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July  2019, 18(4): 1637-1662. doi: 10.3934/cpaa.2019078

New general decay results in a finite-memory bresse system

Salim A. Messaoudi , and  Jamilu Hashim Hassan

Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, P.O. Box 546, Dhahran 31261, Saudi Arabia

* Corresponding author

Received  April 2018 Revised  July 2018 Published  January 2019

Fund Project: This work is funded by KFUPM under Project IN161006.

This paper is concerned with the following memory-type Bresse system
$ \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} $
with homogeneous Dirichlet-Neumann-Neumann boundary conditions, where
$ (x,t) \in (0,L) \times (0, \infty) $
,
$ g $
 is a positive strictly increasing function satisfying, for some nonnegative functions
$ \xi $
 and
$ H $
,
$ g'(t)\leq-\xi(t)H(g(t)),\qquad\forall t\geq0. $
Under appropriate conditions on
$ \xi $
 and
$ H $
, we prove, in cases of equal and non-equal speeds of wave propagation, some new decay results that generalize and improve the recent results in the literature.
Keywords: Bresse system, viscoelastic, general and ptimal decay, equal and non-equal speeds of wave propagation.
Mathematics Subject Classification: Primary: 35B35, 35L05; Secondary: 35B40, 35L20.
Citation: Salim A. Messaoudi, Jamilu Hashim Hassan. New general decay results in a finite-memory bresse system. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1637-1662. doi: 10.3934/cpaa.2019078
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.  Google Scholar

[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. Google Scholar

[3]

V. I. Arnol'd, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

A. Guesmia and M. Kafini, Bresse system with infinite memories, Math. Methods Appl. Sci., 38 (2015), 2389-2402.  doi: 10.1002/mma.3228.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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. Google Scholar

[3]

V. I. Arnol'd, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

A. Guesmia and M. Kafini, Bresse system with infinite memories, Math. Methods Appl. Sci., 38 (2015), 2389-2402.  doi: 10.1002/mma.3228.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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