Finite time blow-up in the higher dimensional parabolic-elliptic-ODE minimal chemotaxis-haptotaxis system

Abstract

In this work, we mainly study nonnegative classical solutions to a Neumann initial-boundary value problem for the following parabolic-elliptic-ODE minimal chemotaxis-haptotaxis system{ut=Δu−χ∇⋅(u∇v)−ξ∇⋅(u∇w),x∈Ω,t>0,0=Δv−v+u,x∈Ω,t>0,wt=−vw,x∈Ω,t>0 in a bounded and smooth domain Ω⊂Rn(n≥1) with χ,ξ≥0, nonnegative initial data (u0,w0) and homogeneous Neumann boundary conditions.In this setup, we first show pure haptotaxis (χ=0) cannot induce any blow-up and pattern for any n≥1, showing haptotaxis is overbalanced by diffusion on boundedness and longtime behavior. Then, in the radial setting with Ω=BR⊂Rn(n≥3), it is well-known that small mass of ‖u0‖L1(Ω) can lead to blow-up in the corresponding Keller-Segel chemotaxis-only model obtained by setting w≡0, known as generic mass blow-up phenomenon [23]. Herein, in the presence of the temporal nonlocality brought by haptotaxis, we show, in an explicit realm of the form χ‖u0‖L1(Ω)>A(n,R)ξ‖w0‖L∞(Ω), that the aforementioned generic mass finite time blow-up phenomenon preserves. This seems to be the first rigorous blow-up result in relevant chemotaxis-haptotaxis models. The blow-up result also suggests that haptotatic cross-diffusion may not be negligible compared to chemotactic aggregation in ≥3D, in contrast to the recently detected 2D negligibility of haptotaxis in a minimal chemotaxis-haptotaxis model [16].