Global asymptotic stability in a two-dimensional chemotaxis model arising from tumor invasion

Abstract

This paper considers the chemotaxis model with density-suppressed motility: ut = ∇·(φ(v)∇u) + ∇·(ψ(v)u∇v) + f(u), vt = Δv + wz, wt = −wz, wt = −wz, zt = Δz − z + u, x ∈ Ω, t > 0 under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂R2. Given that the positive motility function φ(v) has the lower-upper bound, we can conclude that the system possesses a unique bounded classical solution. Moreover, it is proved that the global bounded solution (u, v, w, z) will converge to r/μ1α−1,v̄0+w̄0,0,r/μ1α−1 as t → ∞.