Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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GLOBAL EXISTENCE AND STABILITY FOR EULER-BERNOULLI BEAM EQUATION WITH MEMORY CONDITION AT THE BOUNDARY

  • Park, Jong-Yeoul (Department of Mathematics College of Science Pusan National University) ;
  • Kim, Joung-Ae (Department of Mathematics College of Science Pusan National University)
  • Published : 2005.11.01

Abstract

In this article we prove the existence of the solution to the mixed problem for Euler-Bernoulli beam equation with memory condition at the boundary and we study the asymptotic behavior of the corresponding solutions. We proved that the energy decay with the same rate of decay of the relaxation function, that is, the energy decays exponentially when the relaxation function decay exponentially and polynomially when the relaxation function decay polynomially.

Keywords

global existence;Euler-Bernoulli beam equation;Galerkin method;boundary value problem

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References

  1. P. L. Chow and J. L. Menaldi, Boundary stabilization of a nonlinear string with nerodynamic force, Control of Partial Differential Equations, Lecture notes in pure and Appl. Math. 165 (1994), 63-79
  2. M. Ciarletta, A differential problem for heat equation with a boundary condition with memory, Appl. Math. Lett. 10 (1997), no. 1, 95-191
  3. M. Fabrizio and M. Morro, A boundary condition with memory in Electroma- gretism, Arch. Ration. Mech. Anal. 136 (1996), 359-381 https://doi.org/10.1007/BF02206624
  4. V. Komornik, Rapid boundary stabilization of the wave equation, SIAM J. Control Optim. 29 (1991), 197-208 https://doi.org/10.1137/0329011
  5. V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation, J. Math. Pures Appl. 69 (1990), 33-54
  6. I. Lasiecka, Global uniform decay rates for the solution to the wave equation with nonlinear boundary conditions, Appl. Anal. 47 (1992), 191-212 https://doi.org/10.1080/00036819208840140
  7. J. L. Lions, Quelques Methodes de resolution de problemes aux limites non lineaires, Dunod Gauthiers Villars, Paris, 1969
  8. M. Milla Miranda and L. A. Medeiros, On boundary value problem for wave equa- tions : Existence Uniqueness-Asymptotic behavior, Rev. Math. Apl. 17 (1996), 47-73
  9. J. E. Munoz Rivera and D. Andrade, Exponential decay of nonlinear wave equation with a viscoelastic boundary condition, Math. Methods Appl. Sci. 23 (2000), 41-61 https://doi.org/10.1002/(SICI)1099-1476(20000110)23:1<41::AID-MMA102>3.0.CO;2-B
  10. T. Qin, Breakdown of solutions to nonlinear wave equation with a viscoelastic boundary condition, Arab. J. Sci. Engng. 19 (1994), no. 2A, 195-201
  11. T. Qin, Global solvability of nonlinear wave equation with a viscoelastic boundary condition, Chin. Ann. Math. 14B (1993), no. 3, 335-346
  12. M. Tucsnak, Boundary stabilization for stretched string equation, Differential Integral Equation 6 (1993), no. 4, 925-935
  13. J. Vancostenoble and P. Martinez, Optimality of energy estimate for the wave equation with nonlinear boundary velocity feedbacks, SIAM J. Control Optim. 39 (2000), no. 3, 776-797 https://doi.org/10.1137/S0363012999354211
  14. E. Zuazua, Uniform stabilization of the wave equation by nonlinear boundary feedback, SIAM J. Control Optim. 28 (1990), 466-477 https://doi.org/10.1137/0328025

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