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ON SOLVABILITY OF THE DISSIPATIVE KIRCHHOFF EQUATION WITH NONLINEAR BOUNDARY DAMPING

  • Zhang, Zai-Yun (College of Science National University of Defense Technology, School of Mathematics Hunan Institute of Science and Technology) ;
  • Huang, Jian-Hua (College of Science National University of Defense Technology)
  • Received : 2013.01.31
  • Published : 2014.01.31

Abstract

In this paper, we prove the global existence and uniqueness of the dissipative Kirchhoff equation $$u_{tt}-M({\parallel}{\nabla}u{\parallel}^2){\triangle}u+{\alpha}u_t+f(u)=0\;in\;{\Omega}{\times}[0,{\infty}),\\u(x,t)=0\;on\;{\Gamma}_1{\times}[0,{\infty}),\\{\frac{{\partial}u}{\partial{\nu}}}+g(u_t)=0\;on\;{\Gamma}_0{\times}[0,{\infty}),\\u(x,0)=u_0,u_t(x,0)=u_1\;in\;{\Omega}$$ with nonlinear boundary damping by Galerkin approximation benefited from the ideas of Zhang et al. [33]. Furthermore,we overcome some difficulties due to the presence of nonlinear terms $M({\parallel}{\nabla}u{\parallel}^2)$ and $g(u_t)$ by introducing a new variables and we can transform the boundary value problem into an equivalent one with zero initial data by argument of compacity and monotonicity.

References

  1. M. Aassila, Asymptotic behavior of solutions to a quasilinear hyperbolic equation with nonlinear damping, Electron. J. Qual. Theory Differ. Equ. 1998 (1998), no. 7, 12 pp.
  2. R. A. Adams, Sobolev Space, Acadmic Press, New York, 1975.
  3. A. Arosio and S. Spagnolo, Global solutions of the Cauchy problem for a nonlinear hyperbolic equation, Nonlinear Differential Equations and Their Applications, College de France Seminar, 6, Pitman, London, 1984.
  4. M. M. Cavalcanti, V. N. Domings Cavalcanti, J. S. Prates Filho, and J. A. Soriano, Existence and exponential decay for a Kirchhoff-Carrier model with viscosity, J. Math. Anal. Appl. 226 (1998), no. 1, 20-40.
  5. M. M. Cavalcanti, V. N. Domings Cavalcanti, J. S. Prates Filho, and J. A. Soriano, Existence and uniform decay of solutions of a degenarate equation nonlinear boundary damping and boundary memory source term, Nonlinear Analysis T. M. A. 38 (1999), 281-294. https://doi.org/10.1016/S0362-546X(98)00195-3
  6. M. M. Cavalcanti, V. N. Domings Cavalcanti, J. S. Prates Filho, and J. A. Soriano, Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping, Differential Integral Equations 14 (2001), no. 1, 85-116.
  7. M. M. Cavalcanti, V. N. Domings Cavalcanti, and J. A. Soriano, On existence and asymptotic stability of solutions of the degenerate wave equation with nonlinear boundary conditions, J. Math. Anal. Appl. 281 (2003), no. 1, 108-124. https://doi.org/10.1016/S0022-247X(02)00558-9
  8. F. Gazzola and M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations, Ann. Inst. H. Poincare Anal. Non Lineaire 23 (2006), no. 2, 185-207. https://doi.org/10.1016/j.anihpc.2005.02.007
  9. R. Ikehata, Some remarks on the wave equations with nonlinear damping and source terms, Nonlinear Anal. 54 (1996), no. 10, 1165-1175.
  10. N. I. Karachalios and N. M. Stavrakakis, Global existence and blow up results for some nonlinear wave equations on $R^n$, Adv. Differential Equations 6 (2001), no. 2, 155-174.
  11. G. Kirchhoff, Vorlesungen Uber Mechanik, Teubner, Leipzig, 1883.
  12. J. E. Lagnese, Note on boundary stabilization of wave equations, SIAM J. Control Optim. 26 (1988), no. 5, 1250-1257. https://doi.org/10.1137/0326068
  13. I. Lasiecka and J. Ong, Global sovability and uniform decays of solutions to quasilinear equation with nonlinear boundary conditions, Communications in PDE 24 (1999), 2069-2109. https://doi.org/10.1080/03605309908821495
  14. I. Lasiecka and D. Tataru,Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differential Integral Equations 6 (1993), no. 3, 507-533.
  15. H. A. Levine and S. Park, Global existence and global nonexistence of solutions of the Cauchy problem for a nonlinearly wave equation, J. Math. Anal. Appl. 228 (1998), no. 1, 181-205. https://doi.org/10.1006/jmaa.1998.6126
  16. J. L. Lions, Quelques Methodes Resolution des Problemes aux Limites Non-Lineares, Dunod, Paris, 1969.
  17. J. L. Lions and E. Magenes, Problemes aux limites non homogenes applications, Dunod, Paris, 1, 1968.
  18. T. Matsuyama and R. Ikehata, On global solutions and energy decay for the wave equations of Kirchhoff type with nonlinear damping terms, J. Math. Anal. Appl. 204 (1996), no. 3, 729-753. https://doi.org/10.1006/jmaa.1996.0464
  19. K. Narasimha, Nonlinear vibration of an elastic string, J. Sound Vib. 8 (1968), 134-146. https://doi.org/10.1016/0022-460X(68)90200-9
  20. K. Narasimha and Yamada, On global solutions of some degenerate quasilinear hyperbolic equation with dissipative terms, Funkcialaj Ekvacioj 33 (1990), 151-159.
  21. K. Ono, On global existence, asymtotic stability and blow-up of solutions for some degenerate nonlinear wave equations of Kirchhoff type, Math. Methods. Appl. Sci. 20 (1997), 151-177. https://doi.org/10.1002/(SICI)1099-1476(19970125)20:2<151::AID-MMA851>3.0.CO;2-0
  22. K. Ono, On global solutions and blow-up of solutions of nonlinear Kirchhoff string with nonlinear dissipation, J. Math. Anal. Appl. 216 (1997), 321-342. https://doi.org/10.1006/jmaa.1997.5697
  23. K. Ono, Global existence, decay, and blowup of solutions for some mildly degenerate nonlinear Kirchhoff strings, J. Differential Equations 137 (1997), no. 2, 273-301. https://doi.org/10.1006/jdeq.1997.3263
  24. J. Y. Park and J. J. Bae, On the existence of solutions of strongly damped wave equations, Internat. J. Math. and Math. Sci. 23 (2000), no. 6, 369-382. https://doi.org/10.1155/S0161171200000971
  25. R. Pitts and M. A. Rammaha, Global existence and non-existence theorems for nonlinear wave equations, Indiana Univ. Math. J. 51 (2002), no. 6, 1479-1509. https://doi.org/10.1512/iumj.2002.51.2215
  26. M. A. Rammaha and T. A. Strei, Global existence and nonexistence for nonlinear wave equations with damping and source terms, Trans. Amer. Math. Soc. 354 (2002), no. 9, 3621-3637 (electrolic). https://doi.org/10.1090/S0002-9947-02-03034-9
  27. I. Segal, Nonlinear semigroups, Ann. of Math. 78 (1963), 339-364. https://doi.org/10.2307/1970347
  28. Z. J. Yang, Initial boundary value problem for a class of non-linear strongly damped wave equations, Math. Methods. Appl. Sci. 26 (2003), no. 12, 1047-1066. https://doi.org/10.1002/mma.412
  29. K. Yosida, Fuctional Analysis, Sringer-Verlag, NewYork, 1996.
  30. Z. Y. Zhang,Central manifold for the elastic string with dissipative effect, Pacific Journal of Applied Mathematics 4 (2010), no. 2, 329-343.
  31. Z. Y. Zhang, Z. H. Liu, and X. J. Miao, Estimate on the dimension of global attractor for nonlinear dissipative Kirchhoff equation, Acta Appl. Math. 110 (2010), no. 1, 271-282. https://doi.org/10.1007/s10440-008-9405-1
  32. Z. Y. Zhang, Z. H. Liu, X. J. Miao, and Y. Z. Chen, Global existence and uniform stabilization of a generalized dissipative Klein-Gordon equation type with boundary damping, Journal of Mathematics and Physics 52 (2011), no. 2, 023502, 12 pp.
  33. Z. Y. Zhang and X. J. Miao, Global existence and uniform decay for wave equation with dissipative term and boundary damping, Comput. Math. Appl. 59 (2010), no. 2, 1003-1018. https://doi.org/10.1016/j.camwa.2009.09.008
  34. Z. Y. Zhang and X. J. Miao, Existence and asymptotic behavior of solutions to generalized Kirchhoff equation, Nonlinear Stud. 19 (2012), no. 1, 57-70.
  35. Z. Y. Zhang, X. J. Miao, and D. M. Yu,On solvability and stabilization of a class of hyperbolic hemivariational inequalities in elasticity, Funkcial. Ekvac. 54 (2011), no. 2, 297-314. https://doi.org/10.1619/fesi.54.297

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