A note on the free energy of the Keller–Segel model for subcritical and supercritical cases. (September 2015) - Bian, Shen
Abstract: This paper is devoted to the analysis of a degenerate Keller–Segel model with the diffusion exponent 2 d d + 2 < m < 2 − 2 d and m > 2 − 2 d . For m ≤ 2 − 2 d, this model possesses a scaling invariant space L p norm with p : = d ( 2 − m ) 2 . When m = 2 d d + 2, a result of Chen et al. (2012) shows that the L p norm of the steady states is the critical point of the free energy. For m = 2 − 2 d, the L p norm of the steady states minimizes the free energy (Blanchet et al., 2009). In this paper, we will explore the relationship between the L p norm of the steady states and the free energy with the diffusion exponent 2 d d + 2 < m < 2 − 2 d . Here a modified Hardy–Littlewood–Sobolev inequality plays an important role: ∬ R d × R d u ( x ) u ( y ) | x − y | d − 2 d x d y ≤ C H L S ‖ u ‖ p 2 − m ‖ u ‖ m m . In addition, we will give the upper bound of the support of the steady state solutions for the subcritical case m > 2 − 2 d via the concentration–compactness principle (Lions, 1984).