L~γ-measure Criteria For Boundedness In A Quasilinear Parabolic-elliptic Keller-segel System With Supercritical Sensitivity

This paper studies the parabolic-elliptic Keller-Segel system with supercritical sensitivity: u_t=▽·（D（u）▽u）-▽（S（u）▽v）,0 = ?v-v + u in ? ×（0,T）,where bounded domain ? ? Rⁿ,n ≥ 2,subject to the non-flux boundary conditions,D（u）（?）（u + 1）^-q,S（u）（?）u（u + 1）^α-q-1 with q ∈ R,α ≥2/n.Concerning the suitable measure for initial data to be determined,it is proved that the problem possesses a unique globally bounded solution for α ≥2/n whenever ||u₀||L^γ is sufficiently small with γ >nα/2+q if 0 < q <2/n,or γ >nα/2 if q < 0.