Browse > Article
http://dx.doi.org/10.4134/BKMS.2014.51.1.189

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)
Publication Information
Bulletin of the Korean Mathematical Society / v.51, no.1, 2014 , pp. 189-206 More about this Journal
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.
Keywords
global existence; dissipative Kirchhoff equation; Galerkin approximation; boundary damping;
Citations & Related Records
연도 인용수 순위
  • Reference
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.   DOI   ScienceOn
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.   DOI   ScienceOn
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.   DOI   ScienceOn
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.   DOI   ScienceOn
13 J. L. Lions, Quelques Methodes Resolution des Problemes aux Limites Non-Lineares, Dunod, Paris, 1969.
14 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.   DOI
15 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.
16 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.   DOI   ScienceOn
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.   DOI   ScienceOn
19 K. Narasimha, Nonlinear vibration of an elastic string, J. Sound Vib. 8 (1968), 134-146.   DOI   ScienceOn
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.   DOI
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.   DOI   ScienceOn
23 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).   DOI   ScienceOn
24 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.   DOI   ScienceOn
25 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.   DOI
26 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.   DOI
27 I. Segal, Nonlinear semigroups, Ann. of Math. 78 (1963), 339-364.   DOI
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.   DOI   ScienceOn
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.   DOI
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, 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.   DOI   ScienceOn
34 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.   DOI   ScienceOn
35 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.