New Traffic Flow Model with Nonlinear Anticipation and Intelligent-control Boundary: Existence, Long-time Behavior and Large-relaxation-time Limit
In this paper, we propose a novel physical model for traffic flow incorporating nonlinear anticipation effects and an intelligent-control boundary, mathematically formulated as a damping boundary condition: \begin{align*} \begin{cases} u_t^\tau+v_x^\tau=0, & x\in(0,1),\; t>0,\\[1mm] v_t^\tau+g(u_x^\tau)u_x^\tau=\dfrac{f(u^\tau)-v^\tau}{\tau}, & x\in(0,1),\; t>0,\\[1mm] (u^\tau,v^\tau)(x,0)=(u_0^\tau(x),v_0^\tau(x)), & x\in(0,1),\\[1mm] u_x^\tau(0,t)=0,\quad u_x^\tau(1,t)=-ku_t^\tau(1,t), & k>0,\; t>0, \end{cases} \end{align*} where $f$ and $g$ are smooth functions satisfying suitable structural assumptions, and $\tau>0$ is the relaxation time. The primary objective is to rigorously investigate how the intelligent-control boundary suppresses the stop-and-go phenomenon in the large-relaxation-time regime-a mechanism that has not been mathematically addressed in previous studies. Utilizing the energy method, we establish the global well-posedness and exponential time-decay of solutions for the original system under arbitrarily large initial data with small spatial derivatives. Furthermore, we analyze the asymptotic behavior in the large-relaxation-time limit $\tau\to\infty$, by introducing a novel technique that incorporates constant shifts into the initial data of the limiting system. This approach enables us to construct a modified auxiliary system, through which we successfully obtain the global convergence of the original solutions to the asymptotic profiles for all time $t$ as relaxation time $\tau\to\infty$. Numerical simulations further demonstrate that in the large-relaxation-time regime, the stop-and-go density waves emerging at the early stage are gradually suppressed by the damping boundary. This leads the traffic stream to eventually evolve into an essentially uniform profile, which perfectly validates our theoretical results.