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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 & Continuous Dynamical Systems - A, 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.  doi: 10.1002/(SICI)1099-1476(199712)20:18<1563::AID-MMA925>3.0.CO;2-9.  Google Scholar [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.   Google Scholar [3] C. Buriol, Energy decay rates for the Timoshenko system of thermoelastic plates,, Nonlinear Analysis, 64 (2006), 92.  doi: 10.1016/j.na.2005.06.010.  Google Scholar [4] R. C. Charão, E. Bisognin, V. Bisognin and A. F. Pazoto, Asymptotic behavior for a dissipative plate equation in $\R^N$ with periodic coefficients,, Electronic J. Differential Equations, 2008 (2008).   Google Scholar [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.  doi: 10.1142/S0219891609001824.  Google Scholar [6] Darmawijoyo and W. T. van Horssen, On boundary damping for a weakly nonlinear wave equation,, Nonlinear Dynamics, 30 (2002), 179.  doi: 10.1023/A:1020473930223.  Google Scholar [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.   Google Scholar [8] W. Desch, K. B. Hannsgen and R. L. Wheeler, Passive boundary damping of viscoelastic structures,, J. Integral Equations Appl., 8 (1996), 125.  doi: 10.1216/jiea/1181075934.  Google Scholar [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.   Google Scholar [10] A. D. Drozdov and V. B. Kolmanovskii, "Stability in Viscoelasticity,", North-Holland Series in Applied Mathematics and Mechanics, 38 (1994).   Google Scholar [11] Y. Enomoto, On a thermoelastic plate equation in an exterior domain,, Math. Meth. Appl. Sci., 25 (2002), 443.  doi: 10.1002/mma.290.  Google Scholar [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.  doi: 10.1007/BF00375589.  Google Scholar [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.  doi: 10.1142/S021820250600173X.  Google Scholar [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.  doi: 10.1142/S0218202508002930.  Google Scholar [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.   Google Scholar [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.  doi: 10.1080/03605309908821495.  Google Scholar [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.   Google Scholar [18] W. Liu, Local boundary controllability for the semi-linear plate equation,, Comm. Partial Differential Equations, 23 (1998), 201.   Google Scholar [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.   Google Scholar [20] Z. Liu and S. Zheng, On the exponential stability of linear viscoelasticity and thermo-viscoelasticity,, Quart. Appl. Math., 54 (1996), 21.   Google Scholar [21] Z. Liu and S. Zheng, "Semi-Groups Associated With Dissipative Systems,", Chapman $&$ Hall/CRC Research Notes in Mathematics, 398 (1999).   Google Scholar [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, (2003).   Google Scholar [23] A. Matsumura, On the asymptotic behavior of semi-linear wave equations,, Publ. Res. Inst. Math. Sci., 12 (1976), 169.  doi: 10.2977/prims/1195190962.  Google Scholar [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.  doi: 10.1016/S0022-247X(03)00511-0.  Google Scholar [25] J. E. Muñoz Rivera and R. Racke, Global stability for damped Timoshenko systems,, Discrete Continuous Dynam. Systems, 9 (2003), 1625.   Google Scholar [26] J. E. Muñoz Rivera, Asymptotic behavior in linear viscoelasticity,, Quart. Appl. Math., 52 (1994), 628.   Google Scholar [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.   Google Scholar [28] J. Y. Park, Bilinear boundary optimal control of the velocity terms in a Kirchhoff plate equation,, Trends in Math., 9 (2006), 41.   Google Scholar [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.  doi: 10.1016/j.mcm.2003.10.048.  Google Scholar [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.  doi: 10.1016/S0021-7824(00)00149-5.  Google Scholar [31] Y. Sugitani and S. Kawashima, Decay estimates of solutions to a semi-linear dissipative plate equation,, J. Hyperbolic Differential Equations, ().   Google Scholar [32] R. Teman, "Navier-Stokes Equations,", Studies in Mathematics and Its Applications, 2 (1979).   Google Scholar [33] G. Todorova and B. Yordanov, Critical exponent for a nonlinear wave equation with damping,, J. Differential Equations, 174 (2000), 464.  doi: 10.1006/jdeq.2000.3933.  Google Scholar [34] G. Todorova and B. Yordanov, The energy decay problem for wave equations with nonlinear dissipative terms in $\R^n$,, Indiana University Mathematics Journal, 56 (2007), 389.  doi: 10.1512/iumj.2007.56.2963.  Google Scholar [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.  doi: 10.1016/j.jsv.2005.09.008.  Google Scholar [36] M. A. Zarubinskaya and W. T. van Horssen, On aspects of asymptotic for plate equations,, Nonlinear Dynamics, 41 (2005), 403.  doi: 10.1007/s11071-005-1396-0.  Google Scholar [37] X. Zhang and E. Zuazua, On the optimality of the observability inequalities for Kirchhoff plate systems with potentials in unbounded domains,, in, (2008), 233.  doi: 10.1007/978-3-540-75712-2_19.  Google Scholar

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.  doi: 10.1002/(SICI)1099-1476(199712)20:18<1563::AID-MMA925>3.0.CO;2-9.  Google Scholar [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.   Google Scholar [3] C. Buriol, Energy decay rates for the Timoshenko system of thermoelastic plates,, Nonlinear Analysis, 64 (2006), 92.  doi: 10.1016/j.na.2005.06.010.  Google Scholar [4] R. C. Charão, E. Bisognin, V. Bisognin and A. F. Pazoto, Asymptotic behavior for a dissipative plate equation in $\R^N$ with periodic coefficients,, Electronic J. Differential Equations, 2008 (2008).   Google Scholar [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.  doi: 10.1142/S0219891609001824.  Google Scholar [6] Darmawijoyo and W. T. van Horssen, On boundary damping for a weakly nonlinear wave equation,, Nonlinear Dynamics, 30 (2002), 179.  doi: 10.1023/A:1020473930223.  Google Scholar [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.   Google Scholar [8] W. Desch, K. B. Hannsgen and R. L. Wheeler, Passive boundary damping of viscoelastic structures,, J. Integral Equations Appl., 8 (1996), 125.  doi: 10.1216/jiea/1181075934.  Google Scholar [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.   Google Scholar [10] A. D. Drozdov and V. B. Kolmanovskii, "Stability in Viscoelasticity,", North-Holland Series in Applied Mathematics and Mechanics, 38 (1994).   Google Scholar [11] Y. Enomoto, On a thermoelastic plate equation in an exterior domain,, Math. Meth. Appl. Sci., 25 (2002), 443.  doi: 10.1002/mma.290.  Google Scholar [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.  doi: 10.1007/BF00375589.  Google Scholar [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.  doi: 10.1142/S021820250600173X.  Google Scholar [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.  doi: 10.1142/S0218202508002930.  Google Scholar [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.   Google Scholar [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.  doi: 10.1080/03605309908821495.  Google Scholar [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.   Google Scholar [18] W. Liu, Local boundary controllability for the semi-linear plate equation,, Comm. Partial Differential Equations, 23 (1998), 201.   Google Scholar [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.   Google Scholar [20] Z. Liu and S. Zheng, On the exponential stability of linear viscoelasticity and thermo-viscoelasticity,, Quart. Appl. Math., 54 (1996), 21.   Google Scholar [21] Z. Liu and S. Zheng, "Semi-Groups Associated With Dissipative Systems,", Chapman $&$ Hall/CRC Research Notes in Mathematics, 398 (1999).   Google Scholar [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, (2003).   Google Scholar [23] A. Matsumura, On the asymptotic behavior of semi-linear wave equations,, Publ. Res. Inst. Math. Sci., 12 (1976), 169.  doi: 10.2977/prims/1195190962.  Google Scholar [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.  doi: 10.1016/S0022-247X(03)00511-0.  Google Scholar [25] J. E. Muñoz Rivera and R. Racke, Global stability for damped Timoshenko systems,, Discrete Continuous Dynam. Systems, 9 (2003), 1625.   Google Scholar [26] J. E. Muñoz Rivera, Asymptotic behavior in linear viscoelasticity,, Quart. Appl. Math., 52 (1994), 628.   Google Scholar [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.   Google Scholar [28] J. Y. Park, Bilinear boundary optimal control of the velocity terms in a Kirchhoff plate equation,, Trends in Math., 9 (2006), 41.   Google Scholar [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.  doi: 10.1016/j.mcm.2003.10.048.  Google Scholar [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.  doi: 10.1016/S0021-7824(00)00149-5.  Google Scholar [31] Y. Sugitani and S. Kawashima, Decay estimates of solutions to a semi-linear dissipative plate equation,, J. Hyperbolic Differential Equations, ().   Google Scholar [32] R. Teman, "Navier-Stokes Equations,", Studies in Mathematics and Its Applications, 2 (1979).   Google Scholar [33] G. Todorova and B. Yordanov, Critical exponent for a nonlinear wave equation with damping,, J. Differential Equations, 174 (2000), 464.  doi: 10.1006/jdeq.2000.3933.  Google Scholar [34] G. Todorova and B. Yordanov, The energy decay problem for wave equations with nonlinear dissipative terms in $\R^n$,, Indiana University Mathematics Journal, 56 (2007), 389.  doi: 10.1512/iumj.2007.56.2963.  Google Scholar [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.  doi: 10.1016/j.jsv.2005.09.008.  Google Scholar [36] M. A. Zarubinskaya and W. T. van Horssen, On aspects of asymptotic for plate equations,, Nonlinear Dynamics, 41 (2005), 403.  doi: 10.1007/s11071-005-1396-0.  Google Scholar [37] X. Zhang and E. Zuazua, On the optimality of the observability inequalities for Kirchhoff plate systems with potentials in unbounded domains,, in, (2008), 233.  doi: 10.1007/978-3-540-75712-2_19.  Google Scholar
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