Finite-Time Blow-up in a Quasilinear Degenerate Chemotaxis System with Flux Limitation

This paper deals with the quasilinear degenerate chemotaxis system with flux limitation { u t = ∇ ⋅ ( u p ∇ u u 2 + | ∇ u | 2 ) − χ ∇ ⋅ ( u q ∇ v 1 + | ∇ v | 2 ) , x ∈ Ω , t > 0 , 0 = Δ v − μ + u , x ∈ Ω , t > 0 , where Ω : = B R ( 0 ) ⊂ R n ( n ∈ N ) is a ball with some R > 0 , and χ > 0 , p , q ≥ 1 , μ : = 1 | Ω | ∫ Ω u 0 and u 0 is an initial data of an unknown function u . Bellomo–Winkler (Trans. Am. Math. Soc. Ser. B 4 , 31–67, 2017 ) established existence of an initial data such that the corresponding solution blows up in finite time when p = q = 1 . This paper gives existence of blow-up solutions under some condition for χ and u 0 when 1 ≤ p ≤ q .