965. 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,, \emph{Nonlinearity}, 6 (1993), 457.

[23]

P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation,, \emph{Discrete Cont. Dyn. Sys}, 10 (2004), 211. 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,, \emph{Nonlin. Anal: Real World Applications}, 31 (2016), 227. 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,, \emph{J. Diff. Eqs.}, 254 (2013), 1193. 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, 70 (2014), 565.

[27]

J. K. Hale and G. Raugel, Attractors for dissipative evolutionary equations,, In \emph{International Conference on Differential Equations (Vol. 1, (1993).

[28]

G. Ji and I. Lasiecka, Nonlinear boundary feedback stabilization for a semilinear Kirchhoff plate with dissipation acting only via moments-limiting behavior,, \emph{JMAA}, 229 (1999), 452. doi: 10.1006/jmaa.1998.6170.

[29]

A. Kh. Khanmamedov, Global attractors for von Karman equations with non-linear dissipation,, \emph{J. Math. Anal. Appl}, 318 (2006), 92. 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, (2000).

[32]

I. Lasiecka and R. Triggiani, Sharp trace estimates of solutions to Kirchhoff and Euler-Bernoulli equations,, \emph{Appl. Math Optim}, 28 (1993), 277. doi: 10.1007/BF01200382.

[33]

V. Kalantarov and S. Zelik, Finite-dimensional attractors for the quasi-linear strongly-damped wave equation,, \emph{J. Diff. Equs.}, 247 (2009), 1120. 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, (1989).

[36]

J. Málek and D. Pražak, Large time behavior via the method of $l$-trajectories,, \emph{J. Diff. Eqs.}, 181 (2002), 243. doi: 10.1006/jdeq.2001.4087.

[37]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains,, In \emph{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,, \emph{Nonlinearity}, 19 (2006), 1495. doi: 10.1088/0951-7715/19/7/001.

[39]

J.-P. Puel and M. Tucsnak, Boundary stabilization for the von Karman equations,, \emph{SIAM J. Control and Optim.}, 33 (1995), 255. doi: 10.1137/S0363012992228350.

[40]

D. Pražak, On finite fractal dimension of the global attractor for the wave equation with nonlinear damping,, \emph{J. Dyn. Diff. Eqs.}, 14 (2002), 764. doi: 10.1023/A:1020756426088.

[41]

G. Raugel, Global attractors in partial differential equations,, In \emph{Handbook of Dynamical Systems} (B. Fiedler ed.), (2002). doi: 10.1016/S1874-575X(02)80038-8.

[42]

J. Simon, Compact sets in the space $L^p(0,T;B)$,, \emph{Annali di Matematica pura ed applicata IV}, CXLVI (1987), 65. doi: 10.1007/BF01762360.

[43]

C. P. Vendhan, A study of Berger equations applied to nonlinear vibrations of elastic plates,, \emph{Int. J. Mech. Sci}, 17 (1975), 461.

show all references

References:
[1]

J. P. Aubin, Une théorè de compacité,, \emph{C.R. Acad. Sci. Paris}, 256 (1963), 5042.

[2]

G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system without mechanical dissipation,, \emph{Rend. Istit. Mat. Univ. Trieste}, 28 (1997), 1.

[3]

G. Avalos and I. Lasiecka, Boundary controllability of thermoelastic plates via the free boundary conditions,, \emph{SIAM J. Control. Optim.}, 38 (2000), 337. doi: 10.1137/S0363012998339836.

[4]

A. Babin and M. Vishik, Attractors of Evolution Equations,, North-Holland, (1992).

[5]

J. M. Ball, Global attractors for damped semilinear wave equations,, \emph{Discrete Cont. Dyn. Sys}, 10 (2004), 31. doi: 10.3934/dcds.2004.10.31.

[6]

H. M. Berger, A new approach to the analysis of large deflections of plates,, \emph{J. Appl. Mech.}, 22 (1955), 465.

[7]

V. V. Bolotin, Nonconservative Problems of Elastic Stability,, Pergamon Press, (1963).

[8]

S. C. Brenner and R. Scott, The Mathematical Theory of Finite Element Methods,, \textbf{15}, 15 (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,, \emph{Comm. Pure and Appl. Anal.}, 6 (2007), 113.

[10]

F. Bucci and I. Chueshov, Long-time dynamics of a coupled system of nonlinear wave and thermoelastic plate equations,, \emph{Dynam. Sys.}, 22 (2008), 557. 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,, \emph{J. Diff. Equs.}, 252 (2012), 1229. 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: \emph{Acta}, (2002).

[14]

I. Chueshov, M. Eller and I. Lasiecka, Finite dimensionality of the attractor for a semilinear wave equation with nonlinear boundary dissipation,, \emph{Comm. PDE}, 29 (2004), 1847. doi: 10.1081/PDE-200040203.

[15]

I. Chueshov and I. Lasiecka, Global attractors for von Karman evolutions with a nonlinear boundary dissipation,, \emph{J. Differ. Equs.}, 198 (2004), 196. doi: 10.1016/j.jde.2003.08.008.

[16]

I. Chueshov and I. Lasiecka, Long-time behavior of second-order evolutions with nonlinear damping,, \emph{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,, \emph{J. Differ. Equs.}, 233 (2008), 42. 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,, \emph{J. Dyn. Diff. Equs.}, 21 (2009), 269. 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,, \emph{Comm. in PDE}, 39 (2014), 1965. 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,, \emph{Nonlinearity}, 6 (1993), 457.

[23]

P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation,, \emph{Discrete Cont. Dyn. Sys}, 10 (2004), 211. 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,, \emph{Nonlin. Anal: Real World Applications}, 31 (2016), 227. 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,, \emph{J. Diff. Eqs.}, 254 (2013), 1193. 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, 70 (2014), 565.

[27]

J. K. Hale and G. Raugel, Attractors for dissipative evolutionary equations,, In \emph{International Conference on Differential Equations (Vol. 1, (1993).

[28]

G. Ji and I. Lasiecka, Nonlinear boundary feedback stabilization for a semilinear Kirchhoff plate with dissipation acting only via moments-limiting behavior,, \emph{JMAA}, 229 (1999), 452. doi: 10.1006/jmaa.1998.6170.

[29]

A. Kh. Khanmamedov, Global attractors for von Karman equations with non-linear dissipation,, \emph{J. Math. Anal. Appl}, 318 (2006), 92. 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, (2000).

[32]

I. Lasiecka and R. Triggiani, Sharp trace estimates of solutions to Kirchhoff and Euler-Bernoulli equations,, \emph{Appl. Math Optim}, 28 (1993), 277. doi: 10.1007/BF01200382.

[33]

V. Kalantarov and S. Zelik, Finite-dimensional attractors for the quasi-linear strongly-damped wave equation,, \emph{J. Diff. Equs.}, 247 (2009), 1120. 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, (1989).

[36]

J. Málek and D. Pražak, Large time behavior via the method of $l$-trajectories,, \emph{J. Diff. Eqs.}, 181 (2002), 243. doi: 10.1006/jdeq.2001.4087.

[37]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains,, In \emph{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,, \emph{Nonlinearity}, 19 (2006), 1495. doi: 10.1088/0951-7715/19/7/001.

[39]

J.-P. Puel and M. Tucsnak, Boundary stabilization for the von Karman equations,, \emph{SIAM J. Control and Optim.}, 33 (1995), 255. doi: 10.1137/S0363012992228350.

[40]

D. Pražak, On finite fractal dimension of the global attractor for the wave equation with nonlinear damping,, \emph{J. Dyn. Diff. Eqs.}, 14 (2002), 764. doi: 10.1023/A:1020756426088.

[41]

G. Raugel, Global attractors in partial differential equations,, In \emph{Handbook of Dynamical Systems} (B. Fiedler ed.), (2002). doi: 10.1016/S1874-575X(02)80038-8.

[42]

J. Simon, Compact sets in the space $L^p(0,T;B)$,, \emph{Annali di Matematica pura ed applicata IV}, CXLVI (1987), 65. doi: 10.1007/BF01762360.

[43]

C. P. Vendhan, A study of Berger equations applied to nonlinear vibrations of elastic plates,, \emph{Int. J. Mech. Sci}, 17 (1975), 461.

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