L^σ-measure criteria for boundedness in a quasilinear parabolic-parabolic Keller-Segel system with supercritical sensitivity

This paper studies the parabolic-parabolic Keller-Segel system with supercritical sensitivity: ut = ∇·(ϕ(u)∇u)−∇·(φ(u)∇v), v_t = ∆v−v+u, subject to homogeneous Neumann boundary conditions in a bounded and smooth domain Ω ⊂ ℝⁿ (n ≥ 2), the diffusivity fulfills ϕ(u) ≥ a₀(u + 1)^γ with γ ≥ 0 and a₀ > 0, while the chemotactic sensitivity satisfies 0 ≤ φ(u) ≤ b₀u(u + 1)^α+γ−1 with α > _n ² and b₀ > 0. It is proved that the problem possesses a globally bounded solution for (Formula presented.), whenever (Formula presented.) and (Formula presented.) is sufficiently small. Similarly, the above conclusion still holds for α > 2 provided that (Formula presented.) and (Formula presented.) are small enough.