Energy decay for hyperbolic partial differential equations with variable coefficients and its related topic

从能量衰减率的观点出发，我们发现了具有阻尼项的波动方程在接近无穷远的临界衰减中存在双重结构。其次，在抽象框架下给出了强阻尼波动方程解的渐近轮廓。第三，我们得到了具有非局部项的波动方程解的精确衰减估计，它插值了弱和强阻尼。最后，我们在傅里叶空间构造了一种新的修正能量方法，该方法可以很好地应用于几种具体模型，以获得解的（几乎）最优的衰减估计。

We discovered the existence of the double structure in the wave equations with da-mping terms decaying critically near infinity from the viewpoint of the energy decay rates. Secondly wecaught the asymptotic profile of solutions to the strongly damped wave equations in the abstract framework. Thirdly, we derived precise decay estimates of solutions to the wave equations with non-local terms, wh-ich interpolates the weak and strong damping. Finally, we constructed a new modified energy method in theFourier space, and the method could be well applied to the several concrete models in order to get the (almost) optimal decay estimates of solutions.