Global boundedness and stability analysis of the quasilinear immune chemotaxis system

Abstract

In this paper, we study the following reaction-diffusion-advection system{ut=d1∇⋅(D(u)∇u)−χ∇⋅(S(u)∇v)+F(u,w),(x,t)∈Ω×(0,∞),vt=d2Δv−μ1v+s1g(u)w,(x,t)∈Ω×(0,∞),wt=d3Δw+s2−μ2w−λwh(u)w,(x,t)∈Ω×(0,∞), under homogeneous Neumann boundary conditions in a smoothly bounded domain Ω⊂Rn, which describes a cross-talk between antigens and immune cells via chemokines in the immune system, where d1,d2,d3,s1,s2,λw, μ1,μ2,χ are positive parameters, the functions D,S,g,h∈C2([0,+∞)) and F belongs to C2([0,+∞)×[0,+∞)). When n≥1, we study the global existence and boundedness of classical solutions for the above system under some suitable conditions of the functions D,S,F,g and h. When n≤3, we consider the stability and instability of the above system based on the spectral analysis, energy estimate and bootstrap technique. Moreover, some numerical simulations are carried out to support our theoretical results and also find some new interesting phenomena, such as complex spatial-temporal patterns. Our results not only extend the previous ones of [25], but also involve some new conclusions.