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Global existence and asymptotic behavior of solutions for quasi-linear dissipative plate equation
1. | Faculty of Mathematics, Kyushu University, Fukuoka 819-0395 |
References:
[1] |
P. Bergeret, Classification of smooth solutions to $2\times 2$ hyperbolic systems with boundary damping, Math. Methods Appl. Sci., 20 (1997), 1563-1598.
doi: 10.1002/(SICI)1099-1476(199712)20:18<1563::AID-MMA925>3.0.CO;2-9. |
[2] |
M. E. Bradley and S. Lenhart, Bilinear spatial control of the velocity term in a Kirchhoff plate equation, Electronic J. Differential Equations, 2001 (2001), 1-15. |
[3] |
C. Buriol, Energy decay rates for the Timoshenko system of thermoelastic plates, Nonlinear Analysis, 64 (2006), 92-108.
doi: 10.1016/j.na.2005.06.010. |
[4] |
R. C. Charão, E. Bisognin, V. Bisognin and A. F. Pazoto, Asymptotic behavior for a dissipative plate equation in $\mathbb{R}^{N}$ with periodic coefficients, Electronic J. Differential Equations, 2008 (2008), 23. |
[5] |
C. R. da Luz and R. C. Charão, Asymptotic properties for a semi-linear plate equation in unbounded domains, J. Hyperbolic Differential Equations, 6 (2009), 269-294.
doi: 10.1142/S0219891609001824. |
[6] |
Darmawijoyo and W. T. van Horssen, On boundary damping for a weakly nonlinear wave equation, Nonlinear Dynamics, 30 (2002), 179-191.
doi: 10.1023/A:1020473930223. |
[7] |
R. Denk, R. Racke and Y. Shibata, $L^p$ theory for the linear thermoelastic plate equations in bounded and exterior domains, Adv. Differential Equations, 14 (2009), 685-715. |
[8] |
W. Desch, K. B. Hannsgen and R. L. Wheeler, Passive boundary damping of viscoelastic structures, J. Integral Equations Appl., 8 (1996), 125-171.
doi: 10.1216/jiea/1181075934. |
[9] |
G. G. Doronin, N. A. Lar'kin and A. J. Souza, A hyperbolic problem with nonlinear second-order boundary damping, Electron J. Differential Equations, 1998 (1998), 1-10. |
[10] |
A. D. Drozdov and V. B. Kolmanovskii, "Stability in Viscoelasticity," North-Holland Series in Applied Mathematics and Mechanics, 38, North-Holland Publishing Co., Amsterdam, 1994. |
[11] |
Y. Enomoto, On a thermoelastic plate equation in an exterior domain, Math. Meth. Appl. Sci., 25 (2002), 443-472.
doi: 10.1002/mma.290. |
[12] |
M. Fabrizio and B. Lazzari, On the existence and the asymptotic stability of solutions for linear viscoelastic solids, Arch. Rational Mech. Anal., 116 (1991), 139-152.
doi: 10.1007/BF00375589. |
[13] |
T. Hosono and K. Kawashima, Decay property of regularity-loss type and application to some nonlinear hyperbolic-elliptic system, Math. Models Meth. Appl. Sci., 16 (2006), 1839-1859.
doi: 10.1142/S021820250600173X. |
[14] |
K. Ide and S. Kawashima, Decay property of regularity-loss type and nonlinear effects for dissipative Timoshenko system, Math. Models Meth. Appl. Sci., 18 (2008), 1001-1025.
doi: 10.1142/S0218202508002930. |
[15] |
I. Lasiecka, S. Maad and A. Sasane, Existence and exponential decay of solutions to a quasilinear thermoelastic plate system, Nonlinear Differ. Equ. Appl., 15 (2008), 689-715. |
[16] |
I. Lasiecka and J. Ong, Global solvability and uniform decays of solutions to quasilinear equation with nonlinear boundary dissipation, Comm. Partial Differential Equations, 24 (1999), 2069-2107.
doi: 10.1080/03605309908821495. |
[17] |
H. J. Lee, Uniform decay for solution of the plate equation with a boundary condition of memory type, Trends in Math., 9 (2006), 51-55. |
[18] |
W. Liu, Local boundary controllability for the semi-linear plate equation, Comm. Partial Differential Equations, 23 (1998), 201-221. |
[19] |
Y. Liu and W. Wang, The point-wise estimates of solutions for dissipative wave equation in multi-dimensions, Discrete Continuous Dynam. Systems - A, 20 (2008), 1013-1028. |
[20] |
Z. Liu and S. Zheng, On the exponential stability of linear viscoelasticity and thermo-viscoelasticity, Quart. Appl. Math., 54 (1996), 21-31. |
[21] |
Z. Liu and S. Zheng, "Semi-Groups Associated With Dissipative Systems," Chapman & Hall/CRC Research Notes in Mathematics, 398, Boca Raton, FL, 1999. |
[22] |
J. R. Luyo Sánchez, "O Sistema Dinámico de von Kármán en DomÍNios NÁO Limitados é Globalmente bem Posto no Sentido de Hadamard: Análise do seu Limite Singular," Ph.D Thesis, Institute of Mathematics, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil, 2003. |
[23] |
A. Matsumura, On the asymptotic behavior of semi-linear wave equations, Publ. Res. Inst. Math. Sci., 12 (1976), 169-189.
doi: 10.2977/prims/1195190962. |
[24] |
J. E. Muñoz Rivera, M. G. Naso and F. M. Vegni, Asymptotic behavior of the energy for a class of weakly dissipative second-order systems with memory, J. Math. Anal. Appl., 286 (2003), 692-704.
doi: 10.1016/S0022-247X(03)00511-0. |
[25] |
J. E. Muñoz Rivera and R. Racke, Global stability for damped Timoshenko systems, Discrete Continuous Dynam. Systems, 9 (2003), 1625-1639. |
[26] |
J. E. Muñoz Rivera, Asymptotic behavior in linear viscoelasticity, Quart. Appl. Math., 52 (1994), 628-648. |
[27] |
K. Nishihara, $L^p-L^q$ estimates of solutions to the damped wave equation in $3$-dimensional space and their applications, Math. Z., 244 (2003), 631-649. |
[28] |
J. Y. Park, Bilinear boundary optimal control of the velocity terms in a Kirchhoff plate equation, Trends in Math., 9 (2006), 41-44. |
[29] |
A. F. Pazoto, J. C. Vila Bravo and J. E. Muñoz Rivera, Asymptotic stability of semi-groups associated to linear weak dissipative systems, Math. Comput. Modeling, 40 (2004), 387-392.
doi: 10.1016/j.mcm.2003.10.048. |
[30] |
G. P. Menzala and E. Zuazua, Timoshenko's plate equations as a singular limit of the dynamical von Kármán system, J. Math. Pures Appl., 79 (2000), 73-94.
doi: 10.1016/S0021-7824(00)00149-5. |
[31] |
Y. Sugitani and S. Kawashima, Decay estimates of solutions to a semi-linear dissipative plate equation, J. Hyperbolic Differential Equations, accepted. |
[32] |
R. Teman, "Navier-Stokes Equations," Studies in Mathematics and Its Applications, Vol. 2, Revised Edition, North-Holland, Amsterdam, New York, Oxford, 1979. |
[33] |
G. Todorova and B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Differential Equations, 174 (2000), 464-489.
doi: 10.1006/jdeq.2000.3933. |
[34] |
G. Todorova and B. Yordanov, The energy decay problem for wave equations with nonlinear dissipative terms in $\mathbb{R}^{n}$, Indiana University Mathematics Journal, 56 (2007), 389-416.
doi: 10.1512/iumj.2007.56.2963. |
[35] |
M. A. Zarubinskaya and W. T. van Horssen, On aspects of boundary damping for a rectangular plate, Journal of Sound and Vibration, 292 (2006), 844-853.
doi: 10.1016/j.jsv.2005.09.008. |
[36] |
M. A. Zarubinskaya and W. T. van Horssen, On aspects of asymptotic for plate equations, Nonlinear Dynamics, 41 (2005), 403-413.
doi: 10.1007/s11071-005-1396-0. |
[37] |
X. Zhang and E. Zuazua, On the optimality of the observability inequalities for Kirchhoff plate systems with potentials in unbounded domains, in "Hyperbolic Problems: Theory, Numerics and Applications," Springer, (2008), 233-243.
doi: 10.1007/978-3-540-75712-2_19. |
show all references
References:
[1] |
P. Bergeret, Classification of smooth solutions to $2\times 2$ hyperbolic systems with boundary damping, Math. Methods Appl. Sci., 20 (1997), 1563-1598.
doi: 10.1002/(SICI)1099-1476(199712)20:18<1563::AID-MMA925>3.0.CO;2-9. |
[2] |
M. E. Bradley and S. Lenhart, Bilinear spatial control of the velocity term in a Kirchhoff plate equation, Electronic J. Differential Equations, 2001 (2001), 1-15. |
[3] |
C. Buriol, Energy decay rates for the Timoshenko system of thermoelastic plates, Nonlinear Analysis, 64 (2006), 92-108.
doi: 10.1016/j.na.2005.06.010. |
[4] |
R. C. Charão, E. Bisognin, V. Bisognin and A. F. Pazoto, Asymptotic behavior for a dissipative plate equation in $\mathbb{R}^{N}$ with periodic coefficients, Electronic J. Differential Equations, 2008 (2008), 23. |
[5] |
C. R. da Luz and R. C. Charão, Asymptotic properties for a semi-linear plate equation in unbounded domains, J. Hyperbolic Differential Equations, 6 (2009), 269-294.
doi: 10.1142/S0219891609001824. |
[6] |
Darmawijoyo and W. T. van Horssen, On boundary damping for a weakly nonlinear wave equation, Nonlinear Dynamics, 30 (2002), 179-191.
doi: 10.1023/A:1020473930223. |
[7] |
R. Denk, R. Racke and Y. Shibata, $L^p$ theory for the linear thermoelastic plate equations in bounded and exterior domains, Adv. Differential Equations, 14 (2009), 685-715. |
[8] |
W. Desch, K. B. Hannsgen and R. L. Wheeler, Passive boundary damping of viscoelastic structures, J. Integral Equations Appl., 8 (1996), 125-171.
doi: 10.1216/jiea/1181075934. |
[9] |
G. G. Doronin, N. A. Lar'kin and A. J. Souza, A hyperbolic problem with nonlinear second-order boundary damping, Electron J. Differential Equations, 1998 (1998), 1-10. |
[10] |
A. D. Drozdov and V. B. Kolmanovskii, "Stability in Viscoelasticity," North-Holland Series in Applied Mathematics and Mechanics, 38, North-Holland Publishing Co., Amsterdam, 1994. |
[11] |
Y. Enomoto, On a thermoelastic plate equation in an exterior domain, Math. Meth. Appl. Sci., 25 (2002), 443-472.
doi: 10.1002/mma.290. |
[12] |
M. Fabrizio and B. Lazzari, On the existence and the asymptotic stability of solutions for linear viscoelastic solids, Arch. Rational Mech. Anal., 116 (1991), 139-152.
doi: 10.1007/BF00375589. |
[13] |
T. Hosono and K. Kawashima, Decay property of regularity-loss type and application to some nonlinear hyperbolic-elliptic system, Math. Models Meth. Appl. Sci., 16 (2006), 1839-1859.
doi: 10.1142/S021820250600173X. |
[14] |
K. Ide and S. Kawashima, Decay property of regularity-loss type and nonlinear effects for dissipative Timoshenko system, Math. Models Meth. Appl. Sci., 18 (2008), 1001-1025.
doi: 10.1142/S0218202508002930. |
[15] |
I. Lasiecka, S. Maad and A. Sasane, Existence and exponential decay of solutions to a quasilinear thermoelastic plate system, Nonlinear Differ. Equ. Appl., 15 (2008), 689-715. |
[16] |
I. Lasiecka and J. Ong, Global solvability and uniform decays of solutions to quasilinear equation with nonlinear boundary dissipation, Comm. Partial Differential Equations, 24 (1999), 2069-2107.
doi: 10.1080/03605309908821495. |
[17] |
H. J. Lee, Uniform decay for solution of the plate equation with a boundary condition of memory type, Trends in Math., 9 (2006), 51-55. |
[18] |
W. Liu, Local boundary controllability for the semi-linear plate equation, Comm. Partial Differential Equations, 23 (1998), 201-221. |
[19] |
Y. Liu and W. Wang, The point-wise estimates of solutions for dissipative wave equation in multi-dimensions, Discrete Continuous Dynam. Systems - A, 20 (2008), 1013-1028. |
[20] |
Z. Liu and S. Zheng, On the exponential stability of linear viscoelasticity and thermo-viscoelasticity, Quart. Appl. Math., 54 (1996), 21-31. |
[21] |
Z. Liu and S. Zheng, "Semi-Groups Associated With Dissipative Systems," Chapman & Hall/CRC Research Notes in Mathematics, 398, Boca Raton, FL, 1999. |
[22] |
J. R. Luyo Sánchez, "O Sistema Dinámico de von Kármán en DomÍNios NÁO Limitados é Globalmente bem Posto no Sentido de Hadamard: Análise do seu Limite Singular," Ph.D Thesis, Institute of Mathematics, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil, 2003. |
[23] |
A. Matsumura, On the asymptotic behavior of semi-linear wave equations, Publ. Res. Inst. Math. Sci., 12 (1976), 169-189.
doi: 10.2977/prims/1195190962. |
[24] |
J. E. Muñoz Rivera, M. G. Naso and F. M. Vegni, Asymptotic behavior of the energy for a class of weakly dissipative second-order systems with memory, J. Math. Anal. Appl., 286 (2003), 692-704.
doi: 10.1016/S0022-247X(03)00511-0. |
[25] |
J. E. Muñoz Rivera and R. Racke, Global stability for damped Timoshenko systems, Discrete Continuous Dynam. Systems, 9 (2003), 1625-1639. |
[26] |
J. E. Muñoz Rivera, Asymptotic behavior in linear viscoelasticity, Quart. Appl. Math., 52 (1994), 628-648. |
[27] |
K. Nishihara, $L^p-L^q$ estimates of solutions to the damped wave equation in $3$-dimensional space and their applications, Math. Z., 244 (2003), 631-649. |
[28] |
J. Y. Park, Bilinear boundary optimal control of the velocity terms in a Kirchhoff plate equation, Trends in Math., 9 (2006), 41-44. |
[29] |
A. F. Pazoto, J. C. Vila Bravo and J. E. Muñoz Rivera, Asymptotic stability of semi-groups associated to linear weak dissipative systems, Math. Comput. Modeling, 40 (2004), 387-392.
doi: 10.1016/j.mcm.2003.10.048. |
[30] |
G. P. Menzala and E. Zuazua, Timoshenko's plate equations as a singular limit of the dynamical von Kármán system, J. Math. Pures Appl., 79 (2000), 73-94.
doi: 10.1016/S0021-7824(00)00149-5. |
[31] |
Y. Sugitani and S. Kawashima, Decay estimates of solutions to a semi-linear dissipative plate equation, J. Hyperbolic Differential Equations, accepted. |
[32] |
R. Teman, "Navier-Stokes Equations," Studies in Mathematics and Its Applications, Vol. 2, Revised Edition, North-Holland, Amsterdam, New York, Oxford, 1979. |
[33] |
G. Todorova and B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Differential Equations, 174 (2000), 464-489.
doi: 10.1006/jdeq.2000.3933. |
[34] |
G. Todorova and B. Yordanov, The energy decay problem for wave equations with nonlinear dissipative terms in $\mathbb{R}^{n}$, Indiana University Mathematics Journal, 56 (2007), 389-416.
doi: 10.1512/iumj.2007.56.2963. |
[35] |
M. A. Zarubinskaya and W. T. van Horssen, On aspects of boundary damping for a rectangular plate, Journal of Sound and Vibration, 292 (2006), 844-853.
doi: 10.1016/j.jsv.2005.09.008. |
[36] |
M. A. Zarubinskaya and W. T. van Horssen, On aspects of asymptotic for plate equations, Nonlinear Dynamics, 41 (2005), 403-413.
doi: 10.1007/s11071-005-1396-0. |
[37] |
X. Zhang and E. Zuazua, On the optimality of the observability inequalities for Kirchhoff plate systems with potentials in unbounded domains, in "Hyperbolic Problems: Theory, Numerics and Applications," Springer, (2008), 233-243.
doi: 10.1007/978-3-540-75712-2_19. |
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