四阶非线性波方程整体解的存在唯一性

梁和板振动是重要的物理现象，在数学上通常用四阶非线性波动方程来研究，所以探讨四阶波动方程具有重要的理论价值和实际意义。方程解的存在唯一性是研究方程解的性态和分析解的性质的前提和基础。本文研究了四阶非线性弱阻尼波动方程uσ＋au1＋△^2=f（t，x，u，▽u）的整体解的存在唯一性。利用了空间序列技巧和能量估计方法，验证了当非线性项f（t，x，u，▽u）满足一定条件时，方程存在整体解；并证明了四阶非线性弱阻尼波动方程整体解的唯一性。本文主要扩展了非线性项，在已有文献中的非线性项为丨u丨^p-1u或者为f（u），不含有导数，而本文研究的非线性项为厂（t，z，x，u,▽u），所以适用范围更加广泛。

Existence and Uniqueness of Global Solutions to the Nonlinear Fourth-order Wave Equations

Vibration of beams and plates is an important physical phenomenon, and the physical phenomena are studied by fourth order nonlinear wave equations in mathematics usually, so investigating fourth order wave equation has important theoretical value and practical significance. Existence and uniqueness of solutions for equations are preliminary and foundation to study the behavior and property of solutions. Existence and uniqueness of global solutions to the nonlinear fourth-order wave equations with weak damping uσ＋aut ＋△^2 u= f（t,x,u,▽u） are studied. By using spatial sequence techniques and energy estimate methods, it is obtained the equations have the solutions if the nonlinear terms f（t,x,u, ▽u） satisfy some conditions, and uniqueness of global solutions to the nonlinear fourth-order wave equations with weak damping is proved. This paper extends the nonlinear terms. In the literature, the nonlinear terms are丨 u 丨 ^p-1 u or f（u）, and does not contain derivatives, while the nonlinear terms of this paper are f（t,x, u, ▽u） therefore the concludes have a broader scope.