We analyze an equation that is gradient flow of a functional related to Hardy–Littlewood–Sobolev inequality in whole Euclidean space , . 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.
Mots clés : Degenerate parabolic equation, Energy functional, Gradient flow, Free-energy solutions, Blow-up, Global existence
@article{AIHPC_2017__34_4_961_0, 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://www.numdam.org/articles/10.1016/j.anihpc.2016.08.002/} }
TY - JOUR AU - Ulusoy, Suleyman TI - A Keller–Segel type system in higher dimensions JO - Annales de l'I.H.P. Analyse non linéaire PY - 2017 SP - 961 EP - 971 VL - 34 IS - 4 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2016.08.002/ DO - 10.1016/j.anihpc.2016.08.002 LA - en ID - AIHPC_2017__34_4_961_0 ER -
%0 Journal Article %A Ulusoy, Suleyman %T A Keller–Segel type system in higher dimensions %J Annales de l'I.H.P. Analyse non linéaire %D 2017 %P 961-971 %V 34 %N 4 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2016.08.002/ %R 10.1016/j.anihpc.2016.08.002 %G en %F AIHPC_2017__34_4_961_0
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. http://www.numdam.org/articles/10.1016/j.anihpc.2016.08.002/
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