Blow-up of Solutions for Semilinear Timoshenko System with Damping and Source Terms

Jian Dang; Qingying Hu; Hongwei Zhang

doi:doi:10.11648/j.ijamtp.20160204.13

| Peer-Reviewed

Blow-up of Solutions for Semilinear Timoshenko System with Damping and Source Terms

Jian Dang, Qingying Hu, Hongwei Zhang

Published in International Journal of Applied Mathematics and Theoretical Physics (Volume 2, Issue 4)

Received: 21 August 2016 Accepted: 29 August 2016 Published: 14 October 2016

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

In this paper, we are concerned with one-dimensional Timoshenko model for a beam with nonlinear damping and source terms. We establish a blow-up result when the initial energy is positive and the initial data is not in a potential well. Under arbitrary positive initial energy, we also prove a finite-time blow-up result for a special case.

Published in	International Journal of Applied Mathematics and Theoretical Physics (Volume 2, Issue 4)
DOI	10.11648/j.ijamtp.20160204.13
Page(s)	41-45
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), 2024. Published by Science Publishing Group

Keywords

Timoshenko System, Source Term, Damping Term, Blow-up

References

[1]	S. Timoshenko. On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Philosophical Magazine, 41 (1921), 744-746.
[2]	S. A. Messaoudi, M. I. Mustafa. On the internal and boundary stabilization of Timoshenko beams. Nonlinear Differential Equations and Applications, 15 (2008), 655-671.
[3]	S. A. Messaoudi, B. Said-Houari. Uniform decay in a Timoshenko system with past history. J. Math. Anal. Appl. 360 (2009), 458-475.
[4]	N. E. Tatar. Stabilization of a viscoelastic Timoshenko beam. Applicable Analysis, 92 (1) (2013), 27-43.
[5]	J. U. Kim, Y. Renardy. Boundary control of the Timoshenko beam. SIAM J. Control Optim. 25 (8) (1987), 1417-1429.
[6]	S. A. Messaoudi, A. Soufyame. Boundary stabilization of a nonlinear system of Timoshenko type. Nonlinear Analysis, 67 (2007), 2107-2112.
[7]	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, 2002, 167-179.
[8]	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. 34 (2008), 1-27.
[9]	I. Chueshov, I. Lasiecka. Global attractors for Mindlin-Tomoshenko plate and for their Kirchhoff limits. Milan J. Math. 74 (2006), 117-138.
[10]	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, 39 (2004), 1005-1021.
[11]	A. Soufyame, M. Afilal, T. Aouam. General decay of solutions of a nonlinear Timoshenko system with a boundary control of memory type. Differential and Integral Equations, 22 (2009), 1125-1139.
[12]	P. Pei, M. A. Rammaha, D. Toundykov. Local and global well-posedness of semilinear Reissner-Mindlin-Timoshenko plate equations. Nonlinear Analysis, 105 (2014), 62-85.
[13]	P. Pei, M. A. Rammaha, D. Toundykov. Global well-posedness and stability of semilinear Mindlin-Timoshenko systems. J. Math. Anal. Appl. 418 (2014), 535-568.
[14]	D. H. Sattinger. On global solutions of nonlinear hyperbolic equation. Arch. Rational Mech. Anal. 30 (1968), 108-172.
[15]	I. E. Payne, D. H. Sattinger. Saddle points and instability of nonlinear hyperbolic equations. Israel J. Math. 22 (1975), 273-303.
[16]	K. Agre, M. A. Rammaha. System of nonlinear wave equation with damping and source terms. Differential and Integral Equation, 19 (1) (2006), 1235-1270.
[17]	B. Said-Houari. Global nonexistence of positive initial-energy solutions of a system of nonlinear wave equations with damping and source terms. Differential and Integral Equation, 23 (1-2) (2010), 29-92.
[18]	E. Vitillaro. Global nonexistence theorems for a class of evolution equation with dissipations. Arch. Rational Mech Anal. 149 (1999), 155-182.
[19]	C. O. Alves, M. M. Cavalcanti, V. N. Domingos Cavalcanti, M. A. Rammaha, D. Toundykov. On existence, uniform decay rates and blow up for solutions of system-s of nonlinear wave equations with damping and source terms. Discrete Contin. Dyn. Syst. Ser. S, 2 (3) (2009), 583-608.
[20]	M. A. Rammaha, S. Sakuntasathien. Global existence and blow up of solutions to systems of nonlinear wave equations with degenerate damping and source terms. Nonlinear Analysis: Theory, Methods and Applications, 72 (5) (2005), 2658-2683.
[21]	O. M. Korpusov. Blowup of solutions of strongly dissipative generalized Klein-Gordon equation. Izvestiya: Mathematics, 77 (2) (2013), 325-353.

Cite This Article

Plain Text BibTeX RIS

APA Style

Jian Dang, Qingying Hu, Hongwei Zhang. (2016). Blow-up of Solutions for Semilinear Timoshenko System with Damping and Source Terms. International Journal of Applied Mathematics and Theoretical Physics, 2(4), 41-45. https://doi.org/10.11648/j.ijamtp.20160204.13

Copy | Download

ACS Style

Jian Dang; Qingying Hu; Hongwei Zhang. Blow-up of Solutions for Semilinear Timoshenko System with Damping and Source Terms. Int. J. Appl. Math. Theor. Phys. 2016, 2(4), 41-45. doi: 10.11648/j.ijamtp.20160204.13

Copy | Download

AMA Style

Jian Dang, Qingying Hu, Hongwei Zhang. Blow-up of Solutions for Semilinear Timoshenko System with Damping and Source Terms. Int J Appl Math Theor Phys. 2016;2(4):41-45. doi: 10.11648/j.ijamtp.20160204.13

Copy | Download

@article{10.11648/j.ijamtp.20160204.13,
  author = {Jian Dang and Qingying Hu and Hongwei Zhang},
  title = {Blow-up of Solutions for Semilinear Timoshenko System with Damping and Source Terms},
  journal = {International Journal of Applied Mathematics and Theoretical Physics},
  volume = {2},
  number = {4},
  pages = {41-45},
  doi = {10.11648/j.ijamtp.20160204.13},
  url = {https://doi.org/10.11648/j.ijamtp.20160204.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20160204.13},
  abstract = {In this paper, we are concerned with one-dimensional Timoshenko model for a beam with nonlinear damping and source terms. We establish a blow-up result when the initial energy is positive and the initial data is not in a potential well. Under arbitrary positive initial energy, we also prove a finite-time blow-up result for a special case.},
 year = {2016}
}

Copy | Download

TY  - JOUR
T1  - Blow-up of Solutions for Semilinear Timoshenko System with Damping and Source Terms
AU  - Jian Dang
AU  - Qingying Hu
AU  - Hongwei Zhang
Y1  - 2016/10/14
PY  - 2016
N1  - https://doi.org/10.11648/j.ijamtp.20160204.13
DO  - 10.11648/j.ijamtp.20160204.13
T2  - International Journal of Applied Mathematics and Theoretical Physics
JF  - International Journal of Applied Mathematics and Theoretical Physics
JO  - International Journal of Applied Mathematics and Theoretical Physics
SP  - 41
EP  - 45
PB  - Science Publishing Group
SN  - 2575-5927
UR  - https://doi.org/10.11648/j.ijamtp.20160204.13
AB  - In this paper, we are concerned with one-dimensional Timoshenko model for a beam with nonlinear damping and source terms. We establish a blow-up result when the initial energy is positive and the initial data is not in a potential well. Under arbitrary positive initial energy, we also prove a finite-time blow-up result for a special case.
VL  - 2
IS  - 4
ER  -

Copy | Download

Author Information

Jian Dang

Department of Mathematics, Henan University of Technology, Zhengzhou, China
Qingying Hu

Department of Mathematics, Henan University of Technology, Zhengzhou, China
Hongwei Zhang

Department of Mathematics, Henan University of Technology, Zhengzhou, China

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

Jian Dang, Qingying Hu, Hongwei Zhang. (2016). Blow-up of Solutions for Semilinear Timoshenko System with Damping and Source Terms. International Journal of Applied Mathematics and Theoretical Physics, 2(4), 41-45. https://doi.org/10.11648/j.ijamtp.20160204.13

Copy | Download

ACS Style

Jian Dang; Qingying Hu; Hongwei Zhang. Blow-up of Solutions for Semilinear Timoshenko System with Damping and Source Terms. Int. J. Appl. Math. Theor. Phys. 2016, 2(4), 41-45. doi: 10.11648/j.ijamtp.20160204.13

Copy | Download

AMA Style

Jian Dang, Qingying Hu, Hongwei Zhang. Blow-up of Solutions for Semilinear Timoshenko System with Damping and Source Terms. Int J Appl Math Theor Phys. 2016;2(4):41-45. doi: 10.11648/j.ijamtp.20160204.13

Copy | Download

@article{10.11648/j.ijamtp.20160204.13,
  author = {Jian Dang and Qingying Hu and Hongwei Zhang},
  title = {Blow-up of Solutions for Semilinear Timoshenko System with Damping and Source Terms},
  journal = {International Journal of Applied Mathematics and Theoretical Physics},
  volume = {2},
  number = {4},
  pages = {41-45},
  doi = {10.11648/j.ijamtp.20160204.13},
  url = {https://doi.org/10.11648/j.ijamtp.20160204.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20160204.13},
  abstract = {In this paper, we are concerned with one-dimensional Timoshenko model for a beam with nonlinear damping and source terms. We establish a blow-up result when the initial energy is positive and the initial data is not in a potential well. Under arbitrary positive initial energy, we also prove a finite-time blow-up result for a special case.},
 year = {2016}
}

Copy | Download

TY  - JOUR
T1  - Blow-up of Solutions for Semilinear Timoshenko System with Damping and Source Terms
AU  - Jian Dang
AU  - Qingying Hu
AU  - Hongwei Zhang
Y1  - 2016/10/14
PY  - 2016
N1  - https://doi.org/10.11648/j.ijamtp.20160204.13
DO  - 10.11648/j.ijamtp.20160204.13
T2  - International Journal of Applied Mathematics and Theoretical Physics
JF  - International Journal of Applied Mathematics and Theoretical Physics
JO  - International Journal of Applied Mathematics and Theoretical Physics
SP  - 41
EP  - 45
PB  - Science Publishing Group
SN  - 2575-5927
UR  - https://doi.org/10.11648/j.ijamtp.20160204.13
AB  - In this paper, we are concerned with one-dimensional Timoshenko model for a beam with nonlinear damping and source terms. We establish a blow-up result when the initial energy is positive and the initial data is not in a potential well. Under arbitrary positive initial energy, we also prove a finite-time blow-up result for a special case.
VL  - 2
IS  - 4
ER  -

Copy | Download