On a quasilinear parabolic–elliptic chemotaxis system with logistic source

Global existence and boundedness of classical solutions in a quasilinear parabolic–elliptic chemotaxis system with logistic source

“…For details of the proof, we refer the reader to [25,16]. In order to prove the boundedness of the solution, we need the following lemma, which is given in [15].…”

“…Wang et al[16] studied problem (1.1) under the conditions (1.2) and (1.3) with S(u) = χ u and τ = 0, where χ is some positive constant. Their results improve the recent result in[3], which asserts the boundedness of solutions with γ = 2 under the condition b > χ(1 − 2 n(1−m) + ), or, equivalently, m > 1 − 2χ n(χ −b) + .…”

Boundedness of the solution of a higher-dimensional parabolic–ODE–parabolic chemotaxis–haptotaxis model with generalized logistic source

“…The striking feature of Keller-Segel models is the possibility of blow-up of solutions in a finite (or infinite) time (see, e.g., [1,9,18,39]), which strongly depends on the space dimension. Moreover, some recent studies have shown that the blow-up of solutions can be inhibited by the nonlinear diffusion (see Ishida et al [11] Winkler et al [1,27,36,40]) and the (generalized) logistic damping (see Li and Xiang [14], Tello and Winkler [31], Wang et al [33], Zheng et al [48]). …”

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

“…When ψ(u) = g(u) = u, ϕ(u) ≥ c(u + 1) p with p ∈ R, k = 2 and b > 1 − 2 n(1−p) + χ with χ > 0, Cao and Zheng [4] proved that the simplified parabolic-elliptic model has a unique global classic solution, which is uniformly bounded. Recently, Wang et.al in [32] investigated the boundedness and asymptotic behavior for simplified parabolic-elliptic model with the special case ψ(u) = g(u) = u and ϕ(u) ≥ C D u m−1 (m ≥ 1) under other additional technique conditions. In the recent paper [38], for the case of f (u) = ru − μu 2 with r ≥ 0 and μ > 0, in one-dimensional case, Winkler proved that going beyond carrying capacities actually is a genuinely dynamical feature of the simplified parabolic-elliptic system provided that μ < 1 and diffusion is sufficiently weak, moreover, and investigated global boundedness and finitetime blow-up for a corresponding hyperbolic-elliptic limit problem.…”

“…Inspired by the works in [4,29,32], we extend their approaches to problem (1.1). Assume that the functions ψ(u), g(u) and f ∈ C 0 ([0, ∞)) ∩ C 1 ((0, ∞)) satisfy the following conditions…”