Well-Posedness and Exponential Stability of the Von Kármán Beam With Infinite Memory
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Abstract
In the present work, we consider a one-dimensional Von kármán beam with infinite memory, we establish the well-posedness of the system using semigroup theory and prove the exponential stability under some conditions on the kernel of the infinite memory term.
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References
- T.A. Apalara, General Decay of Solutions in One-Dimensional Porous-Elastic System With Memory, J. Math. Anal. Appl. 469 (2019), 457–471. https://doi.org/10.1016/j.jmaa.2017.08.007.
- T.A. Apalara, On the Stabilization of a Memory-Type Porous Thermoelastic System, Bull. Malays. Math. Sci. Soc. 43 (2019), 1433–1448. https://doi.org/10.1007/s40840-019-00748-2.
- S. Baibeche, L. Bouzettouta, A. Guesmia, M. Abdelli, Well-Posedness and Exponential Stability of Swelling Porous Elastic Soils With a Second Sound and Distributed Delay Term, J. Math. Comput. Sci. 12 (2022), 82. https://doi.org/10.28919/jmcs/7106.
- A. Benabdallah, D. Teniou, Exponential Stability of a Von Karman Model With Thermal Effects, Electron. J. Differ. Equ. 1998 (1998), 7.
- L. Bouzettouta, A. Djebabla, Exponential Stabilization of the Full Von Kármán Beam by a Thermal Effect and a Frictional Damping and Distributed Delay, J. Math. Phys. 60 (2019), 041506. https://doi.org/10.1063/1.5043615.
- L. Bouzettouta, F. Hebhoub, K. Ghennam, S. Benferdi, Exponential Stability for a Nonlinear Timoshenko System With Distributed Delay, Int. J. Anal. Appl. 19 (2021), 77-90. https://doi.org/10.28924/2291-8639-19-2021-77.
- S. Zitouni, K. Zennir, L. Bouzettouta, Uniform Decay for a Viscoelastic Wave Equation With Density and TimeVarying Delay in R n , Filomat. 33 (2019), 961–970. https://doi.org/10.2298/fil1903961z.
- L. Bouzettouta, S. Zitouni, Kh. Zennir, H. Sissaoui, Well-Posedness and Decay of Solutions to Bresse System With Internal Distributed Delay, Int. J. Appl. Math. Stat. 56 (2017), 153–168.
- A. Djebabla, N-e. Tatar, Exponential Stabilization of the Full Von Kármán Beam by a Thermal Effect and a Frictional Damping, Georgian Math. J. 20 (2013), 427–438. https://doi.org/10.1515/gmj-2013-0019.
- A. Favini, M.A. Horn, I. Lasiecka, D. Tataru, Global Existence, Uniqueness and Regularity of Solutions to a Von Karman System With Nonlinear Boundary Dissipation, Differ. Integral Equ. 9 (1996), 267–294.
- A.E. Green, P.M. Naghdi, A Re-Examination of the Basic Postulates of Thermomechanics, Proc. R. Soc. Lond. A. 432 (1991), 171–194. https://doi.org/10.1098/rspa.1991.0012.
- A.E. Green, P.M. Naghdi, On Undamped Heat Waves in an Elastic Solid, J. Thermal Stresses. 15 (1992), 253–264. https://doi.org/10.1080/01495739208946136.
- A.E. Green, P.M. Naghdi, Thermoelasticity Without Energy Dissipation, J. Elasticity. 31 (1993), 189–208. https://doi.org/10.1007/bf00044969.
- M.A. Horn, I. Lasiecka, Uniform Decay of Weak Solutions to a Von Kármán Plate With Nonlinear Boundary Dissipation, Differ. Integral Equ. 7 (1994), 885–908. https://doi.org/10.57262/die/1370267712.
- H.E. Khochemane, L. Bouzettouta, A. Guerouah, Exponential Decay and Well-Posedness for a One-Dimensional Porous-Elastic System With Distributed Delay, Appl. Anal. 100 (2019), 2950–2964. https://doi.org/10.1080/00036811.2019.1703958.
- H.E. Khochemane, A. Djebabla, S. Zitouni, L. Bouzettouta, Well-Posedness and General Decay of a Nonlinear Damping Porous-Elastic System With Infinite Memory, J. Math. Phys. 61 (2020), 021505. https://doi.org/10.1063/1.5131031.
- H. E. Khochemane, S. Zitouni, L. Bouzettouta, Stability Result for a Nonlinear Damping Porous-Elastic System With Delay Term, Nonlinear Stud. 27 (2020), 487–503.
- J.E. Lagnese, Modelling and Stabilization of Nonlinear Plates, In: Estimation and Control of Distributed Parameter Systems (Vorau, 1990), International Series of Numerical Mathematics, Birkhäuser, Basel, Vol. 100, pp. 247–264, (1991).
- J.E. Lagnese, Uniform Asymptotic Energy Estimates for Solutions of the Equations of Dynamic Plane Elasticity With Nonlinear Dissipation at the Boundary, Nonlinear Anal.: Theory Methods Appl. 16 (1991), 35–54. https://doi.org/10.1016/0362-546x(91)90129-o.
- J.E. Lagnese, G. Leugering, Uniform Stabilization of a Nonlinear Beam by Nonlinear Boundary Feedback, J. Differ. Equ. 91 (1991), 355–388. https://doi.org/10.1016/0022-0396(91)90145-y.
- W. Liu, K. Chen, J. Yu, Existence and General Decay for the Full Von Kármán Beam With a Thermo-Viscoelastic Damping, Frictional Dampings and a Delay Term, IMA J. Math. Control Inform. 34 (2017), 521-542. https://doi.org/10.1093/imamci/dnv056.
- G.P. Menzala, E. Zuazua, Timoshenko’s Beam Equation as Limit of a Nonlinear One-Dimensional Von Kármán System, Proc. R. Soc. Edinburgh: Sect. A Math. 130 (2000), 855–875. https://doi.org/10.1017/s0308210500000470.
- A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, New York: Springer, Applied Mathematical Sciences, 44, (1983).
- J.P. Puel, M. Tucsnak, Boundary Stabilization for the von Kármán Equations, SIAM J. Control Optim. 33 (1995), 255–273. https://doi.org/10.1137/s0363012992228350.
- F.D. Araruna, P. Braz e Silva, E. Zuazua, Asymptotics and Stabilization for Dynamic Models of Nonlinear Beams, Proc. Estonian Acad. Sci. 59 (2010), 150–155. https://doi.org/10.3176/proc.2010.2.14.
- S. Zitouni, L. Bouzettouta, K. Zennir, D. Ouchenane. Exponential Decay of Thermo-Elastic Bresse System With Distributed Delay Term, Hacettepe J. Math. Stat. 47 (2018), 1216–1230.