Global boundedness to a chemotaxis–haptotaxis model with nonlinear diffusion

Abstract

This paper deals with the following chemotaxis–haptotaxis model ut=∇⋅(D(u)∇u)−χ∇⋅(u(1+u)α∇v)−ξ∇⋅(u(1+u)β∇w)+u(a−μuk−1−λw),vt=△v−v+u,wt=−vw,under homogeneous Neumann boundary condition in a bounded domain Ω⊂R3, with χ,ξ,μ,λ>0, k>1, a∈R, and D(u)≥CD(u+1)m−1 for CD>0,m∈R. It is shown that if α>2−k and β>76−2k3, the corresponding system possesses a classical solution (u,v,w) which is globally bounded. The result improved the work proposed in Xu et al. (2019), in which, the global boundedness of solutions is established for m>0,α>0,β≥0 and k=2.