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Boundedness in a three-dimensional chemotaxis–haptotaxis model with nonlinear diffusion

Research paper by Xuegang Hu, Liangchen Wang, Chunlai Mu, Ling Li

Indexed on: 11 Apr '17Published on: 27 Dec '16Published in: Comptes Rendus Mathematique



Abstract

The quasilinear chemotaxis–haptotaxis system{ut=∇⋅(D(u)∇u)−χ∇⋅(u∇v)−ξ∇⋅(u∇w)ut=+μu(1−u−w),x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0,wt=−vw,x∈Ω,t>0, is considered under homogeneous Neumann boundary conditions in a bounded and smooth domain Ω⊂R3Ω⊂R3. Here χ>0χ>0, ξ>0ξ>0 and μ>0μ>0, D(u)≥cDum−1D(u)≥cDum−1 for all u>0u>0 with some cD>0cD>0 and D(u)>0D(u)>0 for all u≥0u≥0. It is shown that if the ratio χμ is sufficiently small, then the system possesses a unique global classical solution that is uniformly bounded. Our result is independent of m.