本项目研究来源于阻尼减震材料和流体力学等领域的带记忆项和非线性源的粘弹性方程解的存在性、衰减和有限时刻爆破等性态，着重分析记忆项、不同阻尼项和非线性源分别对解的存在性及其各种性态的影响。拟采用扰动能量估计、凸性方法和加权不等式技巧来处理记忆项的影响，设法得到能量的（最优）衰减速率；借助位势井理论建立适当的稳定集和不稳定集以克服非线性源带来的困难，并结合改进的凸性技巧讨论解在有限时刻爆破的更广泛的条件；分别通过定义加权乘子、获得广义解满足的变分不等式、或构造抽象算子半群并采用谱分析方法考虑更一般耗散、退化阻尼、或不定阻尼的影响。 相关问题的解决不仅对非线性发展方程理论方法的发展有一定的科学意义，而且对粘弹性材料等方面的应用具有参考价值。

This project is concerned with the qualitative properties for some viscoelastic wave equations with both memory term and source term, which come from the fields of vibration damping materials and fluid mechanics. The main works include the global existence, asymptotic behavior and finite time blow-up of solutions. Based on the perturbed energy technique, convexity method and weighted integral inequality technique, we intend to overcome the effect of the memory term so that we can get the (optimal) decay rate of the energy. By using the potential well theory, we establish the stable set and unstable set to deal with the difficulities brought by the source term, so that we can prove the finite time blow-up of solutions with more general conditions. We try to deal with the effects of the general conditions of the dissipation, the degenerate damping and the indefinite damping, via using a weighted multiplier, obtaining a variational equality satisfied by generalized solutions and using the semigroup setting, respectively. The solution of these related problems not only has scientific significances for the development of theories and methods of the nonlinear evolution equations, but also has an important reference value for the applications on both the viscoelastic material and the hydrodynamics.