Global dynamics of a quasilinear chemotaxis model arising from tumor invasion

This paper is concerned with a quasilinear chemotaxis system ut=∇⋅(D(u)∇u)−∇⋅(S(u)∇v),x∈Ω,t>0,vt=Δv+wz,x∈Ω,t>0,wt=−wz,x∈Ω,t>0,zt=Δz−z+u,x∈Ω,t>0,with homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂Rn(n≥1), where D satisfies D(u)>0 for all u≥0 and behaves algebraically as u→∞. It is shown that if S(u)D(u)≤Cuα with some constants C>0 for all u≥1 and α<1+1n,if1≤n≤3,α<4n,ifn≥4,then for sufficiently smooth initial data, the system possesses a unique bounded classical solution (u,v,w,z), which exponentially converges to the equilibrium (ū0,v̄0+w̄0,0,ū0) as t→+∞, where ū0=1|Ω|∫Ωu0(x)dx, v̄0=1|Ω|∫Ωv0(x)dx andw̄0=1|Ω|∫Ωw0(x)dx.