A note on the free energy of the Keller–Segel model for subcritical and supercritical cases. (September 2015)

• Record Type: Journal Article
• Authors: Bian, Shen
• Is Part Of: Nonlinear analysis. Volume 125(2015)
• Page Start: 406
• Page End: 422
• Publication Date: 2015-09
• Subjects: Free energy -- Steady state solutions -- Invariant scaling -- Far-field mass -- Compact support; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2015.05.020
• Languages: English
• ISSNs: 0362-546X

Abstract

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).