本文考虑高维 Burgers 方程外区域问题球对称解的大时间渐近行为, 主要关注在球对称初始扰 动下球对称稳态波的非线性稳定性. 对这一问题, Hashimoto 和 Matsumura (2019) 给出了保证其球对 称稳态波存在性的一个充分条件, 但是由于这一稳态波不再是单调的, 他们只能在更强的假设下证明 其非线性稳定性. 本文的主要目的是在 Hashimoto 和 Matsumura 给出的保证这一稳态波存在的条件 下证明其非线性稳定性. 此外, 还得到了该外区域问题的整体球对称解收敛到上述稳态波的关于时间 变元的代数和指数衰减率估计. 本文的稳定性分析是基于空间加权的能量方法, 问题的关键在于构造 适当的权函数来控制由于稳态波的非单调性及边界条件的出现所导致的困难. 至于关于时间变元的衰 减估计, 除了这一空间加权的能量方法之外, 还利用了由 Kawashima 和 Matsumura 在 1985 年引入的 空间 - 时间加权的能量方法. 

We are concerned with the large-time behavior of radially symmetric solutions to the exterior problem of multidimensional Burgers equation and focus on the nonlinear stability of its radially symmetric stationary waves under radially symmetric initial perturbation. For such a problem, a sufficient condition to guarantee the existence of such a stationary wave is obtained by Hashimoto and Matsumura in 2019, but since the stationary wave is no longer monotonic, its nonlinear stability is justified only for the case where an additional assumption is imposed. The main purpose of this paper is to verify the time asymptotically nonlinear stability of such a stationary wave under the condition imposed by Hashimoto and Matsumura to guarantee its existence. Moreover, we also derive the temporal convergence rates, both algebraically and exponentially, of solutions of the above exterior problem to the stationary wave. Our stability analysis is based on a space weighted energy method with a suitable chosen weight function, while for the temporal decay rates, in addition to such a space weighted energy method, we also use the space-time weighted energy method introduced by Kawashima and Matsumura in 1985.