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

Variational analysis of semilinear plate equation with free boundary conditions

Andrzej Nowakowski 1,

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.
Keywords: Dual variational method, nonlinear force, free boundary conditions, nonlinerity on the boundary., semi-linear plate equation.
Mathematics Subject Classification: Primary: 35L05; Secondary: 35L2.
Citation: Andrzej Nowakowski. Variational analysis of semilinear plate equation with free boundary conditions. Discrete & Continuous Dynamical Systems, 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), 377-411. 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), 2571-2611. 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), 654-683. 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), 2032-2064. 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), 1017-1029. 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), 1-13.  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), 451-466.  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), 119-158. 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), 395-407. 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), 1699-1715. 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), 42-86. 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), viii+183 pp. 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, Noth-Holland Publ. Co., Amsterdam-Oxford, 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), 267-294.  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), 1430-1439. 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, SIAM, 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), 149-192.  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), 1-21.  Google Scholar

[21]

J.-L. Lions, Quelques Méthodes de Résolution de Problémes aux Limites non Linéaires, Dunod, Paris, 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), 1495-1514. 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), 1-8.  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), 77-90. 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), 173-193.  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), 375-395. 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), 1426-1433. 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), 377-411. 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), 2571-2611. 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), 654-683. 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), 2032-2064. 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), 1017-1029. 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), 1-13.  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), 451-466.  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), 119-158. 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), 395-407. 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), 1699-1715. 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), 42-86. 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), viii+183 pp. 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, Noth-Holland Publ. Co., Amsterdam-Oxford, 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), 267-294.  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), 1430-1439. 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, SIAM, 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), 149-192.  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), 1-21.  Google Scholar

[21]

J.-L. Lions, Quelques Méthodes de Résolution de Problémes aux Limites non Linéaires, Dunod, Paris, 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), 1495-1514. 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), 1-8.  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), 77-90. 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), 173-193.  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), 375-395. 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), 1426-1433. doi: 10.1137/S0036141004440198.  Google Scholar

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