A chemotaxis system with signal-dependent motility, indirect signal production and generalized logistic source: Global existence and asymptotic stabilization

Abstract

This work considers a chemotaxis system with signal-dependent motility in a smooth bounded domain Ω⊂Rn. If λ∈R,μ>0 andl>max⁡{n+24,1} are constants, then the system{ut=Δ(γ(v)u)+λu−μul,x∈Ω,t>0,vt=Δv−v+w,x∈Ω,t>0,wt=Δw−w+u,x∈Ω,t>0, with homogeneous Neumann boundary conditions possesses a global solution. Moreover, the solution satisfies‖u(⋅,t)−(λ+μ)1l−1‖L∞+‖v(⋅,t)−(λ+μ)1l−1‖L∞+‖w(⋅,t)−(λ+μ)1l−1‖L∞→0 as t→∞ under some extra hypotheses, where λ+=max⁡{λ,0}.