# American Institute of Mathematical Sciences

October  2014, 19(8): 2603-2616. doi: 10.3934/dcdsb.2014.19.2603

## Variational approach to stability of semilinear wave equation with nonlinear boundary conditions

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

Received  October 2013 Revised  April 2014 Published  August 2014

We discuss solvability for the semilinear equation of the vibrating string $x_{tt}(t,y)-\Delta x(t,y)=F_{x}(t,y,x(t,y))-G_{x}(t,y,x(t,y))$ in bounded domain and same type of nonlinearity on the boundary. To this effect we derive new variational methods one for the boundary equation the second for interior equation. Next we discuss stability of solutions with respect to initial conditions basing on variational approach.
Citation: Andrzej Nowakowski. Variational approach to stability of semilinear wave equation with nonlinear boundary conditions. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2603-2616. doi: 10.3934/dcdsb.2014.19.2603
##### References:
 [1] G. Auchmuty, Variational principles for finite dimensional initial value problems,, Contemporar y Math., 426 (2007), 45. doi: 10.1090/conm/426/08183. Google Scholar [2] 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 [3] 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 [4] L. Bociu and I. Lasiecka, Uniqueness of weak solutions for the semilinear wave equations with supercritical boundary/interior sources and damping,, DCDS, 22 (2008), 835. doi: 10.3934/dcds.2008.22.835. Google Scholar [5] 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 [6] 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 [7] 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 [8] 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 [9] M. M. Cavalcanti, V. N. D. Cavalcanti and I. Lasiecka, Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction,, J. Differential Equations, 236 (2007), 407. doi: 10.1016/j.jde.2007.02.004. Google Scholar [10] 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 [11] I. Ekeland and R. Temam, Convex Analysis and Variational Problems,, Studies Math. Appl. I, (1976). Google Scholar [12] R. T. Glassey, Blow-up theorems for nonlinear wave equations,, Math. Z., 132 (1973), 183. doi: 10.1007/BF01213863. Google Scholar [13] 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 [14] H. A. Levine and G. Todorova, Blow up of solutions of the Cauch problem for a wave equation with nonlinear damping term and source terms and positive initial energy,, Proc. Amer. Math. Soc., 129 (2001), 793. doi: 10.1090/S0002-9939-00-05743-9. Google Scholar [15] 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 [16] A. Nowakowski, Nonhomogeneous boundary value problem for semilinear hyperbolic equation,, Journal of Dynamical and Control Systems, 14 (2008), 537. doi: 10.1007/s10883-008-9050-z. Google Scholar [17] A. Nowakowski, Nonlinear parabolic equations associated with subdifferential operators, periodic problems,, Bull. Polish Acad. Sc. Math., 36 (1998), 615. Google Scholar [18] 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 [19] L. E. Payne and D. Sattinger, Saddle points and instability of nonlinear hyperbolic equations,, Israel Math. J., 22 (1975), 273. doi: 10.1007/BF02761595. Google Scholar [20] 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 [21] G. Todorova, Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms,, Nonlinear Analysis, 41 (2000), 891. doi: 10.1016/S0362-546X(98)00317-4. Google Scholar [22] H. Tsutsumi, On solutions of semilinear differential equations in a Hilbert space,, Math. Japonicea, 17 (1972), 173. Google Scholar [23] E. Vitillaro, Global existence of the wave equation with nonlinear boundary damping and source terms,, J. Differential Equations, 186 (2002), 259. doi: 10.1016/S0022-0396(02)00023-2. Google Scholar [24] 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 [25] 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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##### References:
 [1] G. Auchmuty, Variational principles for finite dimensional initial value problems,, Contemporar y Math., 426 (2007), 45. doi: 10.1090/conm/426/08183. Google Scholar [2] 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 [3] 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 [4] L. Bociu and I. Lasiecka, Uniqueness of weak solutions for the semilinear wave equations with supercritical boundary/interior sources and damping,, DCDS, 22 (2008), 835. doi: 10.3934/dcds.2008.22.835. Google Scholar [5] 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 [6] 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 [7] 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 [8] 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 [9] M. M. Cavalcanti, V. N. D. Cavalcanti and I. Lasiecka, Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction,, J. Differential Equations, 236 (2007), 407. doi: 10.1016/j.jde.2007.02.004. Google Scholar [10] 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 [11] I. Ekeland and R. Temam, Convex Analysis and Variational Problems,, Studies Math. Appl. I, (1976). Google Scholar [12] R. T. Glassey, Blow-up theorems for nonlinear wave equations,, Math. Z., 132 (1973), 183. doi: 10.1007/BF01213863. Google Scholar [13] 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 [14] H. A. Levine and G. Todorova, Blow up of solutions of the Cauch problem for a wave equation with nonlinear damping term and source terms and positive initial energy,, Proc. Amer. Math. Soc., 129 (2001), 793. doi: 10.1090/S0002-9939-00-05743-9. Google Scholar [15] 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 [16] A. Nowakowski, Nonhomogeneous boundary value problem for semilinear hyperbolic equation,, Journal of Dynamical and Control Systems, 14 (2008), 537. doi: 10.1007/s10883-008-9050-z. Google Scholar [17] A. Nowakowski, Nonlinear parabolic equations associated with subdifferential operators, periodic problems,, Bull. Polish Acad. Sc. Math., 36 (1998), 615. Google Scholar [18] 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 [19] L. E. Payne and D. Sattinger, Saddle points and instability of nonlinear hyperbolic equations,, Israel Math. J., 22 (1975), 273. doi: 10.1007/BF02761595. Google Scholar [20] 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 [21] G. Todorova, Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms,, Nonlinear Analysis, 41 (2000), 891. doi: 10.1016/S0362-546X(98)00317-4. Google Scholar [22] H. Tsutsumi, On solutions of semilinear differential equations in a Hilbert space,, Math. Japonicea, 17 (1972), 173. Google Scholar [23] E. Vitillaro, Global existence of the wave equation with nonlinear boundary damping and source terms,, J. Differential Equations, 186 (2002), 259. doi: 10.1016/S0022-0396(02)00023-2. Google Scholar [24] 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 [25] 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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