2018/04/11 by Chunhua Jin, Yifu Wang, Jin, Chunhua +3
Computer Science · Mathematics · Medicine · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Biology Tumor Growth #Mathematical and Theoretical Epidemiology and Ecology Models #math.AP
paper · pdf · doi:10.48550/arxiv.1804.03964
openalex publication_date 2018/04/11 · openalex created_date 2018/04/24 · arxiv created 2018/09/30 · arxiv updated 2018/10/02 · openalex updated_date 2026/07/28
In this paper, we study a nutrient-taxis model with porous medium slow diffusion \ \beginaligned ut=Δum-χ∇⋅(u∇ v)+ξuv-ρu,
vt-Δv=-vu+μv(1-v), \endaligned. in a bounded domain Ω⊂ \mathbb R3 with zero-flux boundary condition. It is shown that for any m>\frac114-√ 3,the problem admits a global weak solution for any large initial datum. We divide the study into three cases,(i) ξμ=0, ρ≥ 0; (ii) ξμρ>0; (iii) ξμ>0, ρ=0. In particular, for Case (i) and Case (ii), the global solutions are uniformly bounded. Subsequently, the large time behavior of these global bounded solutions are also discussed. At last, we also extend the results to the coupled chemotaxis-Stokes system. Important progresses for chemotaxis-Stokes system with m>\frac 76, m>\frac 87 and m>\frac 98 have been carried out respectively by \citeW2, TW2, W3, but leave a gap for 1<m≤ \frac98. Our result for chemotaxis-Stokes system supplements part of the gap (\frac114-√ 3, \frac 98). Here \frac114-√ 3≈ 1.018.