本项目主要研究现代数学物理中出现的具有记忆项和k(x,t)-拉普拉斯算子的非线性伪抛物方程、具有记忆项和变指数型源项的四阶抛物方程的初边值问题，以及具有变指数型边界阻尼和内部源项的粘弹性波动方程的初边值问题。这些方程都具有重要的物理意义，同时在数学方法上也具有很强的挑战性，故而受到数学家和物理学家的高度重视。研究内容为：在各种变指数型广义Sobolev函数空间框架内，从次临界、临界和超临界三种能量级情形分别讨论上述方程的定性问题。利用Galerkin逼近、位势井、单调紧致法、惩罚法、粘性消去法、以及变分不等式等方法证明整体解的存在性；利用改进的凹性方法、位势井、改进的能量扰动法和变分不等式等方法研究整体解的不存在性；特别的，在适当的幂指数形式下还讨论整体解存在与不存在的最佳条件，即门槛结果；利用改进的能量扰动法、乘子方法等研究整体解的渐近性质以及能量衰减率估计。

In this project, we mainly study the initial boundary value problems for the nonlinear pseudo-parabolic equation with memory and k(x,t)-Laplacian, the fourth order parabolic equation with memory and source of variable exponent type, and the viscoelastic wave equation with nonlinear boundary damping and internal source of variable exponent type. All of these equations arise from different branches of modern mathematical 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: In the framework of the generalized Sobolev space with variable exponent type, we will study the qualitative problems for these equations mentioned above with sub-critical initial energy, critical initial energy and sup-critical initial energy cases, respectively. By the Galerkin approximation, potential well theory, monotonicity-compactness method, penalty method, the method of vanishing viscosity and variational inequality etc, we will prove the global existence of solutions for theses equations. Using the improved concavity method, potential well theory, improved energy pertubation method and variational inequality etc, we will consider the nonexistence of global solutions. Especially, the sharp conditions (threshold results) of global existence and nonexistence of solutions for these equations will be discussed under certain conditions for the variable exponent forms. We will give the asymptotic property and explicit decay rate estimates of the energy for these equations by making use of the improved energy pertubation method, multiplier method and other methods.