Asymptotic behavior of solutions to a tumor angiogenesis model with chemotaxis--haptotaxis

2019/03/26 by Pang, Peter Y. H., Wang, Yifu
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1903.10835

Abstract

This paper studies the following system of differential equations modeling tumor angiogenesis in a bounded smooth domain Ω⊂ ℝ^N (N=1,2): \p_t=Δp-∇\cdotp p(\frac α1+c∇ c+ρ∇ w)+λp(1-p), · amp; x∈ Ω, t · gt;0, c_t=Δc-c-μpc, · amp;x∈ Ω, t · gt;0,
w_t= γp(1-w), · amp; x∈ Ω, t · gt;0,. where α, ρ, λ, μ and γ are positive parameters. For any reasonably regular initial data (p₀, c₀, w₀), we prove the global boundedness (L^∞-norm) of p via an iterative method. Furthermore, we investigate the long-time behavior of solutions to the above system under an additional mild condition, and improve previously known results. In particular, in the one-dimensional case, we show that the solution (p,c,w) converges to (1,0,1) with an explicit exponential rate as time tends to infinity.

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