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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 |
References:
[1] |
G. Auchmuty, Variational principles for finite dimensional initial value problems, Contemporar y Math., 426 (2007), 45-56.
doi: 10.1090/conm/426/08183. |
[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), 377-411.
doi: 10.1007/s00526-008-0188-z. |
[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), 2571-2611.
doi: 10.1090/S0002-9947-05-03880-8. |
[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), 835-860.
doi: 10.3934/dcds.2008.22.835. |
[5] |
L. Bociu and I. Lasiecka, Local Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping, J. Differential Equations, 249 (2010), 654-683.
doi: 10.1016/j.jde.2010.03.009. |
[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), 2032-2064.
doi: 10.1002/mana.200910182. |
[7] |
L. Bociu, M. Rammaha and D. Toundykov, Wave equations with super-critical interior and boundary nonlinearities, Mathematics and Computers in Simulation, 82 (2012), 1017-1029.
doi: 10.1016/j.matcom.2011.04.006. |
[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), 451-466. |
[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), 407-459.
doi: 10.1016/j.jde.2007.02.004. |
[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), 119-158.
doi: 10.1016/j.jde.2004.04.011. |
[11] |
I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Studies Math. Appl. I, Noth-Holland Publ. Comp., 1976. |
[12] |
R. T. Glassey, Blow-up theorems for nonlinear wave equations, Math. Z., 132 (1973), 183-203.
doi: 10.1007/BF01213863. |
[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), 1-21. |
[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), 793-805.
doi: 10.1090/S0002-9939-00-05743-9. |
[15] |
I. Lasiecka, J. L. Lions and R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl., 65 (1986), 149-192. |
[16] |
A. Nowakowski, Nonhomogeneous boundary value problem for semilinear hyperbolic equation, Journal of Dynamical and Control Systems, 14 (2008), 537-558.
doi: 10.1007/s10883-008-9050-z. |
[17] |
A. Nowakowski, Nonlinear parabolic equations associated with subdifferential operators, periodic problems, Bull. Polish Acad. Sc. Math., 36 (1998), 615-621. |
[18] |
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. |
[19] |
L. E. Payne and D. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel Math. J., 22 (1975), 273-303.
doi: 10.1007/BF02761595. |
[20] |
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. |
[21] |
G. Todorova, Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms, Nonlinear Analysis, 41 (2000), 891-905.
doi: 10.1016/S0362-546X(98)00317-4. |
[22] |
H. Tsutsumi, On solutions of semilinear differential equations in a Hilbert space, Math. Japonicea, 17 (1972), 173-193. |
[23] |
E. Vitillaro, Global existence of the wave equation with nonlinear boundary damping and source terms, J. Differential Equations, 186 (2002), 259-298.
doi: 10.1016/S0022-0396(02)00023-2. |
[24] |
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. |
[25] |
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. |
show all references
References:
[1] |
G. Auchmuty, Variational principles for finite dimensional initial value problems, Contemporar y Math., 426 (2007), 45-56.
doi: 10.1090/conm/426/08183. |
[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), 377-411.
doi: 10.1007/s00526-008-0188-z. |
[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), 2571-2611.
doi: 10.1090/S0002-9947-05-03880-8. |
[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), 835-860.
doi: 10.3934/dcds.2008.22.835. |
[5] |
L. Bociu and I. Lasiecka, Local Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping, J. Differential Equations, 249 (2010), 654-683.
doi: 10.1016/j.jde.2010.03.009. |
[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), 2032-2064.
doi: 10.1002/mana.200910182. |
[7] |
L. Bociu, M. Rammaha and D. Toundykov, Wave equations with super-critical interior and boundary nonlinearities, Mathematics and Computers in Simulation, 82 (2012), 1017-1029.
doi: 10.1016/j.matcom.2011.04.006. |
[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), 451-466. |
[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), 407-459.
doi: 10.1016/j.jde.2007.02.004. |
[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), 119-158.
doi: 10.1016/j.jde.2004.04.011. |
[11] |
I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Studies Math. Appl. I, Noth-Holland Publ. Comp., 1976. |
[12] |
R. T. Glassey, Blow-up theorems for nonlinear wave equations, Math. Z., 132 (1973), 183-203.
doi: 10.1007/BF01213863. |
[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), 1-21. |
[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), 793-805.
doi: 10.1090/S0002-9939-00-05743-9. |
[15] |
I. Lasiecka, J. L. Lions and R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl., 65 (1986), 149-192. |
[16] |
A. Nowakowski, Nonhomogeneous boundary value problem for semilinear hyperbolic equation, Journal of Dynamical and Control Systems, 14 (2008), 537-558.
doi: 10.1007/s10883-008-9050-z. |
[17] |
A. Nowakowski, Nonlinear parabolic equations associated with subdifferential operators, periodic problems, Bull. Polish Acad. Sc. Math., 36 (1998), 615-621. |
[18] |
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. |
[19] |
L. E. Payne and D. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel Math. J., 22 (1975), 273-303.
doi: 10.1007/BF02761595. |
[20] |
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. |
[21] |
G. Todorova, Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms, Nonlinear Analysis, 41 (2000), 891-905.
doi: 10.1016/S0362-546X(98)00317-4. |
[22] |
H. Tsutsumi, On solutions of semilinear differential equations in a Hilbert space, Math. Japonicea, 17 (1972), 173-193. |
[23] |
E. Vitillaro, Global existence of the wave equation with nonlinear boundary damping and source terms, J. Differential Equations, 186 (2002), 259-298.
doi: 10.1016/S0022-0396(02)00023-2. |
[24] |
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. |
[25] |
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. |
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