Nonlinear Pedestrian-flow Model: Uniform Well-posedness and Global Existence

This paper is concerned with the existence, uniqueness and propagation of monotonous properties for a certain class of Cauchy problems for first-order Hamilton-Jacobi equations for which initial data is a gradient function. In particular, we show that the monotonicity properties are propagated under certain assumptions. The main result is gained by the method of characteristics and a priori estimates under a monotonicity criterion. Applications of equations studying here refer to control problems in crowd dynamics.