DECAY ESTIMATE AND GLOBAL EXISTENCE OF SEMILINEAR THERMOELASTIC TIMOSHENKO SYSTEM WITH TWO DAMPING EFFECTS

doi:10.1007/s10473-019-0601-z

数学物理学报(英文版) ›› 2019, Vol. 39 ›› Issue (6): 1461-1486.doi: 10.1007/s10473-019-0601-z

• 论文 • 下一篇

DECAY ESTIMATE AND GLOBAL EXISTENCE OF SEMILINEAR THERMOELASTIC TIMOSHENKO SYSTEM WITH TWO DAMPING EFFECTS

王维克¹, 薛锐²

1. School of Mathematical Sciences & Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, China;
2. School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China

收稿日期:2018-03-29 修回日期:2019-04-02 出版日期:2019-12-25 发布日期:2019-12-30
通讯作者: Rui XUE,E-mail:xuemantian008@sjtu.edu.cn E-mail:xuemantian008@sjtu.edu.cn
作者简介:Weike WANG,E-mail:wkwang@sjtu.edu.cn
基金资助:
The first author is supported by National Natural Science Foundation of China (11771284).

DECAY ESTIMATE AND GLOBAL EXISTENCE OF SEMILINEAR THERMOELASTIC TIMOSHENKO SYSTEM WITH TWO DAMPING EFFECTS

Weike WANG¹, Rui XUE²

1. School of Mathematical Sciences & Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, China;
2. School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China

Received:2018-03-29 Revised:2019-04-02 Online:2019-12-25 Published:2019-12-30
Contact: Rui XUE,E-mail:xuemantian008@sjtu.edu.cn E-mail:xuemantian008@sjtu.edu.cn
Supported by:
The first author is supported by National Natural Science Foundation of China (11771284).

摘要/Abstract

摘要： In this paper, we study a semilinear Timoshenko system with heat conduction having two damping effects. The observation that two damping effects might lead to smaller decay rates for solutions in comparison to one damping effect is rigorously proved here in providing optimality results. Moreover, the global well-posedness for small data is presented.

关键词: semilinear Timoshenko system, heat conduction, damping, optimal decay rate, eigenvalue expansion method

Abstract: In this paper, we study a semilinear Timoshenko system with heat conduction having two damping effects. The observation that two damping effects might lead to smaller decay rates for solutions in comparison to one damping effect is rigorously proved here in providing optimality results. Moreover, the global well-posedness for small data is presented.

Key words: semilinear Timoshenko system, heat conduction, damping, optimal decay rate, eigenvalue expansion method

中图分类号:

35A01

王维克, 薛锐. DECAY ESTIMATE AND GLOBAL EXISTENCE OF SEMILINEAR THERMOELASTIC TIMOSHENKO SYSTEM WITH TWO DAMPING EFFECTS[J]. 数学物理学报(英文版), 2019, 39(6): 1461-1486.

Weike WANG, Rui XUE. DECAY ESTIMATE AND GLOBAL EXISTENCE OF SEMILINEAR THERMOELASTIC TIMOSHENKO SYSTEM WITH TWO DAMPING EFFECTS[J]. Acta mathematica scientia,Series B, 2019, 39(6): 1461-1486.

参考文献

[1] Ammar-Khodja F, Benabdallah A, Muñoz Rivera J E, et al. Energy decay for Timoshenko systems of memory type. J Differential Equations, 2003, 194(1):82-115
[2] Chen J, Wang W K. The point-wise estimates for the solution of damped wave equation with nonlinear convection in multi-dimensional space. Commun Pure Appl Anal, 2014, 13(1):307-330
[3] Fernández Sare H D, Racke R. On the stability of damped Timoshenko systems:Cattaneo versus Fourier law. Arch Ration Mech Anal, 2009, 194(1):221-251
[4] Guesmia A, Messaoudi S A. A general stability result in a Timoshenko system with infinite memory:a new approach. Math Methods Appl Sci, 2014, 37(3):384-392
[5] Ide K, Haramoto K, and Kawashima S. Decay property of regularity-loss type for dissipative Timoshenko system. Math Models Methods Appl Sci, 2008, 18(5):647-667
[6] Kafini M, Messaoudi S A, Mustafa M I. Energy decay rates for a Timoshenko-type system of thermoelasticity of type III with constant delay. Appl Anal, 2014, 93(6):1201-1216
[7] Kirane M, Said-Houari B, Anwar M N. Stability result for the Timoshenko system with a time-varying delay term in the internal feedbacks. Commun Pure Appl Anal, 2011, 10(2):667-686
[8] Li F B, Wang W K. The pointwise estimate of solutions to the parabolic conservation law in multidimensions. Nonlinear Differ Equ Appl, 2014, 21(1):87-103
[9] Liu Y Q. Asymptotic behavior of solutions to a nonlinear plate equation with memory. Commun Pure Appl Anal, 2017, 16(2):533-556
[10] Liu Y Q, Wang W K. The pointwise estimates of solutions for dissipative wave equation in multi-dimensions. Discrete Contin Dyn Syst, 2008, 20(4):1013-1028
[11] Mao S K, Liu Y Q. Decay property for solutions to plate type equations with variable coefficients. Kinet Relat Models, 2017, 10(3):785-797
[12] Matsumura A. On the asymptotic behavior of solutions of semi-linear wave equations. Publ Res Inst Math Sci, 1976/77, 12(1):169-189
[13] Messaoudi S A, Mustafa M I. A stability result in a memory-type Timoshenko system. Dynam Systems Appl, 2009, 18(3/4):457-468
[14] Messaoudi S A, Pokojovy M, Said-Houari B. Nonlinear damped Timoshenko systems with second sound-global existence and exponential stability. Math Methods Appl Sci, 2009, 32(5):505-534
[15] Messaoudi S A, Said-Houari B. Energy decay in a Timoshenko-type system of thermoelasticity of type III. J Math Anal Appl, 2008, 348(1):298-307
[16] Mori N, Kawashima S. Decay property for the Timoshenko system with Fourier's type heat conduction. J Hyperbolic Differ Equ, 2014, 11(1):135-157
[17] Muñoz Rivera J E, Racke R. Global stability for damped Timoshenko systems. Discrete Contin Dyn Syst, 2003, 9(6):1625-1639
[18] Muñoz Rivera J E, Fernández Sare H D. Stability of Timoshenko systems with past history. J Math Anal Appl, 2008, 339(1):482-502
[19] Muñoz Rivera J E, Racke R. Mildly dissipative nonlinear Timoshenko systems-global existence and exponential stability. J Math Anal Appl, 2002, 276(1):248-278
[20] Pachpatte B G. Inequalities for Differential and Integral Equations. volume 197 of Mathematics in Science and Engineering. San Diego, CA:Academic Press, Inc, 1998
[21] Pei P, Rammaha M A, Toundykov D. Global well-posedness and stability of semilinear Mindlin-Timoshenko system. J Math Anal Appl, 2014, 418(2):535-568
[22] Pei P, Rammaha M A, Toundykov D. Local and global well-posedness of semilinear Reissner-MindlinTimoshenko plate equations. Nonlinear Anal, 2014, 105:62-85
[23] Racke R. Lectures on Nonlinear Evolution Equations-Initial Value Problems. 2nd ed. Cham:Birkhäuser/Springer, 2015
[24] Racke R, Said-Houari B. Global existence and decay property of the Timoshenko system in thermoelasticity with second sound. Nonlinear Anal, 2012, 75(13):4957-4973
[25] Racke R, Said-Houari B. Decay rates and global existence for semilinear dissipative Timoshenko systems. Quart Appl Math, 2013, 71(2):229-266
[26] Racke R, Wang W K, Xue R. Optimal decay rates and global existence for a semilinear timoshenko system with two damping effects. Math Methods Appl Sci, 2017, 40(1):210-222
[27] Said-Houari B, Kasimov A. Damping by heat conduction in the Timoshenko system:Fourier and Cattaneo are the same. J Differential Equations, 2013, 255(4):611-632
[28] Said-Houari B, Laskri Y. A stability result of a Timoshenko system with a delay term in the internal feedback. Appl Math Comput, 2010, 217(6):2857-2869
[29] Santos M L, Almeida Júnior D S, Rodrigues J H etal. Decay rates for Timoshenko system with nonlinear arbitrary localized damping. Differential Integral Equations, 2014, 27(1/2):1-26
[30] Segal I. Quantization and dispersion for nonlinear relativistic equations//Mathematical Theory of Elementary Particles. Cambridge, Mass:MIT Press, 1966:79-108
[31] Soufyane A, Said-Houari B. The effect of the wave speeds and the frictional damping terms on the decay rate of the bresse system. Evolution Equations and Control Theory, 2014, 3(4):713-738
[32] Soufyane A, Wehbe A. Uniform stabilization for the Timoshenko beam by a locally distributed damping. Electron J Differential Equations, 2003, pages No. 29, 14 pp
[33] Timoshenko S. On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Philisophical Magazine, 1921, 41:744-746
[34] Ueda Y, Duan R J, Kawashima S. Decay structure for symmetric hyperbolic systems with non-symmetric relaxation and its application. Arch Ration Mech Anal, 2012, 205(1):239-266
[35] Wang W K, Yang T. The pointwise estimates of solutions for euler equations with damping in multidimensions. J Differential Equations, 2001, 173(2):410-450
[36] Wu Z G, Wang W K. Pointwise estimate of solution for non-isentropic Navier-Stokes-Poisson equations in multi-dimensions. Acta Math Sci, 2012, 32B(5):1681-1702
[37] Wu Y H, Xue X P. Boundary feedback stabilization of Kirchhoff-type Timoshenko system. J Dyn Control Syst, 2014, 20(4):523-538

DECAY ESTIMATE AND GLOBAL EXISTENCE OF SEMILINEAR THERMOELASTIC TIMOSHENKO SYSTEM WITH TWO DAMPING EFFECTS

DECAY ESTIMATE AND GLOBAL EXISTENCE OF SEMILINEAR THERMOELASTIC TIMOSHENKO SYSTEM WITH TWO DAMPING EFFECTS

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 13

Metrics

本文评价

推荐阅读 8

[1]	丁时进, 黄丙远, 黎泉荣. GLOBAL EXISTENCE AND DECAY ESTIMATES FOR THE CLASSICAL SOLUTIONS TO A COMPRESSIBLE FLUID-PARTICLE INTERACTION MODEL[J]. 数学物理学报(英文版), 2019, 39(6): 1525-1537.
[2]	李秀文, 刘振海, 李景, Chris TISDELL. EXISTENCE AND CONTROLLABILITY FOR NONLINEAR FRACTIONAL CONTROL SYSTEMS WITH DAMPING IN HILBERT SPACES[J]. 数学物理学报(英文版), 2019, 39(1): 229-242.
[3]	孔德兴, 刘琦. GLOBAL EXISTENCE OF CLASSICAL SOLUTIONS TO THE HYPERBOLIC GEOMETRY FLOW WITH TIME-DEPENDENT DISSIPATION[J]. 数学物理学报(英文版), 2018, 38(3): 745-755.
[4]	侯飞. ON THE GLOBAL EXISTENCE OF SMOOTH SOLUTIONS TO THE MULTI-DIMENSIONAL COMPRESSIBLE EULER EQUATIONS WITH TIME-DEPENDING DAMPING IN HALF SPACE[J]. 数学物理学报(英文版), 2017, 37(4): 949-964.
[5]	袁益让, 杨青, 李长峰, 孙同军. NUMERICAL METHOD OF MIXED FINITE VOLUME-MODIFIED UPWIND FRACTIONAL STEP DIFFERENCE FOR THREE-DIMENSIONAL SEMICONDUCTOR DEVICE TRANSIENT BEHAVIOR PROBLEMS[J]. 数学物理学报(英文版), 2017, 37(1): 259-279.
[6]	Aissa GUESMIA, Salim MESSAOUDI. SOME STABILITY RESULTS FOR TIMOSHENKO SYSTEMS WITH COOPERATIVE FRICTIONAL AND INFINITE-MEMORY DAMPINGS IN THE DISPLACEMENT[J]. 数学物理学报(英文版), 2016, 36(1): 1-33.
[7]	吴舜堂. GENERAL DECAY OF SOLUTIONS FOR A VISCOELASTIC EQUATION WITH BALAKRISHNAN-TAYLOR DAMPING AND NONLINEAR BOUNDARY DAMPING-SOURCE INTERACTIONS[J]. 数学物理学报(英文版), 2015, 35(5): 981-994.
[8]	吴云顺, 谭忠. ASYMPTOTIC BEHAVIOR OF THE STOKES APPROXIMATION EQUATIONS FOR COMPRESSIBLE FLOWS IN R³[J]. 数学物理学报(英文版), 2015, 35(3): 746-760.
[9]	张映辉, 吴国春. GLOBAL EXISTENCE AND ASYMPTOTIC BEHAVIOR FOR THE 3D COMPRESSIBLE NON–ISENTROPIC EULER EQUATIONS WITH DAMPING[J]. 数学物理学报(英文版), 2014, 34(2): 424-434.
[10]	陈化, 刘功伟. GLOBAL EXISTENCE, UNIFORM DECAY AND EXPONENTIAL GROWTH FOR A CLASS OF SEMI-LINEAR WAVE EQUATION WITH STRONG DAMPING[J]. 数学物理学报(英文版), 2013, 33(1): 41-58.
[11]	Abderrahmane ZARAI, Nasser-eddine TATAR, Salem ABDELMALEK. ELASTIC MEMBRANE EQUATION WITH MEMORY TERM AND NONLINEAR BOUNDARY DAMPING: GLOBAL EXISTENCE, DECAY AND BLOWUP OF THE SOLUTION[J]. 数学物理学报(英文版), 2013, 33(1): 84-106.
[12]	蔡晓静, 雷利华. L² DECAY OF THE INCOMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DAMPING[J]. 数学物理学报(英文版), 2010, 30(4): 1235-1248.
[13]	李胜家. TIME DEPENDENT ELASTIC SYSTEM[J]. 数学物理学报(英文版), 1995, 15(S1): 39-52.