Threshold value for a quasilinear Keller–Segel chemotaxis system with the intermediate exponent in a bounded domain

Abstract

We consider a quasilinear chemotaxis model ut=∇⋅(D(u)∇u)−∇⋅(S(u)∇v),τvt=Δv−v+u,with nonlinear diffusion function D∈C2([0,∞)) and chemotactic sensitivity S∈C2([0,∞)) in a bounded domain Ω⊂Rd(d≥3). Here the rate D(s)/S(s) grows like s2−m with 2d/(d+2)<m<2−2/d as s→∞, and τ=0,1.It is first shown that there exists a M∗>0 such that if free energy with initial data is suitably small and ‖u0‖L1(Ω)α‖u0‖Lm(Ω)β<M∗ with α=2/(2−m)−d/m>0 and β=d−2/(2−m)>0, then the classical solutions to the above system are uniformly-in-time bounded. Second, in radially symmetric settings we can find M∗>0 such that ‖u0‖L1(Ω)α‖u0‖Lm(Ω)β>M∗ and the corresponding solution must be unbounded. These results show that the global behavior of classical solutions is classified by the combination of norms of initial data when 2d/(d+2)<m<2−2/d.