Global Existence and Asymptotic Behavior of Classical Solutions for a 3D Two-Species Keller-Segel-Stokes System with Competitive Kinetics

Abstract

This paper deals with the two-species Keller-Segel-Stokes system with competitive kinetics                (n 1 ) t + u · ∇n 1 = ∆n 1 − χ 1 ∇ · (n 1 ∇c) + µ 1 n 1 (1 − n 1 − a 1 n 2 ), x ∈ Ω, t > 0, (n 2 ) t + u · ∇n 2 = ∆n 2 − χ 2 ∇ · (n 2 ∇c) + µ 2 n 2 (1 − a 2 n 1 − n 2 ), x ∈ Ω, t > 0,under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ R 3 with smooth boundary. Many mathematicians study chemotaxis-fluid systems and two-species chemotaxis systems with competitive kinetics. However, there are not many results on coupled two-species chemotaxis-fluid systems which have difficulties of the chemotaxis effect, the competitive kinetics and the fluid influence. Recently, in the two-species chemotaxis-Stokes system, where −c + αn 1 + βn 2 is replaced with −(αn 1 + βn 2 )c in the above system, global existence and asymptotic behavior of classical solutions were obtained in the 3-dimensional case under the condition that µ 1 , µ 2 are sufficiently large ([5]). Nevertheless, the above system has not been studied yet; we cannot apply the same argument as in the previous works because of lacking the L ∞ -information of c. The main purpose of this paper is to obtain global existence and stabilization of classical solutions to the above system in the 3-dimensional case under the largeness conditions for µ 1 , µ 2 .2010Mathematics Subject Classification. Primary: 35K45; Secondary: 92C17; 35Q35.

Global existence and asymptotic behavior of classical solutions for a 3D two‐species chemotaxis‐Stokes system with competitive kinetics

“…The first theorem gives global existence and boundedness in under smallness conditions of the chemotactic effect and largeness conditions of the logistic power. This result provides some guide to methods and results for the 3‐dimensional Keller‐Segel‐Stokes system, which is the case that −( α n 1 + β n 2 ) c is replaced with − c + α n 1 + β n 2 in .…”

Global solutions to a two-species chemotaxis system with singular sensitivity and logistic source

“…Before we go to the details of our analysis, let us point out that the global existence, boundedness and stabilization of (weak) solutions to the two-species chemotaxis-fluid system have also been established (see e.g. [2,5,6,12,19]).…”

“…By the choice of r, rμ 1 p < 0 and (p(p-1)χ 1 +2pr) 2 4p(p-1) -(prχ 1 + r(r + 1)) < 0, because r ∈ (r -, r + ) ⇒ r 2 -(p -1)r + p(p -1) 2…”

Boundedness and stabilization in a two-dimensional two-species chemotaxis-Navier–Stokes system with competitive kinetics

“…On the other hand, in the case that n 2 = 0 and Ω ⊂ R 2 , global existence and boundedness of classical solutions to (1.1) have been attained ( [4]). Moreover, in the case that κ = 0 in (1.1), which namely means that the fourth equation in (1.1) is the Stokes equation, global existence and stabilization can be found in [2]; in the case that κ = 0 in (1.1) and that −(αn 1 + βn 2 )c is replaced with +αn 1 + βn 2 − c, global existence and boundedness of classical solutions to the Keller-Segel-Stokes system and their asymptotic behaviour are found in [3].…”