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WEAK SOLUTIONS OF THE EQUATION OF MOTION OF MEMBRANE WITH STRONG VISCOSITY

  • Published : 2007.03.31

Abstract

We study the equation of a membrane with strong viscosity. Based on the variational formulation corresponding to the suitable function space setting, we have proved the fundamental results on existence, uniqueness and continuous dependence on data of weak solutions.

Keywords

References

  1. H. T. Banks, R. C. Smith, and Y. Wang, Smart Material Structures, Modeling, Estimation and Control, RAM, John Wiley and Sons, Masson, 1996
  2. R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 5, Evolution Problems I, Springer-Verlag, 1992
  3. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-Heidelberg-New York, 1977
  4. J. Greenberg, On the existence, uniqueness and stability of equation $P_{{\rho}0}X_{tt}=E(X_x)X_{xx }+;X_{xxt}$, J. Math. Anal. Appl. 25 (1969), 575-591 https://doi.org/10.1016/0022-247X(69)90257-1
  5. J. Greenberg, R. MacCamy, and V. Mizel, On the existence, uniqueness and stability of equation ${\sigma}(u_x)u_{tt}\;+\;{\lambda}u_{xxt}\;=\;_{{\rho}0}u_{tt}$, J. Math. Mech. 17 (1967/68), 707-728
  6. J. Ha and S. Nakagiri, Existence and regularity of weak solutions for semilinear second order evolution equations, Funcial. Ekvac. 41 (1998), no. 1, 1-24
  7. J. S. Hwang and S. Nakagiri, Optimal control problems for the equation of motion of membrane with strong viscosity, J. Math. Anal. Appl. 321 (2006), no. 1, 327-342 https://doi.org/10.1016/j.jmaa.2005.07.015
  8. K. Kikuchi, An analysis of the nonlinear equation of motion of a vibrating membrane in the space of BV functions, J. Math. Soc. Japan 52 (2000), no. 4, 741-766 https://doi.org/10.2969/jmsj/05240741
  9. T. Kobayashi, H. Pecher, and Y. Shibata, On a global in time existence theorem of smooth solutions to nonlinear wave equation with viscosity, Math. Ann. 296 (1993), no.2, 215-234 https://doi.org/10.1007/BF01445103
  10. R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physis, Second Edition, Applied Mathematical Sciences. Vol. 68, Springer-Verlag, Berlin-Heidelberg-New York, 1997

Cited by

  1. Parameter identification problem for the equation of motion of membrane with strong viscosity vol.342, pp.1, 2008, https://doi.org/10.1016/j.jmaa.2007.11.029
  2. SOLUTIONS OF QUASILINEAR WAVE EQUATION WITH STRONG AND NONLINEAR VISCOSITY vol.48, pp.4, 2011, https://doi.org/10.4134/JKMS.2011.48.4.867
  3. Weak and Strong Solutions for a Strongly Damped Quasilinear Membrane Equation vol.2017, 2017, https://doi.org/10.1155/2017/4529847