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New general decay results in a finite-memory bresse system
Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, P.O. Box 546, Dhahran 31261, Saudi Arabia |
$ \begin{array}{ll} \rho_1\varphi_{tt}-k_1(\varphi_x+\psi+lw)_x-lk_3(w_x-l\varphi) = 0,\\ \rho_2\psi_{tt}-k_2\psi_{xx}+k_1(\varphi_x+\psi+lw)+ \int_0^tg(t-s)\psi_{xx}(\cdot,s)ds = 0,\\ \rho_1w_{tt}-k_3(w_x-l\varphi)_x+lk_1(\varphi_x+\psi+lw) = 0, \end{array} $ |
$ (x,t) \in (0,L) \times (0, \infty) $ |
$ g $ |
$ \xi $ |
$ H $ |
$ g'(t)\leq-\xi(t)H(g(t)),\qquad\forall t\geq0. $ |
$ \xi $ |
$ H $ |
References:
[1] |
M. O. Alves, L. H. Fatori, M. A. Jorge Silva and R. N. Monteriro,
Stability and optimality of decay rate for a weakly dissipative Bresse system, Math. Methods Appl. Sci., 38 (2015), 898-908.
doi: 10.1002/mma.3115. |
[2] |
M. S. Alves, O. Vera, J. Muñoz-Rivera and A. Rambaud, Exponential stability to the Bresse system with boundary dissipation conditions, (2015), arXiv: 1506.01657. |
[3] |
V. I. Arnol'd, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1989.
doi: 10.1007/978-1-4757-2063-1. |
[4] |
F. Dell'Oro,
Asymptotic stability of thermoelastic systems of Bresse type, J. Differ. Equ., 258 (2015), 3902-3927.
doi: 10.1016/j.jde.2015.01.025. |
[5] |
L. H. Fatori and R. N. Monteiro,
The optimal decay rate for a weak dissipative Bresse system, Appl. Math. Lett., 25 (2012), 600-604.
doi: 10.1016/j.aml.2011.09.067. |
[6] |
A. Guesmia and M. Kafini,
Bresse system with infinite memories, Math. Methods Appl. Sci., 38 (2015), 2389-2402.
doi: 10.1002/mma.3228. |
[7] |
A. Guesmia and S. A. Messaoudi,
On the stabilization of Timoshenko systems with memory and different speeds of wave propagation, Appl. Math. Comput., 219 (2013), 9424-9437.
doi: 10.1016/j.amc.2013.03.105. |
[8] |
T. F. Ma and R. N. Monteiro,
Singular limit and long-time dynamics of Bresse systems, SIAM J. Math. Anal., 49 (2017), 2468-2495.
doi: 10.1137/15M1039894. |
[9] |
M. I. Mustafa,
General decay result for nonlinear viscoelastic equations, J. Math. Anal. Appl., 457 (2018), 134-152.
doi: 10.1016/j.jmaa.2017.08.019. |
[10] |
J. A. Soriano, J. E. Muñoz Rivera and L. H. Fatori,
Bresse system with indefinite damping, J. Math. Anal. Appl., 387 (2012), 284-290.
doi: 10.1016/j.jmaa.2011.08.072. |
[11] |
A. Soufyane and B. Said-Houari,
The effect of the wave speeds and the frictional damping terms on the decay rate of the bresse system, Evol. Equations Control Theory, 3 (2014), 713-738.
doi: 10.3934/eect.2014.3.713. |
[12] |
A. Wehbe and W. Youssef,
Exponential and polynomial stability of an elastic Bresse system with two locally distributed feedbacks, J. Math. Phys., 51 (2010), 1-17.
doi: 10.1063/1.3486094. |
show all references
References:
[1] |
M. O. Alves, L. H. Fatori, M. A. Jorge Silva and R. N. Monteriro,
Stability and optimality of decay rate for a weakly dissipative Bresse system, Math. Methods Appl. Sci., 38 (2015), 898-908.
doi: 10.1002/mma.3115. |
[2] |
M. S. Alves, O. Vera, J. Muñoz-Rivera and A. Rambaud, Exponential stability to the Bresse system with boundary dissipation conditions, (2015), arXiv: 1506.01657. |
[3] |
V. I. Arnol'd, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1989.
doi: 10.1007/978-1-4757-2063-1. |
[4] |
F. Dell'Oro,
Asymptotic stability of thermoelastic systems of Bresse type, J. Differ. Equ., 258 (2015), 3902-3927.
doi: 10.1016/j.jde.2015.01.025. |
[5] |
L. H. Fatori and R. N. Monteiro,
The optimal decay rate for a weak dissipative Bresse system, Appl. Math. Lett., 25 (2012), 600-604.
doi: 10.1016/j.aml.2011.09.067. |
[6] |
A. Guesmia and M. Kafini,
Bresse system with infinite memories, Math. Methods Appl. Sci., 38 (2015), 2389-2402.
doi: 10.1002/mma.3228. |
[7] |
A. Guesmia and S. A. Messaoudi,
On the stabilization of Timoshenko systems with memory and different speeds of wave propagation, Appl. Math. Comput., 219 (2013), 9424-9437.
doi: 10.1016/j.amc.2013.03.105. |
[8] |
T. F. Ma and R. N. Monteiro,
Singular limit and long-time dynamics of Bresse systems, SIAM J. Math. Anal., 49 (2017), 2468-2495.
doi: 10.1137/15M1039894. |
[9] |
M. I. Mustafa,
General decay result for nonlinear viscoelastic equations, J. Math. Anal. Appl., 457 (2018), 134-152.
doi: 10.1016/j.jmaa.2017.08.019. |
[10] |
J. A. Soriano, J. E. Muñoz Rivera and L. H. Fatori,
Bresse system with indefinite damping, J. Math. Anal. Appl., 387 (2012), 284-290.
doi: 10.1016/j.jmaa.2011.08.072. |
[11] |
A. Soufyane and B. Said-Houari,
The effect of the wave speeds and the frictional damping terms on the decay rate of the bresse system, Evol. Equations Control Theory, 3 (2014), 713-738.
doi: 10.3934/eect.2014.3.713. |
[12] |
A. Wehbe and W. Youssef,
Exponential and polynomial stability of an elastic Bresse system with two locally distributed feedbacks, J. Math. Phys., 51 (2010), 1-17.
doi: 10.1063/1.3486094. |
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