Lγ-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: ut = ∇ · (D(u)∇u) − ∇ · (S(u)∇v), 0 = ∆v − v + u in Ω × (0, T ), where the bounded domain Ω ⊂ R ⁿ , n ≥ 2, sub ject to the non-flux boundary conditions, D(u) ≥ a0 (u + 1) ^−q , 0 ≤ S(u) ≤ b ₀ u(u + 1) ^α −q− ¹ with q ∈ R, α > _n ² , and a0, b ₀ > 0. It is proved that the problem possesses a unique globally bounded solution for α > _n ² whenever ku0 k _L nα ₂ is sufficiently small. In addition, we establish the large-time behavior of solutions when q = 0.