Large time behavior of solution to a quasilinear chemotaxis model describing tumor angiogenesis with/without logistic source

• Published in: Nonlinear analysis: real world applications Vol. 81; p. 104214
• Main Authors: Xiao, Min, Zhao, Jie, He, Qiurong
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.02.2025
• Subjects: Angiogenesis; Asymptotic behavior; Chemotaxis; Convection; Logistic source; Angiogenesis; 35A01; Asymptotic behavior; Logistic source; 92C17; 35B40; 35K57; Chemotaxis; 35K55; Convection; 35Q92
• ISSN: 1468-1218
• DOI: 10.1016/j.nonrwa.2024.104214

In this paper, we deal with the following Neumann-initial boundary value problem for a quasilinear chemotaxis model describing tumor angiogenesis: ut=∇⋅(D(u)∇u)−χ∇⋅(u∇v)+ξ∇⋅(u∇w)+μu−μu2,x∈Ω,t>0,vt=Δv+∇⋅(v∇w)−v+u,x∈Ω,t>0,0=Δw−w+u,x∈Ω,t>0,∂u∂ν=∂v∂ν=∂w∂ν=0,x∈∂Ω,t>0,u(x,0)=u0(x),v(x,0)=v0(x),x∈Ω,in a bounded smooth domain Ω⊂Rn(n≤3), where the parameter χ,ξ>0,μ≥0, D(u) is supposed to satisfy the behind property D(u)≥(u+1)αwithα>0.Assume that either μ≥0,α>1 or μ=0,ξ≥λ1∗χ2, where the parameter λ1∗=λ1∗(u0,v0,Ω)>0, then the system admits a global classical solution (u,v,w) by subtle energy estimates. Moreover, when μ=0, it is asserted that the corresponding solution exponentially converges to the constant stationary solution (u0¯,u0¯,u0¯) provided the initial data u0 is sufficiently small, where u0¯=∫Ωu0|Ω|. Finally, when μ>0, it can be proved that the corresponding solution of the system decays to (1,1,1) exponentially for suitable large μ.