Well-Posedness and Exponential Stability of the Von Kármán Beam With Infinite Memory

Main Article Content

Abdelkader Dibes, Lamine Bouzettouta, Manel Abdelli, Salah Zitouni


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.

Article Details


  1. 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.
  2. 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.
  3. 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.
  4. A. Benabdallah, D. Teniou, Exponential Stability of a Von Karman Model With Thermal Effects, Electron. J. Differ. Equ. 1998 (1998), 7.
  5. 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.
  6. 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.
  7. 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.
  8. 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.
  9. 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.
  10. 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.
  11. 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.
  12. 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.
  13. A.E. Green, P.M. Naghdi, Thermoelasticity Without Energy Dissipation, J. Elasticity. 31 (1993), 189–208. https://doi.org/10.1007/bf00044969.
  14. 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.
  15. 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.
  16. 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.
  17. 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.
  18. 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).
  19. 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.
  20. 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.
  21. 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.
  22. 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.
  23. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, New York: Springer, Applied Mathematical Sciences, 44, (1983).
  24. 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.
  25. 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.
  26. 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.