# American Institute of Mathematical Sciences

July  2015, 35(7): 3133-3154. doi: 10.3934/dcds.2015.35.3133

## Variational analysis of semilinear plate equation with free boundary conditions

 1 University of Lodz, Faculty of Math & Computer Sciences, Banacha 22, 90-238 Lodz

Received  September 2013 Revised  November 2014 Published  January 2015

We present a variational analysis for the semilinear equation of the vibrating plate $x_{t t}(t,y)+\Delta ^{2}x(t,y)+l(t,y,x(t,y))=0$ in a bounded domain and a free nonlinear boundary condition $\Delta x(t,y)=H_{x}(t,y,x(t,y))-Q_{x}(t,y,x(t,y))$. In this context new dual variational methods are developed. Applying a variational approach we discuss a stability of solutions with respect to initial conditions.
Citation: Andrzej Nowakowski. Variational analysis of semilinear plate equation with free boundary conditions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3133-3154. doi: 10.3934/dcds.2015.35.3133
##### References:
 [1] C. O. Alves and M. M. Cavalcanti, On existence, uniform decay rates and blow up for solutions of the 2-D wave equation with exponential source,, Calc. Var. Partial Differential Equations, 34 (2009), 377.  doi: 10.1007/s00526-008-0188-z.  Google Scholar [2] V. Barbu, I. Lasiecka and M. A. Rammaha, On nonlinear wave equations with degenerate damping and source terms,, Trans. Amer. Math. Soc., 357 (2005), 2571.  doi: 10.1090/S0002-9947-05-03880-8.  Google Scholar [3] L. Bociu and I. Lasiecka, Local Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping,, J. Differential Equations, 249 (2010), 654.  doi: 10.1016/j.jde.2010.03.009.  Google Scholar [4] L. Bociu, M. Rammaha and D. Toundykov, On a wave equation with supercritical interior and boundary sources and damping terms,, Mathematische Nachrichten, 284 (2011), 2032.  doi: 10.1002/mana.200910182.  Google Scholar [5] L. Bociu, M. Rammaha and D. 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I, (1976).   Google Scholar [15] A. Favini, I. Lasiecka, M. A. Horn and D. Tataru, Global existence, uniqueness and regularity of solutions to a von Karman system with nonlinear boundary dissipation,, Differential Integral Equations, 9 (1996), 267.   Google Scholar [16] J. R. Kang, Global attractor for an extensible beam equation with localized nonlinear damping and linear memory,, Math. Methods Appl. Sci., 34 (2011), 1430.  doi: 10.1002/mma.1450.  Google Scholar [17] J. Lagnese, Boundary Stabilization of Thin Plates,, SIAM, (1989).  doi: 10.1137/1.9781611970821.  Google Scholar [18] I. Lasiecka, Mathematical Control Theory of Coupled PDE's,, CMBS-NSF Lecture Notes, (2002).  doi: 10.1137/1.9780898717099.  Google Scholar [19] I. Lasiecka, J. L. Lions and R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators,, J. Math. Pures Appl., 65 (1986), 149.   Google Scholar [20] H. A. Levine, Instability and nonexistence of global solutions of nonlinear wave equations of the form $Pu_{t t}$ = $Au+F(u)$,, Trans. Amer. Math. Soc., 192 (1974), 1.   Google Scholar [21] J.-L. Lions, Quelques Méthodes de Résolution de Problémes aux Limites non Linéaires,, Dunod, (1969).   Google Scholar [22] A. Nowakowski and D. O'Regan, Periodic solutions for forced vibrations of beam equation with nonmonotone nonlinearities,, Multidimensional case, ().   Google Scholar [23] A. Nowakowski, Solvability and stability of a semilinear wave equation with nonlinear boundary conditions,, Nonlinear Anal., 73 (2010), 1495.  doi: 10.1016/j.na.2010.04.035.  Google Scholar [24] R. Parreira da Silva, Lower semicontinuity of pullback attractors for a singularly nonautonomous plate equation,, Electronic Journal of Differential Equations, (2012), 1.   Google Scholar [25] M. A. Rammaha, The influence of damping and source terms on solutions of nonlinear wave equations,, Bol. Soc. Parana. Mat., 25 (2007), 77.  doi: 10.5269/bspm.v25i1-2.7427.  Google Scholar [26] H. Tsutsumi, On solutions of semilinear differential equations in a Hilbert space,, Math. Japonicea, 17 (1972), 173.   Google Scholar [27] E. Vitillaro, A potential well theory for the wave equation with nonlinear source and boundary damping terms,, Glasg. Math. J., 44 (2002), 375.  doi: 10.1017/S0017089502030045.  Google Scholar [28] B. Yordanov and Q. S. Zhang, Finite-time blowup for wave equations with a potential,, SIAM J. Math. Anal., 36 (2005), 1426.  doi: 10.1137/S0036141004440198.  Google Scholar

show all references

##### References:
 [1] C. O. Alves and M. M. Cavalcanti, On existence, uniform decay rates and blow up for solutions of the 2-D wave equation with exponential source,, Calc. Var. Partial Differential Equations, 34 (2009), 377.  doi: 10.1007/s00526-008-0188-z.  Google Scholar [2] V. Barbu, I. Lasiecka and M. A. Rammaha, On nonlinear wave equations with degenerate damping and source terms,, Trans. Amer. Math. Soc., 357 (2005), 2571.  doi: 10.1090/S0002-9947-05-03880-8.  Google Scholar [3] L. Bociu and I. Lasiecka, Local Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping,, J. Differential Equations, 249 (2010), 654.  doi: 10.1016/j.jde.2010.03.009.  Google Scholar [4] L. Bociu, M. Rammaha and D. Toundykov, On a wave equation with supercritical interior and boundary sources and damping terms,, Mathematische Nachrichten, 284 (2011), 2032.  doi: 10.1002/mana.200910182.  Google Scholar [5] L. Bociu, M. Rammaha and D. Toundykov, Wave equations with super-critical interior and boundary nonlinearities,, Mathematics and Computers in Simulation, 82 (2012), 1017.  doi: 10.1016/j.matcom.2011.04.006.  Google Scholar [6] V. L. Carbone, M. J. D. Nascimento, K. Schiabel-Silva and R. P. Silva, Pullback attractors for a singularly nonautonomous plate equation,, Electronic Journal of Differential Equations, (2011), 1.   Google Scholar [7] M. M. Cavalcanti, V. N. D. Cavalcanti, J. S. P. Filho and J. A. Soriano, Existence and uniform decay of solutions of a parabolic-hyperbolic equation with nonlinear boundary damping and boundary source term,, Comm. Anal. Geom., 10 (2002), 451.   Google Scholar [8] M. M. Cavalcanti, V. N. D. Cavalcanti and P. Martinez, Existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term,, J. Differential Equations, 203 (2004), 119.  doi: 10.1016/j.jde.2004.04.011.  Google Scholar [9] V. V. Chepyzhov, M. I. Vishik and S. V. Zelik, Strong trajectory attractors for dissipative Euler equations,, J. Math. Pures Appl., 96 (2011), 395.  doi: 10.1016/j.matpur.2011.04.007.  Google Scholar [10] I. Chueshov, Convergence of solutions of von Karman evolution equations to equilibria,, Appl. Anal., 91 (2012), 1699.  doi: 10.1080/00036811.2011.577930.  Google Scholar [11] I. Chueshov and I. Lasiecka, Long-time dynamics of von Karman semi-flows with nonlinear boundary-interior damping,, J. Differential Equations, 233 (2007), 42.  doi: 10.1016/j.jde.2006.09.019.  Google Scholar [12] I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with non-linear damping,, Mem. Amer. Math. Soc., 195 (2008).  doi: 10.1090/memo/0912.  Google Scholar [13] I. Chueshov and I. Lasiecka, Von Karman Evolution Equations,, Springer-Verlag, (2010).  doi: 10.1007/978-0-387-87712-9.  Google Scholar [14] I. Ekeland and R. Temam, Convex Analysis and Variational Problems,, Studies Math. Appl. I, (1976).   Google Scholar [15] A. Favini, I. Lasiecka, M. A. Horn and D. Tataru, Global existence, uniqueness and regularity of solutions to a von Karman system with nonlinear boundary dissipation,, Differential Integral Equations, 9 (1996), 267.   Google Scholar [16] J. R. Kang, Global attractor for an extensible beam equation with localized nonlinear damping and linear memory,, Math. Methods Appl. Sci., 34 (2011), 1430.  doi: 10.1002/mma.1450.  Google Scholar [17] J. Lagnese, Boundary Stabilization of Thin Plates,, SIAM, (1989).  doi: 10.1137/1.9781611970821.  Google Scholar [18] I. Lasiecka, Mathematical Control Theory of Coupled PDE's,, CMBS-NSF Lecture Notes, (2002).  doi: 10.1137/1.9780898717099.  Google Scholar [19] I. Lasiecka, J. L. Lions and R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators,, J. Math. Pures Appl., 65 (1986), 149.   Google Scholar [20] H. A. Levine, Instability and nonexistence of global solutions of nonlinear wave equations of the form $Pu_{t t}$ = $Au+F(u)$,, Trans. Amer. Math. Soc., 192 (1974), 1.   Google Scholar [21] J.-L. Lions, Quelques Méthodes de Résolution de Problémes aux Limites non Linéaires,, Dunod, (1969).   Google Scholar [22] A. Nowakowski and D. O'Regan, Periodic solutions for forced vibrations of beam equation with nonmonotone nonlinearities,, Multidimensional case, ().   Google Scholar [23] A. Nowakowski, Solvability and stability of a semilinear wave equation with nonlinear boundary conditions,, Nonlinear Anal., 73 (2010), 1495.  doi: 10.1016/j.na.2010.04.035.  Google Scholar [24] R. Parreira da Silva, Lower semicontinuity of pullback attractors for a singularly nonautonomous plate equation,, Electronic Journal of Differential Equations, (2012), 1.   Google Scholar [25] M. A. Rammaha, The influence of damping and source terms on solutions of nonlinear wave equations,, Bol. Soc. Parana. Mat., 25 (2007), 77.  doi: 10.5269/bspm.v25i1-2.7427.  Google Scholar [26] H. Tsutsumi, On solutions of semilinear differential equations in a Hilbert space,, Math. Japonicea, 17 (1972), 173.   Google Scholar [27] E. Vitillaro, A potential well theory for the wave equation with nonlinear source and boundary damping terms,, Glasg. Math. J., 44 (2002), 375.  doi: 10.1017/S0017089502030045.  Google Scholar [28] B. Yordanov and Q. S. Zhang, Finite-time blowup for wave equations with a potential,, SIAM J. Math. Anal., 36 (2005), 1426.  doi: 10.1137/S0036141004440198.  Google Scholar
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