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Saturated fronts in crowds dynamics. (English) Zbl 1469.35128

Summary: We consider a degenerate scalar parabolic equation, in one spatial dimension, of flux-saturated type. The equation also contains a convective term. We study the existence and regularity of traveling-wave solutions; in particular we show that they can be discontinuous. Uniqueness is recovered by requiring an entropy condition, and entropic solutions turn out to be the vanishing-diffusion limits of traveling-wave solutions to the equation with an additional non-degenerate diffusion. Applications to crowds dynamics, which motivated the present research, are also provided.

MSC:

35K65 Degenerate parabolic equations
35C07 Traveling wave solutions
35D30 Weak solutions to PDEs
35K15 Initial value problems for second-order parabolic equations
35K57 Reaction-diffusion equations
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