This paper is devoted to the study of the global stability of classical solutions to an initial and boundary value problem (IBVP) of a nonlinear PDE system, converted from a model of reinforced random walks, subject to time-dependent boundary conditions. It is shown that under certain integrability conditions on the boundary data, classical solutions to the IBVP exist globally in time and the differences between the solutions and their corresponding ansatz, determined by the initial and boundary conditions, converge to zero, as time goes to infinity. Though the final states of the boundary functions are required to match, the boundary values do not necessarily equal to each other at any finite time. In addition, there is no smallness restriction on the magnitude of the initial perturbations. Furthermore, numerical simulations are performed to investigate the long-time dynamics of the IBVP when the final states of the boundary functions do not match or the boundary functions are periodic in time.

本文致力于研究非线性 PDE 系统的初始和边值问题 (IBVP) 的经典解的全局稳定性，该系统是从增强随机游走模型转换而来的，受时间相关的边界条件影响。结果表明，在边界数据的某些可积性条件下，IBVP 的经典解在时间上全局存在，并且由初始和边界条件确定的解与其对应的 ansatz 之间的差异随着时间趋于无穷大而收敛为零. 尽管边界函数的最终状态需要匹配，但边界值不一定在任何有限时间内彼此相等。此外，对初始扰动的大小没有限制。此外，