Boundedness in a two-species chemotaxis system with nonlinear sensitivity and signal secretion

Abstract

The fully parabolic two-species chemotaxis system with nonlinear sensitivity, Lotka-Volterra competitive kinetics, and signal secretion{ut=Δu−∇⋅(S1(u)∇v)+μ1u(1−uα1−a1w),x∈Ω,t>0,vt=Δv−v+g1(w),x∈Ω,t>0,wt=Δw−∇⋅(S2(w)∇z)+μ2w(1−wα2−a2u),x∈Ω,t>0,zt=Δz−z+g2(u),x∈Ω,t>0 is considered in a bounded domain Ω⊂R3 with homogeneous Neumann boundary conditions and the parameters μi,αi,ai>0(i=1,2). The chemotactic sensitivity function |Si(s)|≤|χi|sqi for all s>0 with χi∈R and qi>0(i=1,2). The nonlinear signal secretion function gi(s)≤kisγi for all s>0 with ki>0 and γi>0(i=1,2). When the initial data satisfy the appropriate regularization assumptions, we prove that the system has a global bounded classical solution under different parameter conditions. Our results partially improve on results reported by Ren and Liu (2019) [22].