June  2011, 4(2): 531-547. doi: 10.3934/krm.2011.4.531

Decay property for a plate equation with memory-type dissipation

1. 

Department of Mathematics, North China Electric Power University, Beijing 102208, China

2. 

Faculty of Mathematics, Kyushu University, Fukuoka 819-0395

Received  September 2010 Revised  December 2010 Published  April 2011

In this paper we focus on the initial value problem of the semi-linear plate equation with memory in multi-dimensions $(n\geq1)$, the decay structure of which is of regularity-loss property. By using Fourier transform and Laplace transform, we obtain the fundamental solutions and thus the solution to the corresponding linear problem. Appealing to the point-wise estimate in the Fourier space of solutions to the linear problem, we get estimates and properties of solution operators, by exploiting which decay estimates of solutions to the linear problem are obtained. Also by introducing a set of time-weighted Sobolev spaces and using the contraction mapping theorem, we obtain the global in-time existence and the optimal decay estimates of solutions to the semi-linear problem under smallness assumption on the initial data.
Citation: Yongqin Liu, Shuichi Kawashima. Decay property for a plate equation with memory-type dissipation. Kinetic & Related Models, 2011, 4 (2) : 531-547. doi: 10.3934/krm.2011.4.531
References:
[1]

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

[2]

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

[3]

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, 46 (2008). Google Scholar

[4]

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

[5]

P. M. N. Dharmawardane, J. E. Muñoz Rivera and S. Kawashima, Decay property for second order hyperbolic systems of viscoelastic materials,, J. Math. Anal. Appl., 366 (2010), 621. doi: 10.1016/j.jmaa.2009.12.019. Google Scholar

[6]

A. D. Drozdov and V. B. Kolmanovskii, "Stability in Viscoelasticity,", North-Holland Publishing Co., Amsterdam, 1994., 38 (). Google Scholar

[7]

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

[8]

T. Hosono and S. 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

[9]

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

[10]

Y. Liu and S. Kawashima, Global existence and asymptotic behavior of solutions for quasi-linear dissipative plate equation,, Discrete Contin. Dyn. Syst., 29 (2011), 1113. Google Scholar

[11]

Z. Liu and S. Zheng, On the exponential stability of linear viscoelasticity and thermo-viscoelasticity,, Quart. Appl. Math., 54 (1996), 21. Google Scholar

[12]

Z. Liu and S. Zheng, "Semi-Groups Associated with Dissipative Systems,", Chapman $&$ Hall/CRC, (1999). Google Scholar

[13]

J. E. Muñoz Rivera, Asymptotic behavior in linear viscoelasticity,, Quart. Appl. Math., 52 (1994), 628. Google Scholar

[14]

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

[15]

J. E. Muñoz Rivera and R. Racke, Global stability for damped Timoshenko systems,, Discrete Contin. Dyn. Syst., 9 (2003), 1625. doi: 10.3934/dcds.2003.9.1625. Google Scholar

[16]

A. F. Pazoto, J. C. Vila Bravo and J. E. Muñoz Rivera, Asymptotic stability of semi-groups associated to linear weak dissipative systems,, Math. Computer Modeling, 40 (2004), 387. doi: 10.1016/j.mcm.2003.10.048. Google Scholar

[17]

Y. Sugitani and S. Kawashima, Decay estimates of solutions to a semi-linear dissipative plate equation,, J. Hyperbolic Differential Equations, 7 (2010), 471. doi: 10.1142/S0219891610002207. Google Scholar

[18]

X. Zhang and E. Zuazua, On the optimality of the observability inequalities for Kirchhoff plate systems with potentials in unbounded domains,, Springer, 2008, 233--243., (). doi: 10.1007/978-3-540-75712-2_19. Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

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, 46 (2008). Google Scholar

[4]

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

[5]

P. M. N. Dharmawardane, J. E. Muñoz Rivera and S. Kawashima, Decay property for second order hyperbolic systems of viscoelastic materials,, J. Math. Anal. Appl., 366 (2010), 621. doi: 10.1016/j.jmaa.2009.12.019. Google Scholar

[6]

A. D. Drozdov and V. B. Kolmanovskii, "Stability in Viscoelasticity,", North-Holland Publishing Co., Amsterdam, 1994., 38 (). Google Scholar

[7]

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

[8]

T. Hosono and S. 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

[9]

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

[10]

Y. Liu and S. Kawashima, Global existence and asymptotic behavior of solutions for quasi-linear dissipative plate equation,, Discrete Contin. Dyn. Syst., 29 (2011), 1113. Google Scholar

[11]

Z. Liu and S. Zheng, On the exponential stability of linear viscoelasticity and thermo-viscoelasticity,, Quart. Appl. Math., 54 (1996), 21. Google Scholar

[12]

Z. Liu and S. Zheng, "Semi-Groups Associated with Dissipative Systems,", Chapman $&$ Hall/CRC, (1999). Google Scholar

[13]

J. E. Muñoz Rivera, Asymptotic behavior in linear viscoelasticity,, Quart. Appl. Math., 52 (1994), 628. Google Scholar

[14]

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

[15]

J. E. Muñoz Rivera and R. Racke, Global stability for damped Timoshenko systems,, Discrete Contin. Dyn. Syst., 9 (2003), 1625. doi: 10.3934/dcds.2003.9.1625. Google Scholar

[16]

A. F. Pazoto, J. C. Vila Bravo and J. E. Muñoz Rivera, Asymptotic stability of semi-groups associated to linear weak dissipative systems,, Math. Computer Modeling, 40 (2004), 387. doi: 10.1016/j.mcm.2003.10.048. Google Scholar

[17]

Y. Sugitani and S. Kawashima, Decay estimates of solutions to a semi-linear dissipative plate equation,, J. Hyperbolic Differential Equations, 7 (2010), 471. doi: 10.1142/S0219891610002207. Google Scholar

[18]

X. Zhang and E. Zuazua, On the optimality of the observability inequalities for Kirchhoff plate systems with potentials in unbounded domains,, Springer, 2008, 233--243., (). doi: 10.1007/978-3-540-75712-2_19. Google Scholar

[1]

Yongqin Liu. The point-wise estimates of solutions for semi-linear dissipative wave equation. Communications on Pure & Applied Analysis, 2013, 12 (1) : 237-252. doi: 10.3934/cpaa.2013.12.237

[2]

Priyanjana M. N. Dharmawardane. Decay property of regularity-loss type for quasi-linear hyperbolic systems of viscoelasticity. Conference Publications, 2013, 2013 (special) : 197-206. doi: 10.3934/proc.2013.2013.197

[3]

Jiao Chen, Weike Wang. The point-wise estimates for the solution of damped wave equation with nonlinear convection in multi-dimensional space. Communications on Pure & Applied Analysis, 2014, 13 (1) : 307-330. doi: 10.3934/cpaa.2014.13.307

[4]

Shikuan Mao, Yongqin Liu. Decay of solutions to generalized plate type equations with memory. Kinetic & Related Models, 2014, 7 (1) : 121-131. doi: 10.3934/krm.2014.7.121

[5]

Jianhai Bao, Xing Huang, Chenggui Yuan. New regularity of kolmogorov equation and application on approximation of semi-linear spdes with Hölder continuous drifts. Communications on Pure & Applied Analysis, 2019, 18 (1) : 341-360. doi: 10.3934/cpaa.2019018

[6]

Gang Luo, Qingzhi Yang. The point-wise convergence of shifted symmetric higher order power method. Journal of Industrial & Management Optimization, 2017, 13 (5) : 0-0. doi: 10.3934/jimo.2019115

[7]

Li Ma, Lin Zhao. Regularity for positive weak solutions to semi-linear elliptic equations. Communications on Pure & Applied Analysis, 2008, 7 (3) : 631-643. doi: 10.3934/cpaa.2008.7.631

[8]

Út V. Lê. Contraction-Galerkin method for a semi-linear wave equation. Communications on Pure & Applied Analysis, 2010, 9 (1) : 141-160. doi: 10.3934/cpaa.2010.9.141

[9]

Shaokuan Chen, Shanjian Tang. Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. Mathematical Control & Related Fields, 2015, 5 (3) : 401-434. doi: 10.3934/mcrf.2015.5.401

[10]

Petteri Harjulehto, Peter Hästö, Juha Tiirola. Point-wise behavior of the Geman--McClure and the Hebert--Leahy image restoration models. Inverse Problems & Imaging, 2015, 9 (3) : 835-851. doi: 10.3934/ipi.2015.9.835

[11]

Xiaojie Wang. Weak error estimates of the exponential Euler scheme for semi-linear SPDEs without Malliavin calculus. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 481-497. doi: 10.3934/dcds.2016.36.481

[12]

Jason R. Morris. A Sobolev space approach for global solutions to certain semi-linear heat equations in bounded domains. Conference Publications, 2009, 2009 (Special) : 574-582. doi: 10.3934/proc.2009.2009.574

[13]

Henri Schurz. Analysis and discretization of semi-linear stochastic wave equations with cubic nonlinearity and additive space-time noise. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 353-363. doi: 10.3934/dcdss.2008.1.353

[14]

Enrique Fernández-Cara, Arnaud Münch. Numerical null controllability of semi-linear 1-D heat equations: Fixed point, least squares and Newton methods. Mathematical Control & Related Fields, 2012, 2 (3) : 217-246. doi: 10.3934/mcrf.2012.2.217

[15]

Hongjie Dong, Dapeng Du. Global well-posedness and a decay estimate for the critical dissipative quasi-geostrophic equation in the whole space. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1095-1101. doi: 10.3934/dcds.2008.21.1095

[16]

Romain Aimino, Huyi Hu, Matthew Nicol, Andrei Török, Sandro Vaienti. Polynomial loss of memory for maps of the interval with a neutral fixed point. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 793-806. doi: 10.3934/dcds.2015.35.793

[17]

Barbara Brandolini, Francesco Chiacchio, Jeffrey J. Langford. Estimates for sums of eigenvalues of the free plate via the fourier transform. Communications on Pure & Applied Analysis, 2020, 19 (1) : 113-122. doi: 10.3934/cpaa.2020007

[18]

Yongqin Liu. Asymptotic behavior of solutions to a nonlinear plate equation with memory. Communications on Pure & Applied Analysis, 2017, 16 (2) : 533-556. doi: 10.3934/cpaa.2017027

[19]

Tuan Anh Dao, Michael Reissig. $ L^1 $ estimates for oscillating integrals and their applications to semi-linear models with $ \sigma $-evolution like structural damping. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5431-5463. doi: 10.3934/dcds.2019222

[20]

Nguyen Thieu Huy, Vu Thi Ngoc Ha, Pham Truong Xuan. Boundedness and stability of solutions to semi-linear equations and applications to fluid dynamics. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2103-2116. doi: 10.3934/cpaa.2016029

2018 Impact Factor: 1.38

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (16)

Other articles
by authors

[Back to Top]