Traveling waves for degenerate diffusive equations on networks

Andrea Corli; Lorenzo di Ruvo; Luisa Malaguti; Massimiliano D. Rosini; Andrea Corli; Lorenzo di Ruvo; Luisa Malaguti; Massimiliano D. Rosini

doi:10.3934/nhm.2017015

Networks and Heterogeneous Media

2017, Volume 12, Issue 3: 339-370. doi: 10.3934/nhm.2017015

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Traveling waves for degenerate diffusive equations on networks

1.
Department of Mathematics and Computer Science, University of Ferrara, I-44121 Italy
2.
Department of Sciences and Methods for Engineering, University of Modena and Reggio Emilia, I-42122 Italy
3.
Department of Mathematics, Maria Curie-Skłodowska-University, PL-20031 Poland

Received: 01 December 2016 Revised: 01 April 2017
Primary: 35K65; Secondary: 35C07, 35K55, 35K57

In this paper we consider a scalar parabolic equation on a star graph; the model is quite general but what we have in mind is the description of traffic flows at a crossroad. In particular, we do not necessarily require the continuity of the unknown function at the node of the graph and, moreover, the diffusivity can be degenerate. Our main result concerns a necessary and sufficient algebraic condition for the existence of traveling waves in the graph. We also study in great detail some examples corresponding to quadratic and logarithmic flux functions, for different diffusivities, to which our results apply.
- Parabolic equations,
- wavefront solutions,
- traffic flow on networks
Citation: Andrea Corli, Lorenzo di Ruvo, Luisa Malaguti, Massimiliano D. Rosini. Traveling waves for degenerate diffusive equations on networks[J]. Networks and Heterogeneous Media, 2017, 12(3): 339-370. doi: 10.3934/nhm.2017015

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Abstract

In this paper we consider a scalar parabolic equation on a star graph; the model is quite general but what we have in mind is the description of traffic flows at a crossroad. In particular, we do not necessarily require the continuity of the unknown function at the node of the graph and, moreover, the diffusivity can be degenerate. Our main result concerns a necessary and sufficient algebraic condition for the existence of traveling waves in the graph. We also study in great detail some examples corresponding to quadratic and logarithmic flux functions, for different diffusivities, to which our results apply.

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