Boundedness of solutions in a chemotaxis system with nonlinear sensitivity and logistic source

Abstract

Please cite this article in press as: P. Zheng et al., Boundedness of solutions in a chemotaxis system with nonlinear sensitivity and logistic source, AbstractThis paper deals with a parabolic-elliptic chemotaxis system with nonlinear sensitivity and logistic sourceunder homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ R n (n ≥ 1), where χ > 0, the function ψ(u) is the chemotactic sensitivity, g(u)is the production rate of the chemoattractant and f (u) is the logistic source. Under some suitable assumptions on the nonlinearities ψ(u), g (u) and logistic source f (u), we study the global boundedness of solutions for the problem.

Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues

“…[147][148][149]209 Likewise, logistic-type sources may suppress blowup also in systems with nonlinear cell diffusion and variants in the cross-diffusive terms. 38,42,204,231 On the other hand, a mathematical caveat shows that finite-time blow-up may occur in a corresponding parabolic-elliptic version of (3.43) if n ≥ 5 and α < 3 2 + 1 2n−2 . 213 As indicated by numerical simulations, 158 see also Ref.…”

Existence of traveling wave solutions of parabolic–parabolic chemotaxis systems

“…In literature, (1.1) is called the Keller-Segel model or a chemotaxis model. Since the works by Keller and Segel, a rich variety of mathematical models for studying chemotaxis has appeared (see [3], [9], [10], [16], [19], [29], [37], [38], [39], [41], [45], [46], [47], [48], [49], [50], [51], and the references therein). In the current paper and the further coming papers, we will investigate various dynamical aspects of the following parabolic-parabolic Keller-Segel systems, u t = ∆u − χ∇(u∇v) + u(a − bu), x ∈ R N τ v t = ∆v − v + u, x ∈ R N .…”

Spreading speeds and traveling waves of a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^N$

“…Global existence and asymptotic behavior of solutions of (1.2) on bounded domain Ω have been extensively studied by many authors. The reader is referred to [3], [9], [16], [39], [41], [45], [46], [47], [48], [49], [50], [51], and references therein for the studies of (1.2) on bounded domain with Neumann or Dirichlet boundary conditions and with f (u, v) being logistic type source function or 0 and m(u), χ(u, v), and g(u, v) being various kinds of functions.…”