Global boundedness to a chemotaxis system with singular sensitivity and logistic source

Abstract

In this paper, we study the parabolic-elliptic Keller-Segel system with singular sensitivity and logistic-type source: u t = ∆u − χ∇·( u v ∇v)+ ru − µu k , 0 = ∆v − v + u under the non-flux boundary conditions in a smooth bounded convex domain Ω ⊂ R n , χ, r, µ > 0, k > 1 and n ≥ 2. It is shown that the system possesses a globally bounded classical solution if k > 3n−2 n , and r > χ 2 4 for 0 < χ ≤ 2, or r > χ − 1 for χ > 2. In addition, under the same condition for r, χ, the system admits a global generalized solution when k ∈ (2 − 1 n , 3n−2 n ], moreover this global generalized solution should be globally bounded provided r µ and the initial data u 0 suitably small.

Asymptotic behavior to a chemotaxis consumption system with singular sensitivity

“…We begin with the local existence of classical solutions to by the standard contraction argument like that in Zhao and Zheng ,. Lemma 2.1…”

“…Theorem IV 5.3 Moreover, the strictly positivity of ( u , v ) can be obtained by the strong maximum principle. The uniqueness of solutions to in can be derived as that in Zhao and Zheng □…”

“…The uniqueness of solutions to (1) in Ω × (0, T max ) can be derived as that in Zhao and Zheng. 13 ◻ Next, we give some a priori estimate of the solution to (1). For convenience, denote T ∶= T max , and…”

“…We refer to previous studies [10][11][12][13] for more results on chemotaxis models with singular sensitivity. Now recall the chemotaxis consumption system with general diffusion and sensitivity…”

A new result for 2D boundedness of solutions to a chemotaxis–haptotaxis model with/without sub-logistic source

“…Indeed, for n = 1, 2, even arbitrarily small µ > 0 is enough to prevent blow-up by ensuring all solutions to (1.1) are global-in-time and uniformly bounded for all reasonably initial data [8,17,28,53]. This is even true for a two-dimensional chemotaxis system with singular sensitivity [7,58]. A very recent subtle study from [55] shows that logistic damping is not the weakest damping to ensure boundedness for (1.1) in 2-D. More precisely, with the logistic source κu − µu 2 in (1.1) replaced by a locally bounded kinetic term f (u) satisfying f (0) ≥ 0 as well as Evidently, besides the standard logistic source, f covers sub-logistic sources like:…”