Asymptotic stability in a quasilinear chemotaxis-haptotaxis model with general logistic source and nonlinear signal production

Abstract

This paper deals with the quasilinear chemotaxis-haptotaxis model of cancer invasion{ut=∇⋅(D(u)∇u)+∇⋅(S1(u)∇v)+∇⋅(S2(u)∇w)+f(u,w),x∈Ω,t>0,τvt=Δv−v+g1(w)g2(u),x∈Ω,t>0,wt=−vw,x∈Ω,t>0 in a bounded smooth domain Ω⊂RN(N≥1) with zero-flux boundary conditions, where τ∈{0,1}, the functions D(u),S1(u),S2(u)∈C2([0,∞)), f(u,w)∈C1([0,∞)2),g1(w),g2(u)∈C1([0,∞)) fulfill D(u)≥CD(u+1)−α,S1(u)≤χu(u+1)β−1,S2(u)≤ξu(u+1)γ−1, f(u,w)≤u(a−μur−1−λw),f(0,w)≥0,g1(w)≥0,0≤g2(u)≤Kuκ with CD,χ,ξ,μ,K,κ>0, λ≥0, r>1 and α,β,γ,a∈R. Under specific parameters conditions, it is shown that for any appropriately regular initial data, the associated initial-boundary value problem admits a globally bounded classical solution. Moreover, when f=u(a−μur−1−λw), g1(w)≡1 and g2(u)=uκ, the asymptotic stability of solutions is also investigated. Specifically, for some a0,μ0>0 independent of (u0,v0), the bounded classical solution (u,v,w) exponentially converges to ((aμ)1r−1,(aμ)κr−1,0) in Lp(Ω)×L∞(Ω)×W1,∞(Ω) for any p≥2 if a>a0 and μ>μ0. These results improve or extend previously known ones, and partial results are new.