Eventual smoothness and asymptotic behaviour of solutions to a chemotaxis system perturbed by a logistic growth

Abstract

In this paper we study the chemotaxis-system ut = ∆u − χ∇ • (u∇v) + g(u) x ∈ Ω, t > 0, vt = ∆v − v + u x ∈ Ω, t > 0, defined in a convex smooth and bounded domain Ω of R n , n ≥ 1, with χ > 0 and endowed with homogeneous Neumann boundary conditions. The source g behaves similarly to the logistic function and satisfies g(s) ≤ a − bs α , for s ≥ 0, with a ≥ 0, b > 0 and α > 1. Continuing the research initiated in [33], where for appropriate 1 < p < α < 2 and (u 0 , v 0) ∈ C 0 (Ω) × C 2 (Ω) the global existence of very weak solutions (u, v) to the system (for any n ≥ 1) is shown, we principally study boundedness and regularity of these solutions after some time. More precisely, when n = 3, we establish that-for all τ > 0 an upper bound for a b , ||u 0 || L 1 (Ω) , ||v 0 || W 2,α (Ω) can be prescribed in a such a way that (u, v) is bounded and Hölder continuous beyond τ ;-for all (u 0 , v 0), and sufficiently small ratio a b , there exists a T > 0 such that (u, v) is bounded and Hölder continuous beyond T. Finally, we illustrate the range of dynamics present within the chemotaxis system in one, two and three dimensions by means of numerical simulations.

Large time behavior of solution to quasilinear chemotaxis system with logistic source

“…When µ > 0, a number of dynamical properties have been detected both numerically and also analytically. For example, the exceedance of corresponding carrying capacities seems possible (see [7,11,12,19,23,31,34,39]). For example, when µ > 0 is small, the chemotactic cross diffusion was shown to enforce the occurrence of solutions which attain possibly finite but arbitrarily large values (see [31]).…”

“…In [19], the authors have proved that the population as a whole always persists in the sense that for any nonnegative global classical solution, there exists a lower bound for mass. In [23], it is showed that logistic dampening may prevent blow-up of solutions. Besides, when τ = 0, r = 2 and D(u) ≡ 1, the main results in [22] showed the prevention of blow up under the conditions n ≤ 2, µ > 0 or n ≥ 3, µ > n−2 n χ.…”

On the logarithmic Keller-Segel-Fisher/KPP system

“…Moreover, Cauchy problem is also considered in R 2 , with regular sensitivity and (1.17), in [31]. There are also recent works dealing with the existence of large data weak solutions to Keller-Segel type models with logistic growth, and we refer the reader to [18,36,37,38,39,41,44,46,51] and the references therein for more information in this direction. However, we notice that with the logarithmic sensitivity function and density-dependent production rate, no result concerning the qualitative behavior of large data solutions is available in the knowledge base.…”

Boundedness in a parabolic‐elliptic chemotaxis system with nonlinear diffusion and sensitivity and logistic source

“…23 Additionally, in the case (v) = > 0, but with source term g(s) ≃ k− s r , for s ≥ 0, and with some r > 1, global existence of very weak solutions, as well their boundedness properties and long time behaviour are discussed in other works. [24][25][26] Finally, in order to better define the purpose of this present investigation, let us frame model (1), in its parabolic-elliptic representation, in the existent literature. For m = = 1 and (v) = > 0, it is proved in Tello and Winkler 27 that when > (n − 2) ∕n the solutions are globally bounded, whilst for m ∈ R, = 1 and (v) = > 0 the same result is achieved in Cao and Zheng 28 under the assumption > (1 − 2∕(n(2 − m) + )) .…”