本项目主要研究带有强阻尼项和源项的双色散粘弹性波动方程以及带有非线性边界阻尼项和内部源项的粘弹性波动方程的初边值问题，同时还讨论带有非线性阻尼项的电报方程的Cauchy问题。这些方程都有重要的物理意义，同时在数学研究上也有很强的挑战性，故而受到了物理学家和数学家的高度重视。研究内容为：利用Galerkin逼近、势井和单调紧致等方法的结合或通过Galerkin逼近序列和一系列周期初边值问题整体解的极限过程的结合来证明整体解的存在性，利用改进的凹性方法、势井和能量扰动法等来研究整体解的不存在性，即解在有限的时间内的爆破(blow-up)。特别的，在适当的条件假设下还讨论整体解存在与不存在的临界性门槛结果。利用能量扰动法、乘子方法和积分-微分不等式等来研究方程解的能量的衰减率估计。

In this project, we mainly study the initial boundary value problems for the double dispersive-dissipative viscoelastic wave equation with nonlinear strong damping and source terms，the viscoelastic wave equation with nonlinear boundary damping and internal source terms and the Cauchy problem for a class of telegraph equation with nonlinear damping damping term. All of these equations arise from different branches of physics and have strong physical background. The methods which study these equations are very complicated and important to us.The outline of project as follows: we will study the global existence of solutions to some higher-order nonlinear wave equations by the Galerkin approximation, potential well theory and monotonicity-Compactness method or by a Galerkin approximation scheme combined with a limiting process of the solutions of a series of periodic boundary value problem. Using the improved the concavity method, potential well theory and energy pertubation method etc, we will consider the nonexistence of global solutions for these equations which is called the "blow-up". Especially, the threshold results of global existence and nonexistence of solutions for these equations will be discussed under certain conditions. We will give the explicit decay rate estimates of the energy for these equations by making use of the energy pertubation method, multiplier method, Differential-Integral Inequalities and other methods.