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Asymptotics in shallow water waves
Variational analysis of semilinear plate equation with free boundary conditions
1. | University of Lodz, Faculty of Math & Computer Sciences, Banacha 22, 90-238 Lodz |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[13] |
I. Chueshov and I. Lasiecka, Von Karman Evolution Equations, Springer-Verlag, 2010.
doi: 10.1007/978-0-387-87712-9. |
[14] |
I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Studies Math. Appl. I, Noth-Holland Publ. Co., Amsterdam-Oxford, 1976. |
[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. |
[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. |
[17] |
J. Lagnese, Boundary Stabilization of Thin Plates, SIAM, 1989.
doi: 10.1137/1.9781611970821. |
[18] |
I. Lasiecka, Mathematical Control Theory of Coupled PDE's, CMBS-NSF Lecture Notes, SIAM, 2002.
doi: 10.1137/1.9780898717099. |
[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. |
[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. |
[21] |
J.-L. Lions, Quelques Méthodes de Résolution de Problémes aux Limites non Linéaires, Dunod, Paris, 1969. |
[22] |
A. Nowakowski and D. O'Regan, Periodic solutions for forced vibrations of beam equation with nonmonotone nonlinearities, Multidimensional case, submmited. |
[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. |
[24] |
R. Parreira da Silva, Lower semicontinuity of pullback attractors for a singularly nonautonomous plate equation, Electronic Journal of Differential Equations, (2012), 1-8. |
[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. |
[26] |
H. Tsutsumi, On solutions of semilinear differential equations in a Hilbert space, Math. Japonicea, 17 (1972), 173-193. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[13] |
I. Chueshov and I. Lasiecka, Von Karman Evolution Equations, Springer-Verlag, 2010.
doi: 10.1007/978-0-387-87712-9. |
[14] |
I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Studies Math. Appl. I, Noth-Holland Publ. Co., Amsterdam-Oxford, 1976. |
[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. |
[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. |
[17] |
J. Lagnese, Boundary Stabilization of Thin Plates, SIAM, 1989.
doi: 10.1137/1.9781611970821. |
[18] |
I. Lasiecka, Mathematical Control Theory of Coupled PDE's, CMBS-NSF Lecture Notes, SIAM, 2002.
doi: 10.1137/1.9780898717099. |
[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. |
[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. |
[21] |
J.-L. Lions, Quelques Méthodes de Résolution de Problémes aux Limites non Linéaires, Dunod, Paris, 1969. |
[22] |
A. Nowakowski and D. O'Regan, Periodic solutions for forced vibrations of beam equation with nonmonotone nonlinearities, Multidimensional case, submmited. |
[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. |
[24] |
R. Parreira da Silva, Lower semicontinuity of pullback attractors for a singularly nonautonomous plate equation, Electronic Journal of Differential Equations, (2012), 1-8. |
[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. |
[26] |
H. Tsutsumi, On solutions of semilinear differential equations in a Hilbert space, Math. Japonicea, 17 (1972), 173-193. |
[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. |
[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. |
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