Global stability under dynamic boundary conditions of a nonlinear PDE model arising from reinforced random walks

Abstract

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.