JOURNAL BROWSE
Search
Advanced SearchSearch Tips
ON SOLVABILITY OF THE DISSIPATIVE KIRCHHOFF EQUATION WITH NONLINEAR BOUNDARY DAMPING
facebook(new window)  Pirnt(new window) E-mail(new window) Excel Download
 Title & Authors
ON SOLVABILITY OF THE DISSIPATIVE KIRCHHOFF EQUATION WITH NONLINEAR BOUNDARY DAMPING
Zhang, Zai-Yun; Huang, Jian-Hua;
  PDF(new window)
 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)
 Keywords
global existence;dissipative Kirchhoff equation;Galerkin approximation;boundary damping;
 Language
English
 Cited by
1.
Existence and general decay for nondissipative distributed systems with boundary frictional and memory dampings and acoustic boundary conditions, Zeitschrift für angewandte Mathematik und Physik, 2015, 66, 4, 1595  crossref(new windwow)
2.
General decay for a viscoelastic Kirchhoff equation with Balakrishnan-Taylor damping, dynamic boundary conditions and a time-varying delay term, Evolution Equations and Control Theory, 2017, 6, 2, 239  crossref(new windwow)
3.
General energy decay of solutions for a weakly dissipative Kirchhoff equation with nonlinear boundary damping, Acta Mathematicae Applicatae Sinica, English Series, 2017, 33, 2, 401  crossref(new windwow)
4.
Existence and general decay for nondissipative hyperbolic differential inclusions with acoustic/memory boundary conditions, Mathematische Nachrichten, 2016, 289, 2-3, 300  crossref(new windwow)
 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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

19.
K. Narasimha, Nonlinear vibration of an elastic string, J. Sound Vib. 8 (1968), 134-146. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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). crossref(new window)

27.
I. Segal, Nonlinear semigroups, Ann. of Math. 78 (1963), 339-364. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)