带有热记忆的非均匀柔性结构的长时间动力行为
Long-time Dynamics of a Non-uniform Flexible Structure with Thermal Memory
本文研究了带有热效应的非均匀柔性结构方程,并且该热效应符合Coleman-Gurtin定律.利用半群方法,建立了系统的整体适定性.主要结论是该系统的长时间动力行为.本文证明了系统的拟稳定性,整体吸引子的存在性以及整体吸引子具有有限的分形维数.此外,还证明了指数吸引子的存在性.
This paper is concerned with a non-uniform flexible structure with thermal effect governed by Coleman-Gurtin law. By using semi-group method, we establish the global well-posedness of the system. The main result is the long-time dynamics of the system. We prove the quasi-stability property of the system and obtain the existence of a global attractor, and the global attractor has finite fractal dimension. The existence of exponential attractors is also proved.
柔性结构 / 记忆项 / 拟稳定性 / 整体吸引子 / 指数吸引子 {{custom_keyword}} /
flexible structure / memory / quasi-stability / global attractor / exponential attractor {{custom_keyword}} /
[1] Alves M. S., Gamboa P., Gorain G. C., et al., Asymptotic behavior of a flexible structure with Cattaneo type of thermal effect, Indag. Math., 2016, 27:821-834.
[2] Araujo R. O., Ma T. F., Long-time behavior of quasilinear viscoelastic equation with past history, J. Differential Equations, 2013, 254:4066-4087.
[3] Barbosa A. R. A., Ma T. F., long-time dynamics of an extensible plate equation with thermal memory, J. Math. Anal. Appl., 2014, 416:143-165.
[4] Carlson D. E., Linear Thermoelasticity, in:Handbook of Physics, Springer-Verlag, Berlin, 1972.
[5] Cavalcanti M. M., Fatori L. H., Ma T. F., Attractors for wave equations with degenerate memory, J. Differential Equations, 2016, 260:56-83.
[6] Chueshov I. D., Introduction to the Theory of Infinite Dimensional Dissipative Systems, Acta Scientific Publishing House, Kharkiv, Ukraine, 2002.
[7] Chueshov I. D., Dynamics of Quasi-stable Dissipative Systems, Springer, New York, 2015.
[8] Chueshov I. D., Lasiecka I., Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping, American Mathematical Society, Providence, RI, 2008.
[9] Chueshov I. D., Lasiecka I., Von Karman Evolution Equations, Springer-Verlag, Berlin, 2012.
[10] Chueshov I. D., Lasiecka I., Existence, uniqueness of weak solutions and global attractors for a class of nonlinear 2D Kirchhoff-Boussinesq models, Discrete Contin. Dyn. Syst., 2006, 15:777-809.
[11] Chueshov I. D., Lasiecka I., On global attractors for 2D Kirchhoff-Boussinesq model with supercritical nonlinearity, Comm. Partial Diff. Equa., 2011, 36:67-99.
[12] Dafermos C. M., Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 1970, 37:297-308.
[13] Fatori H., Jorge Silva M. A., Ma T. F., et al., Long-time behavior of a class of thermoelastic plates with nonlinear strain, J. Differential Equations, 2015, 259:4831-4862.
[14] Feng B., Long-time dynamics of a plate equation with memory and time delay, Bull. Braz. Math. Soc. New Series, 2018, 49:395-418.
[15] Feng B., Yang X. G., Qin Y., Uniform attractors for a nonautonomous extensible plate equation with a strong damping, Math. Meth. Appl. Sci., 2017, 40:3479-3492.
[16] Giorgi C., Grasseli M., Pata V., Well-posedness and longtime behavior of the phase-field model with memory in a history space setting, Q. Appl. Math., 2001, 59:701-736.
[17] Giorgi C., Marzocchi A., Pata V., Asymptotic behavior of a semilinear problem in heat conduction with memory, NoDEA:Nonlinear Differ. Equ. Appl., 1998, 5:333-354.
[18] Giorgi C., Naso M. G., Pata V., Exponential stability in linear heat conduction with memory:a semigroup approach, Commun. Appl. Anal., 2001, 5:121-133.
[19] Gorain G. C., Exponential stabilization of longitudinal vibrations of an inhomogeneous beam, J. Math. Sci., 2014, 198:245-251.
[20] Hale J. K., Asymptotic Behavior of Dissipative Systems, Math. Surveys Monogr., American Mathematical Society, Providence, RI, 1988.
[21] Jorge Silva M. A., Ma T. F., Long-time dynamics for a class of Kirchhoff models with memory, J. Math. Phys., 2013, 54, 021505.
[22] Liu K., Liu Z., Exponential decay of energy of the Euler-Bernoulli beam with locally distributed Kelvin-Voigt damping, SIAM J. Control Optim., 1998, 36:1086-1098.
[23] Ma T. F., Pelicer M. L., Attractors for weakly damped beam equations with p-Laplacian, Discrete Contin. Dyn. Sys., 2013, Supplement:513-522.
[24] Ma T. F., Narciso V., Global attractor for a model of extensible beam with nonlinear damping and source terms, Nonlinear Anal., 2010, 73:3402-3412.
[25] Misra S., Alves M., Gorain G. C., et al., Stability of the vibrations of an inhomogeneous flexible structure with thermal effect, Int. J. Dynam. Control, 2015, 3:354-362.
[26] Nandi P. K., Gorain G. C., Kar S., A note on stability of longitudinal vibrations of an inhomogeneous beam, Appl. Math., 2012, 3:19-23.
[27] Pazy A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
[28] Robinson J. C., Infinite-Dimensional Dynamical Systems:An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, England, 2001.
[29] Temam R., Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988.
[30] Yang Z., longtime behavior for a nonlinear wave equation arising in elasto-plastic flow, Math. Methods Appl. Sci., 2009, 32:1082-1104.
[31] Yang Z., Finite-dimensional attractors for the Kirchhoff models, J. Math. Phys., 2010, 51, 092703.
[32] Yang Z., Global attractor and their Hausdorff dimensions for a class of Kirchhoff models, J. Math. Phys., 2010, 51, 032701.
国家自然科学基金(11701465,11701012,61761002);宁夏自然科学基金(2020AAC03233)北方民族大学重大专项(ZDZX201901);北方民族大学校级科研项目(2018XYZSX02)
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