[1] Chen Caisheng, Ren Lei,. Weak solution for a fourth-order nonlinear wave equation [J]. Journal of Southeast University (English Edition), 2005, 21 (3): 369-374. [doi:10.3969/j.issn.1003-7985.2005.03.024]

Copy

Weak solution for a fourth-order nonlinear wave equation()

Share：

Journal of Southeast University (English Edition)[ISSN:1003-7985/CN:32-1325/N]

Volumn:

21

Issue:

2005 3

Page:

369-374

Research Field:

Mathematics, Physics, Mechanics

Publishing date:

2005-09-30

Info

Title:

Weak solution for a fourth-order nonlinear wave equation

Author(s):

Chen Caisheng;  Ren Lei

College of Science, Hohai University, Nanjing 210098, China

Keywords:

nonlinear wave equation;  uniqueness;  energy decay estimate;  blow up

PACS:

O175.29

DOI:

10.3969/j.issn.1003-7985.2005.03.024

Abstract:

The existence and the nonexistence, the uniqueness and the energy decay estimate of solution for the fourth-order nonlinear wave equation u_tt+αΔ^{2 u-b}2Δu_t-βΔu+u_t|u_t|^r+g(u)=0 in Ω×(0, ∞)are studied with the boundary condition u=(əu)/(əυ)=0 on əΩ and the initial condition u(x, 0)=u₀0(x), u_t(x, 0)=u₁1(x, 0)in bounded domain Ω⊂Rⁿ , n≥1.The energy decay rate of the global solution is estimated by the multiplier method.The blow-up result of the solution in finite time is established by the ideal of a potential well theory, and the existence of the solution is gotten by the Galekin approximation method.

References:

[1] Varlamo V.On the damped Boussinesq equation in a circle [J].Nonlinear Analysis, 1999, 38(4):447-470.
[2] Guesmia A.Existence globale et stabilisation interne nonlinéaire d’un systeme de Petrovsky [J].Bull Belg Math Soc Simon Stevin, 1998, 5(4):583-594.
[3] Guesmia A.Energy decay for a damped nonliner coupled system [J]. J of Math Anal and Appl, 1999, 239(1):38-48.
[4] Komornik V.Exact controllability and stabilization:the multiplier method [M].Paris:Masson-Wiley, 1994.
[5] Payne L, Sattinger D.Saddle points and instability on nonlinear hyperbolic equations [J].Israel Math J, 1991, 22:273-303.
[6] Lions J L.Quelques Méthodes de Résolution des Problèmes aux Limites Nonlinéaires [A]. In: Gauthier-Villars[C]. Paris: Dunod, 1969.
[7] Todorola G.Stable and unstable sets for the Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms [J].J of Math Anal and Appl, 1999, 239(2):213-226.

Memo

Memo:

Biography: Chen Caisheng(1956—), male, professor, cshengchen@hhu.edu.cn.

Common functions