Geodesic
A Keller–Segel type system in higher dimensions
Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 4, pp. 961-971

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We analyze an equation that is gradient flow of a functional related to Hardy–Littlewood–Sobolev inequality in whole Euclidean space Rd, d≥3. Under the hypothesis of integrable initial data with finite second moment and energy, we show local-in-time existence for any mass of “free-energy solutions”, namely weak solutions with some free energy estimates. We exhibit that the qualitative behavior of solutions is decided by a critical value. Actually, there is a critical value of a parameter in the equation below which there is a global-in-time energy solution and above which there exist blowing-up energy solutions.

DOI : 10.1016/j.anihpc.2016.08.002
Classification : 35K65, 35B45, 35J20
Keywords: Degenerate parabolic equation, Energy functional, Gradient flow, Free-energy solutions, Blow-up, Global existence 
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     author = {Ulusoy, Suleyman},
     title = {A {Keller{\textendash}Segel} type system in higher dimensions},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {961--971},
     publisher = {Elsevier},
     volume = {34},
     number = {4},
     year = {2017},
     doi = {10.1016/j.anihpc.2016.08.002},
     mrnumber = {3661866},
     zbl = {1435.35205},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2016.08.002/}
}
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Ulusoy, Suleyman. A Keller–Segel type system in higher dimensions. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 4, pp. 961-971. doi: 10.1016/j.anihpc.2016.08.002

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