# American Institute of Mathematical Sciences

September  2017, 12(3): 339-370. doi: 10.3934/nhm.2017015

## 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

* Corresponding author

Received  December 2016 Revised  April 2017 Published  September 2017

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.

Citation: Andrea Corli, Lorenzo di Ruvo, Luisa Malaguti, Massimiliano D. Rosini. Traveling waves for degenerate diffusive equations on networks. Networks & Heterogeneous Media, 2017, 12 (3) : 339-370. doi: 10.3934/nhm.2017015
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##### References:
A star graph
A flux $f_h$ satisfying (f), solid curve, and the corresponding reduced flux $g_h$ defined in (3.4), dashed curve, in the case $c_h<0$, left, and in the case $c_h>0$, right
Values $\max\{f_j(\ell_j^-), f_j(\ell_j^+)\}$ and $\min\{f_j(\ell_j^-), f_j(\ell_j^+)\}$ equal the right-hand side of (4.2) and (4.3), respectively; the lines have slope $c_j\ne0$. Left: $c_j>0$. Right: $c_j<0$
A network with $m=1$
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