[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]
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Weak solution for a fourth-order nonlinear wave equation(
)
一类四阶波动方程的弱解
)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
- 一类四阶波动方程的弱解
- 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 utt+αΔ2 u-b2Δut-βΔu+ut|ut|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), ut(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.
- 研究下列初边值问题:utt+αΔ2u-b2Δut+ut|ut|r+g(u)=0 in Ω×(0, ∞);u(x, 0)=u00(x), ut(x, 0)=u11(x, 0), x∈Ω; u=(əu)/(əυ), x∈əΩ的整体解的存在性和不存在性, 以及整体解的惟一性和能量估计.这里Ω是Rn(n≥1)中的有界区域.借助于乘子方程, 推出了整体解的能量衰减率.借助于势井理论, 得到了在有限时刻内爆破的充分条件.由Galekin 近似方法得到解的存在性.
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.