On a parabolic-elliptic chemotaxis-growth system with nonlinear diffusion

Abstract

This paper considers the following parabolic-elliptic chemotaxisgrowth system with nonlinear diffusion ut = ∇ • (D(u)∇u) − ∇ • (χu q ∇v) + µu(1 − u α), x ∈ Ω, t > 0, 0 = ∆v − v + u γ , x ∈ Ω, t > 0 under homogeneous Neumann boundary conditions for some constants q ≥ 1, α > 0 and γ ≥ 1, where D(u) ≥ c D u m−1 (m ≥ 1) for all u > 0 and D(u) > 0 for all u ≥ 0, and Ω ⊂ R N (N ≥ 1) is a bounded domain with smooth boundary. It is shown that when m > q + γ − 2 N , or α > q + γ − 1, or α = q + γ − 1 and µ > µ * , where µ * = (α+1−m)N −2 (α+1−m)N +2(α−γ) χ, if m ≤ q + γ − 2 N , 0, if m > q + γ − 2 N , then the above system possesses a global bounded classical solution for any sufficiently smooth initial data. The results improve the results by Wang et al.

Influence of nonlinear production on the global solvability of an attraction‐repulsion chemotaxis system

“…(We also mention [16, 28] for the critical analysis on both blowing‐up and global solutions to Keller–Segel systems with super and sublinear stimulus production.) Further, for B , C , D and E as above but for $$A(u,v,w)\simeq u^{q-1}$$, $$q\ge 1$$, in [25] it is shown that there exists $$\mu ^*=\mu ^*(m,q,\chi ,\alpha ,n)\ge 0$$ such that when $$q >m + \gamma -\frac{2}{n}$$ or $$\alpha > m + \gamma -1$$ or $$\alpha = m + \gamma -1$$ and $$\mu >\mu ^*$$ system (1.1) possesses a global bounded classical solution for any sufficiently smooth initial data.…”

Global existence of solutions to Keller-Segel chemotaxis system with heterogeneous logistic source and nonlinear secretion

“…The proof of Lemma 2.2 is similar to the proof of [15,Lemma 2.2]. Furthermore, to make the paper self-contained we provide some basic inequalities.…”

Global boundedness and asymptotic behavior of solutions for a quasilinear chemotaxis model of multiple sclerosis with nonlinear signal secretion

“…We should mention that recently, some global existence results and large time behavior of solutions have also been established for the classical Keller-Segel model, both for the quasilinear parabolic-parabolic and parabolic-elliptic chemotaxis-growth system with nonlinear signal production, one can refer to the literatures [10,36,38,37]. However, to the best of our knowledge, until now there are no results for stabilization of quasilinear chemotaxis model of multiple sclerosis with nonlinear signal secretion, especially, for the case τ = 0.…”

“…Boundedness for multiple sclerosis chemotaxis model with τ = 0. In this section, based on the ideas of [36] and [37], we proceed to derive the main step towards our boundedness proof with τ = 0. Let us first concentrate on the case that m < 1 + 2 N − l − r 2 .…”