Boundedness in a quasilinear two-species chemotaxis system with two chemicals in higher dimensions

Abstract

In this paper, a nonlinear diffusion two-species chemotaxis model with two chemicals{ut=∇⋅(D(u)∇u)−∇⋅(u∇v),τvt=Δv−v+w,wt=∇⋅(S(w)∇w)−∇⋅(w∇z),τzt=Δz−z+u, is considered in a bounded domain x∈Ω⊂Rd(d≥2) with homogeneous Neumann boundary conditions, where t≥0, and τ=0 or τ=1. The diffusion functions (D,S)∈(C2([0,∞)))2 are assumed to take forms like D(u)=(u+1)m1−1 and S(w)=(w+1)m2−1 with some m1,m2>1, respectively. In the fully parabolic case τ=1, it is proved that global classical bounded solutions exist if m1>2−2/d and m2>2−2/d. While for the case τ=0, by the means of the Lyapunov functional, the range of m1 and m2 to ensure the global existence of solutions can be extended to m1m2+2m1/d>m1+m2 or m1m2+2m2/d>m1+m2.