Global weak solutions of a parabolic-elliptic Keller-Segel system with gradient dependent chemotactic coefficients

Abstract

We consider the following Keller-Segel system with gradient dependent chemotactic coefficient:ut = ∆u − χ∇ · (uf(|∇v|)∇v), 0 = ∆v − v + g(u), in smooth bounded domains Ω ⊂ Rn, n ≥ 1 with f(ξ) = (ξp-2(1+ξp) q−p/p), 1 <q ≤ p < ∞ and g(ξ) = (1+ξ)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 < n/n−1, n>=2