# 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 & 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. Google Scholar [2] G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system without mechanical dissipation,, \emph{Rend. Istit. Mat. Univ. Trieste}, 28 (1997), 1. Google Scholar [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. Google Scholar [4] A. Babin and M. Vishik, Attractors of Evolution Equations,, North-Holland, (1992). Google Scholar [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. Google Scholar [6] H. M. Berger, A new approach to the analysis of large deflections of plates,, \emph{J. Appl. Mech.}, 22 (1955), 465. Google Scholar [7] V. V. Bolotin, Nonconservative Problems of Elastic Stability,, Pergamon Press, (1963). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [11] I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems,, Springer, (2015). doi: 10.1007/978-3-319-22903-4. Google Scholar [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. Google Scholar [13] I. Chueshov, Introduction to the Theory of Infinite Dimensional Dissipative Systems, Acta, Kharkov, 1999, in Russian;, English translation: \emph{Acta}, (2002). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [18] I. Chueshov and I. Lasiecka, Von Karman Evolution Equations,, Springer-Verlag, (2010). doi: 10.1007/978-0-387-87712-9. Google Scholar [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. Google Scholar [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. Google Scholar [21] P. Ciarlet and P. Rabier, Les Equations de Von Karman,, {Springer}, (1980). Google Scholar [22] A. Eden and A. J. Milani, Exponential attractors for extensible beam equations,, \emph{Nonlinearity}, 6 (1993), 457. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [27] J. K. Hale and G. Raugel, Attractors for dissipative evolutionary equations,, In \emph{International Conference on Differential Equations (Vol. 1, (1993). Google Scholar [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. Google Scholar [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. Google Scholar [30] J. Lagnese, Boundary Stabilization of Thin Plates,, SIAM, (1989). doi: 10.1137/1.9781611970821. Google Scholar [31] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations,, Cambridge University Press, (2000). Google Scholar [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. Google Scholar [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. Google Scholar [34] J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations,, Springer, (1971). Google Scholar [35] J. L. Lions, Contrôlabilité exacte, perturbations et stabilization de systèmes distribués,, Vol. I, (1989). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [43] C. P. Vendhan, A study of Berger equations applied to nonlinear vibrations of elastic plates,, \emph{Int. J. Mech. Sci}, 17 (1975), 461. Google Scholar

show all references

##### References:
 [1] J. P. Aubin, Une théorè de compacité,, \emph{C.R. Acad. Sci. Paris}, 256 (1963), 5042. Google Scholar [2] G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system without mechanical dissipation,, \emph{Rend. Istit. Mat. Univ. Trieste}, 28 (1997), 1. Google Scholar [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. Google Scholar [4] A. Babin and M. Vishik, Attractors of Evolution Equations,, North-Holland, (1992). Google Scholar [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. Google Scholar [6] H. M. Berger, A new approach to the analysis of large deflections of plates,, \emph{J. Appl. Mech.}, 22 (1955), 465. Google Scholar [7] V. V. Bolotin, Nonconservative Problems of Elastic Stability,, Pergamon Press, (1963). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [11] I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems,, Springer, (2015). doi: 10.1007/978-3-319-22903-4. Google Scholar [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. Google Scholar [13] I. Chueshov, Introduction to the Theory of Infinite Dimensional Dissipative Systems, Acta, Kharkov, 1999, in Russian;, English translation: \emph{Acta}, (2002). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [18] I. Chueshov and I. Lasiecka, Von Karman Evolution Equations,, Springer-Verlag, (2010). doi: 10.1007/978-0-387-87712-9. Google Scholar [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. Google Scholar [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. Google Scholar [21] P. Ciarlet and P. Rabier, Les Equations de Von Karman,, {Springer}, (1980). Google Scholar [22] A. Eden and A. J. Milani, Exponential attractors for extensible beam equations,, \emph{Nonlinearity}, 6 (1993), 457. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [27] J. K. Hale and G. Raugel, Attractors for dissipative evolutionary equations,, In \emph{International Conference on Differential Equations (Vol. 1, (1993). Google Scholar [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. Google Scholar [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. Google Scholar [30] J. Lagnese, Boundary Stabilization of Thin Plates,, SIAM, (1989). doi: 10.1137/1.9781611970821. Google Scholar [31] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations,, Cambridge University Press, (2000). Google Scholar [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. Google Scholar [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. Google Scholar [34] J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations,, Springer, (1971). Google Scholar [35] J. L. Lions, Contrôlabilité exacte, perturbations et stabilization de systèmes distribués,, Vol. I, (1989). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [43] C. P. Vendhan, A study of Berger equations applied to nonlinear vibrations of elastic plates,, \emph{Int. J. Mech. Sci}, 17 (1975), 461. Google Scholar
 [1] Francesca Bucci, Igor Chueshov, Irena Lasiecka. Global attractor for a composite system of nonlinear wave and plate equations. Communications on Pure & Applied Analysis, 2007, 6 (1) : 113-140. doi: 10.3934/cpaa.2007.6.113 [2] Moncef Aouadi, Alain Miranville. Quasi-stability and global attractor in nonlinear thermoelastic diffusion plate with memory. Evolution Equations & Control Theory, 2015, 4 (3) : 241-263. doi: 10.3934/eect.2015.4.241 [3] Azer Khanmamedov, Sema Simsek. Existence of the global attractor for the plate equation with nonlocal nonlinearity in $\mathbb{R} ^{n}$. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 151-172. doi: 10.3934/dcdsb.2016.21.151 [4] Yongqin Liu, Shuichi Kawashima. Global existence and asymptotic behavior of solutions for quasi-linear dissipative plate equation. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1113-1139. doi: 10.3934/dcds.2011.29.1113 [5] I. D. Chueshov, Iryna Ryzhkova. A global attractor for a fluid--plate interaction model. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1635-1656. doi: 10.3934/cpaa.2013.12.1635 [6] Nikos I. Karachalios, Nikos M. Stavrakakis. Estimates on the dimension of a global attractor for a semilinear dissipative wave equation on $\mathbb R^N$. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 939-951. doi: 10.3934/dcds.2002.8.939 [7] Wided Kechiche. Regularity of the global attractor for a nonlinear Schrödinger equation with a point defect. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1233-1252. doi: 10.3934/cpaa.2017060 [8] Yongqin Liu, Shuichi Kawashima. Decay property for a plate equation with memory-type dissipation. Kinetic & Related Models, 2011, 4 (2) : 531-547. doi: 10.3934/krm.2011.4.531 [9] Muhammad I. Mustafa. Viscoelastic plate equation with boundary feedback. Evolution Equations & Control Theory, 2017, 6 (2) : 261-276. doi: 10.3934/eect.2017014 [10] Vladimir V. Chepyzhov, Monica Conti, Vittorino Pata. Totally dissipative dynamical processes and their uniform global attractors. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1989-2004. doi: 10.3934/cpaa.2014.13.1989 [11] Moez Daoulatli, Irena Lasiecka, Daniel Toundykov. Uniform energy decay for a wave equation with partially supported nonlinear boundary dissipation without growth restrictions. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 67-94. doi: 10.3934/dcdss.2009.2.67 [12] Wen Tan. The regularity of pullback attractor for a non-autonomous p-Laplacian equation with dynamical boundary condition. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 529-546. doi: 10.3934/dcdsb.2018194 [13] Boyan Jonov, Thomas C. Sideris. Global and almost global existence of small solutions to a dissipative wave equation in 3D with nearly null nonlinear terms. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1407-1442. doi: 10.3934/cpaa.2015.14.1407 [14] Sébastien Court. Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part II: The nonlinear system.. Evolution Equations & Control Theory, 2014, 3 (1) : 83-118. doi: 10.3934/eect.2014.3.83 [15] Dominique Blanchard, Nicolas Bruyère, Olivier Guibé. Existence and uniqueness of the solution of a Boussinesq system with nonlinear dissipation. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2213-2227. doi: 10.3934/cpaa.2013.12.2213 [16] Kangsheng Liu, Xu Liu, Bopeng Rao. Eventual regularity of a wave equation with boundary dissipation. Mathematical Control & Related Fields, 2012, 2 (1) : 17-28. doi: 10.3934/mcrf.2012.2.17 [17] Rogério Martins. One-dimensional attractor for a dissipative system with a cylindrical phase space. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 533-547. doi: 10.3934/dcds.2006.14.533 [18] Jun Zhou. Global existence and energy decay estimate of solutions for a class of nonlinear higher-order wave equation with general nonlinear dissipation and source term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1175-1185. doi: 10.3934/dcdss.2017064 [19] Marek Fila, Kazuhiro Ishige, Tatsuki Kawakami. Convergence to the Poisson kernel for the Laplace equation with a nonlinear dynamical boundary condition. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1285-1301. doi: 10.3934/cpaa.2012.11.1285 [20] Boling Guo, Zhengde Dai. Attractor for the dissipative Hamiltonian amplitude equation governing modulated wave instabilities. Discrete & Continuous Dynamical Systems - A, 1998, 4 (4) : 783-793. doi: 10.3934/dcds.1998.4.783

2018 Impact Factor: 0.925