Nonlinear Diﬀer. Equ. Appl.
2017 Springer International Publishing AG
Nonlinear Diﬀerential Equations
and Applications NoDEA
Sharp proﬁles in models of collective
Dedicated to Alberto Bressan on the occasion of his 60th birthday.
, Lorenzo di Ruvo and Luisa Malaguti
Abstract. We consider a parabolic partial diﬀerential equation that can
be understood as a simple model for crowds ﬂows. Our main assumption
is that the diﬀusivity and the source/sink term vanish at the same point;
the nonhomogeneous term is diﬀerent from zero at any other point and
so the equation is not monostable. We investigate the existence, regular-
ity and monotone properties of semi-wavefront solutions as well as their
convergence to wavefront solutions.
Mathematics Subject Classiﬁcation. Primary 35K65; Secondary 35C07,
Keywords. Degenerate parabolic equations, Semi-wavefront solutions,
Collective movements, Crowd dynamics.
In this paper we consider the scalar advection–reaction–diﬀusion equation
+ g(ρ),t≥ 0,x∈ R, (1.1)
for the unknown function ρ = ρ(x, t). We assume that f ∈ C
[0, ρ], f(0) = 0,
g ∈ C[0,
ρ], D ∈ C
[0, ρ] and denote for short h(ρ):=f
(ρ); here, ρ is a positive
constant. All along the paper we consider several diﬀerent conditions on both
D and g but we mainly focus on the case that D satisﬁes
ρ)=0andD(ρ) > 0forρ ∈ (0, ρ).
About the forcing term g we mainly deal with the following assumption:
(g) g(ρ) > 0forρ ∈ [0,
This article is part of the topical collection “Hyperbolic PDEs, Fluids, Transport and Appli-
cations: Dedicated to Alberto Bressan for his 60th birthday” guest edited by Fabio Ancona,
Stefano Bianchini, Pierangelo Marcati, Andrea Marson.