GLOBAL WEAK SOLUTIONS OF A PARABOLIC-ELLIPTIC KELLER-SEGEL SYSTEM WITH GRADIENT DEPENDENT CHEMOTACTIC COEFFICIENTS

We consider the following Keller-Segel system with gradient dependent chemotactic coefficient: {u<inf>t</inf> = ∆u − χ∇ · (uf(|∇v|)∇v), 0 = ∆v − v + g(u), in smooth bounded domains Ω ⊂ Rⁿ, n ≥ 1 with f(ξ) = (ξ^p−²(1+ξ^p)^{q− p/p}), 1 < q ≤ p < ∞ and g(ξ) = ξ/(1+ξ<inf>)</inf>^1-^β, ξ ≥ 0, β ∈ [0, 1]. We show the existence of a global weak solution, bounded in L^∞-norm, if 1 < q ≤ p {< ∞, n = 1, 1 < q < <inf>n−</inf>ⁿ<inf>1</inf> , n ≥ 2.