这篇论文主要研究了在 n-维空间中一类带有时滞和非局部反应项的 Lotka-Volterra 竞争系统平面波的稳定性. 我们证明了, 当波速 $c>c^{*}$ 时, 所有的平面波在 $L^{\infty}(\RR^n )$ 空间中依 $t^{-\frac{n}{2}}\ee^{-\epsilon_{\tau}\sigma t}$ 形式指数稳定, 其中 $\sigma >0$, $\tau >0$, $\epsilon_{\tau}=\epsilon(\tau)\in (0,1)$ 是关于时滞 $\tau$ 的减函数. 我们也知道, 时滞会引起解的衰减率的降低. 当波速 $c=c^{*}$ 时, 我们证明了平面波依 $t^{-\frac{n}{2}}$ 形式代数稳定. 论文主要采用的方法有比较原理, 加权能量法和傅里叶变换. 我们通过傅里叶变换将偏微分方程的求解转化为了时滞微分方程的求解问题, 并且通过加权 $L^{1}(\RR^{n})$ 空间和加权 $W^{2,1}(\RR^{n})$ 空间中的估计得到了主要的结果.

In this dissertation, we mainly study the multidimensional stability of planar waves for a class of Lotka-Volterra competition systems with the time delay and nonlocal reaction term in $n$--dimensional space. It is proved that, all planar traveling waves with speed $c>c^{*}$ are exponentially stable in $L^{\infty}(\RR^n )$ in the form of $t^{-\frac{n}{2}}\ee^{-\epsilon_{\tau}\sigma t}$, where $\sigma $ and $\tau $ are positive constants, and $\epsilon_{\tau}=\epsilon(\tau)\in (0,1)$ is a decreasing function for the time delay $\tau$. It is also realized that, time delay essentially causes the decay rate of the solution to slow down. While, for the planar traveling waves with speed $c=c^{*}$, we show that they are algebraically stable in the form of $t^{-\frac{n}{2}}$. The approach adopted is the comparison principle, weighted energy method and Fourier transform. The key to solve the problem is to transform the partial differential equation to a delay differential equation by using the Fourier transform. We establish some estimates in $L^{1}_{w}(\RR^n)$ space and $W^{2,1}_{w}(\RR^n)$ space to obtain the main results.