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This paper deals with the one-dimensional formulation of Hughes model forpedestrian flows in the setting of entropy solutions, which authorizesnon-classical shocks at the location of the so-called turning curve. Weconsider linear cost functions, whose slopes $\alpha$ 0 correspond to differentcrowd behaviours. We prove existence and partial well-posedness results in theframework of entropy solutions. The proofs of existence are based on a asharply formulated many-particle approximation scheme with careful treatment ofinteractions of particles with the turning curve, and on local reductions tothe well-known Lighthill-Whitham-Richards model. For the special case ofBV-regular entropy solutions without non-classical shocks, locally Lipschitzcontinuous dependence of such solutions on the initial datum $\rho$ and on thecost parameter $\alpha$ is proved. Differently from the stability argument andfrom existence results available in the literature, our existence result allowsfor the possible presence of non-classical shocks. First, we exploreconvergence of the many-particle approximations under the assumption of uniformspace variation control. Next, by a local compactness argument that permits tocircumvent the possible absence of global BV bounds, we obtain existence ofsolutions for general measurable data. Finally, we illustrate numerically thatthe model is able to reproduce typical behaviours in case of evacuation.Special attention is devoted to the impact of the parameter $\alpha$ on theevacuation time.

On existence, stability and many-particle approximation of solutions of 1D Hughes model with linear costs