Existence and boundedness of solutions for a Keller-Segel system with gradient dependent chemotactic sensitivity

Abstract

We consider the Keller-Segel system with gradient dependent chemotactic sensitivity ut = Δu - ∇ ∙ (u|∇v|p-2∇v), x ∈ Ω, t > 0, vt = Δv - v + u, x ∈ Ω, t > 0, ∂u/∂v = ∂ν/∂ν = 0, x ∈ ∂Ω, t > 0, u(x, 0) = u0(x), v(x, 0) = v0(x), x ∈ Ω in a smooth bounded domain Ω ⊂ ℝn, n ≥ 2. We shown that for all reasonably regular initial data u0 ≥ 0 and v0 ≥ 0, the corresponding Neumann initial-boundary value problem possesses a global weak solution which is uniformly bounded provided that 1 < p n/(n - 1).