In this paper, we are concerned with one-dimensional Timoshenko model for a beam with nonlinear damping and source terms. Under suitable conditions on the initial data, the theorem of global existence is proved by potential well method combined Galerkin procedure, and decay estimates of the energy is established by means of Nakao’s inequality.
| Published in | International Journal of Theoretical and Applied Mathematics (Volume 2, Issue 1) |
| DOI | 10.11648/j.ijtam.20160201.11 |
| Page(s) | 1-6 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2016. Published by Science Publishing Group |
Timoshenko System, Source Term, Damping Term, Global Existence, Stability
| [1] | S. Timoshenko. On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Philosophical Magazine, vol. 41, pp.744–746, 1921. |
| [2] | S. A. Messaoudi, M. I. Mustafa. On the internal and boundary stabilization of Timoshenko beams. Nonlinear Differential Equations and Applications, vol. 15, pp.655– 671, 2008. |
| [3] | F. Alaban-Boussouira. Asympotatic behavior for Timoshenko beams subject to a single nonlinear feedback control. Nonlinear Differential Equations and Applications, vol. 14, pp.643–669, 2007. |
| [4] | F. Ammar-Khodja, A. Benabdallah, J. E. Munoz-Rivera, R. Racke. Energy decay for Timoshenko system of memory type. J. Diff. Equa, vol. 194, no.1, pp.82–115, 2003. |
| [5] | S. A. Messaoudi, B. Said-Houari. Uniform decay in a Timoshenko system with past history. J. Math. Anal. Appl., vol. 360, pp. 458–475, 2009. |
| [6] | N.E. Tatar. Stabilization of a viscoelastic Timoshenko beam. Applicable Analysis, vol. 92, no.1, pp. 27–43, 2013. |
| [7] | J. U. Kim, Y. Renardy. Boundary control of the Timoshenko beam. SIAM J. Contral Optim., vol. 25, no. 8, pp. 1417–1429, 1987. |
| [8] | S. A. Messaoudi, A. Soufyame. Boundary stabilization of a nonlinear system of Timoshenko type. Nonlinear Analysis, vol. 67, pp. 2107–2112, 2007. |
| [9] | A. Parentz, M. Milla Miranda, L. P. San Gil Jutuca. On local solution for a nonlinear Timoshenko system. Proceedings of the 55 SBA. UFU. Uberlandia-MG-Brazil, pp.167–179, 2002. |
| [10] | F. D. Araruua, J. E. S. Borges. Existence and boundary stabilization of the semilinear Mindlin-Timoshenko system. Electromic J. of Qualitative theory of Diff. Equa., vol. 34, pp. 1–27, 2008. |
| [11] | I. Chueshov, I. Lasiecka. Global attractors for Mindlin-Tomoshenko plate and for their Kirchhoff limits. Milan J. Math., vol. 74, pp. 117–138, 2006. |
| [12] | C.Gorgi and F. M. Vegni. Uniform energy estimates for a semilinear evolution equation of the Mindlin-Timoshenko beam with memory. Mathematical and Computer Modelling, vol.39, pp. 1005-1021, 2004. |
| [13] | A. Soufyame, M. Afilal, T. Aouam. General decy of solutions of a nonlinear Timoshenko system with a boundary control of memory type. Differential and Entegrd Equations, vol. 22, pp. 1125–1139, 2009. |
| [14] | P. Pei, M. A. Rammaha, D. Toundykov. Local and global well-posedness of semilinear Reissner-Mindlin-Timoshenko plate equations. Nonlinear Analysis, vol. 105, pp. 62 – 85, 2014. |
| [15] | P. Pei, M. A. Rammaha, D. Toundykov. Global well-posedness and stability of semilinear Mindlin-Timoshenko systems. J. Math. Anal. Appl., vol. 418, pp. 535–568, 2014. |
| [16] | D. H. Sattinger. On global solutions of nonlinear hyperbolic equation. Arch. Rational Mach. Anal, vol. 30, pp. 108 – 172, 1968. |
| [17] | I. E. Payne, D. H. Sattinger. Saddle points and instability of nonlinear hyperbolic equations. Israel J. Math., vol. 22, pp. 273–303, 1975. |
| [18] | M. Nakao. Asymptotic stability of the bounded or almost periodic solution of the wave equation with nonlinear dissipative term. J. Math. Anal. Appl., vol. 56, pp. 336–347, 1977. |
| [19] | K. Agre, M. A. Rammaha. System of nonlinear wave equation with damping and source terms. Differential and Integral Equation, vol. 19, no. 1, pp. 1235–1270, 2006. |
| [20] | R.Adams. Sobolev Spaces, Second Ed., Academic Press, Elsevier Science, 2003. |
| [21] | S. L. Sobolev. Applications of functional analysis in mathematical physics, Izdat. Leningrad. Gos. Univ., Leningrad, 1950; English trans., Amer. Math. Soc., Providence, R.I., 1963. |
| [22] | Lions, J.-L., Quelques methodes de resolution des problemas aux limites nolineaires, Paris, Dunod, 1969. |
| [23] | Mahammad A. Nurmammadov. The existence and uniqueness of a new boundary value problem (type of problem ‘‘E’’) for linear system equations of the mixed hyperbolic-elliptic type in the multivariate dimension with the changing time direction. Abstract and Applied Analysis, Volume 2015, Article ID 703652, pp. 1-10, 2015. |
| [24] | Mahammad A. Nurmammadov. The solvability of a new boundary value problem with derivatives on the boundary conditions for forward-backward linear systems mixed of Keldysh type in multivariate dimension. International Journal of Theoretical and Applied Mathematics, vol. 1, no. 1, pp. 1-9, 2015. |
| [25] | Mahammad A. Nurmammadov. The solvability of a new boundary value problem with derivatives on the boundary conditions for forward-backward semi linear systems of mixed equations of Keldysh type in multivariate dimension. International Journal of Theoretical and Applied Mathematics, vol. 1, no. 1, pp. 10-20, 2015. |
APA Style
Qingying Hu, Jian Dang, Hongwei Zhang. (2016). Existence and Stability of Solutions for Semilinear Timoshenko System with Damping and Source Terms. International Journal of Theoretical and Applied Mathematics, 2(1), 1-6. https://doi.org/10.11648/j.ijtam.20160201.11
ACS Style
Qingying Hu; Jian Dang; Hongwei Zhang. Existence and Stability of Solutions for Semilinear Timoshenko System with Damping and Source Terms. Int. J. Theor. Appl. Math. 2016, 2(1), 1-6. doi: 10.11648/j.ijtam.20160201.11
Share This Article
Twitter
Linked In
Facebook
Copy
Download@article{10.11648/j.ijtam.20160201.11,
author = {Qingying Hu and Jian Dang and Hongwei Zhang},
title = {Existence and Stability of Solutions for Semilinear Timoshenko System with Damping and Source Terms},
journal = {International Journal of Theoretical and Applied Mathematics},
volume = {2},
number = {1},
pages = {1-6},
doi = {10.11648/j.ijtam.20160201.11},
url = {https://doi.org/10.11648/j.ijtam.20160201.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20160201.11},
abstract = {In this paper, we are concerned with one-dimensional Timoshenko model for a beam with nonlinear damping and source terms. Under suitable conditions on the initial data, the theorem of global existence is proved by potential well method combined Galerkin procedure, and decay estimates of the energy is established by means of Nakao’s inequality.},
year = {2016}
}
TY - JOUR T1 - Existence and Stability of Solutions for Semilinear Timoshenko System with Damping and Source Terms AU - Qingying Hu AU - Jian Dang AU - Hongwei Zhang Y1 - 2016/09/30 PY - 2016 N1 - https://doi.org/10.11648/j.ijtam.20160201.11 DO - 10.11648/j.ijtam.20160201.11 T2 - International Journal of Theoretical and Applied Mathematics JF - International Journal of Theoretical and Applied Mathematics JO - International Journal of Theoretical and Applied Mathematics SP - 1 EP - 6 PB - Science Publishing Group SN - 2575-5080 UR - https://doi.org/10.11648/j.ijtam.20160201.11 AB - In this paper, we are concerned with one-dimensional Timoshenko model for a beam with nonlinear damping and source terms. Under suitable conditions on the initial data, the theorem of global existence is proved by potential well method combined Galerkin procedure, and decay estimates of the energy is established by means of Nakao’s inequality. VL - 2 IS - 1 ER -
Information
Email: