Geodesic
Traveling waves for the Keller–Segel system with Fisher birth terms
Gregoire Nadin1 ; Benoît Perthame2 ; Lenya Ryzhik3
1Ecole Normale Superieure, Paris, France
2Université Pierre et Marie Curie, Paris, France
3Stanford University, United States
Interfaces and free boundaries, Tome 10 (2008) no. 4, pp. 517-538

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DOI

We consider the traveling wave problem for the one dimensional Keller-Segel system with a birth term of either a Fisher/KPP type or with a truncation for small population densities. We prove that there exists a solution under some stability conditions on the coefficients which enforce an upper bound on the solution and H ̇1(R) estimates. Solutions in the KPP case are built as a limit of traveling waves for the truncated birth rates (similar to ignition temperature in combustion theory).
DOI : 10.4171/ifb/200
Classification : 35-XX, 65-XX, 76-XX, 92-XX
Mots-clés : Chemotaxis, traveling waves, Keller–Segel system, reaction-diffusion systems, nonlinear stability

Gregoire Nadin  1   ; Benoît Perthame  2   ; Lenya Ryzhik  3

1 Ecole Normale Superieure, Paris, France
2 Université Pierre et Marie Curie, Paris, France
3 Stanford University, United States 
Gregoire Nadin; Benoît Perthame; Lenya Ryzhik. Traveling waves for the Keller–Segel system with Fisher birth terms. Interfaces and free boundaries, Tome 10 (2008) no. 4, pp. 517-538. doi: 10.4171/ifb/200
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     author = {Gregoire Nadin and Beno{\^\i}t Perthame and Lenya Ryzhik},
     title = {Traveling waves for the {Keller{\textendash}Segel} system with {Fisher} birth terms},
     journal = {Interfaces and free boundaries},
     pages = {517--538},
     year = {2008},
     volume = {10},
     number = {4},
     doi = {10.4171/ifb/200},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/ifb/200/}
}
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