New general decay results in a finite-memory bresse system

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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 and 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.
[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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