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July  2011, 29(3): 1113-1139. doi: 10.3934/dcds.2011.29.1113

## Global existence and asymptotic behavior of solutions for quasi-linear dissipative plate equation

 1 Faculty of Mathematics, Kyushu University, Fukuoka 819-0395

Received  January 2010 Revised  June 2010 Published  November 2010

In this paper we focus on the initial value problem for quasi-linear dissipative plate equation in multi-dimensional space $(n\geq2)$. This equation verifies the decay property of the regularity-loss type, which causes the difficulty in deriving the global a priori estimates of solutions. We overcome this difficulty by employing a time-weighted $L^2$ energy method which makes use of the integrability of $||$(∂$^2_xu_t,$∂$^3_xu)(t)||_{L^{\infty}}$. This $L^\infty$ norm can be controlled by showing the optimal $L^2$ decay estimates for lower-order derivatives of solutions. Thus we obtain the desired a priori estimate which enables us to prove the global existence and asymptotic decay of solutions under smallness and enough regularity assumptions on the initial data. Moreover, we show that the solution can be approximated by a simple-looking function, which is given explicitly in terms of the fundamental solution of a fourth-order linear parabolic equation.
Citation: Yongqin Liu, Shuichi Kawashima. Global existence and asymptotic behavior of solutions for quasi-linear dissipative plate equation. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 1113-1139. doi: 10.3934/dcds.2011.29.1113
##### 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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