A study on asymptotic behavior of solutions for evolution equations

我们首先讨论了由于Ikehata-Matsuyama(1999年最新发展的)装置应用于无界区域中的一些发展方程的可能性。为此，我们得到了几个关于具有Fujita型功率非线性的阻尼波方程整体适定性的结果。其次，我们得到了在N维半空间中处理的(半线性)阻尼波方程的新的衰减估计；第三，利用上面的Ikehata-Matsuyama和Todorova-Yordanov的两种方法，我们成功地去掉了初值的紧性，从而得到了具有星形补的外部区域中波动方程的局部能量衰减估计。这一结果完全推广了Morawetz在1961年得到的结果。需要指出的是，用来推导局部能量衰减结果的方法在外部区域的双曲型方程中有着广泛的应用。

We first discussed the possibility of applications of the device due to Ikehata-Matsuyama (which was newly developed in 1999) to some evolution equations in unbounded domains. For this we have obtained several results concerning the global well-posedness result of damped wave equations with the Fujita type power nonlinearity. And also by the device we have obtained the new result concerning the diffusion phenomenon of abstract second-order evolution equations.Second we have found the new decay estimates of (semilinear) damped wave equations, which are dealt with in the N-dimensional half space.Third, by the two methods due to Ikehata-Matsuyama above and Todorova-Yordanov we have succeeded in removing the compactness of the support on the initial data in order to obtain local energy decay estimates of the wave equation in an exterior domain with a star-shaped complement. This result completely generalizes that obtained by Morawetz in 1961. It should be pointed out that the method used to derive local enegy decay results has many wide applications to hyperbolic equations in exterior domains.