Boundedness and stabilization in a two-species chemotaxis system with two chemicals

Abstract

This paper deals with the two-species chemotaxis system with two chemicalsx ∈ Ω, t > 0, under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ R n (n ≥ 1), where the parameters d 1 , d 2 , d 3 , d 4 > 0, µ 1 , µ 2 > 0, a 1 , a 2 > 0 and α, β > 0. The chemotactic function χ i (i = 1, 2) and the signal production function f i (i = 1, 2) are smooth. If n = 2, it is shown that this system possesses a unique global bounded classical solution provided that |χ i | (i = 1, 2) are bounded. If n ≤ 3, this system possesses a unique global bounded classical solution provided that µ i (i = 1, 2) are sufficiently large. Specifically, we first obtain an explicit formula µ i0 > 0 such that this system has no blow-up whenever µ i > µ i0 . Moreover, by constructing suitable energy functions, it is shown that:• If a 1 , a 2 ∈ (0, 1) and µ 1 and µ 2 are sufficiently large, then any global bounded solution exponentially converges to• If a 1 > 1 > a 2 > 0 and µ 2 is sufficiently large, then any global bounded solution exponentially converges to (0, f 1 (1)/α, 1, 0) as t → ∞;• If a 1 = 1 > a 2 > 0 and µ 2 is sufficiently large, then any global bounded solution algebraically converges to (0, f 1 (1)/α, 1, 0) as t → ∞. 2010 Mathematics Subject Classification. Primary: 35K35, 35A01, 35B44; Secondary: 35B35, 92C17.

A new result for boundedness and stabilization in a two-species chemotaxis system with two chemicals

“…When µ i > 0 (i = 1, 2) and τ = 1, if n = 2, global boundedness solution has been studied by Black [4] for any µ 1 , µ 2 > 0 and asymptotic stabilization of arbitrary global bounded solution was considered provided that µ1 are sufficiently large. Recently, this results were extended and partially improved by Wang et al [39] (see also [27]). When µ i > 0 (i = 1, 2) and τ = 0, global bounded solution was constructed in [53] under that χ 1 χ 2 < µ 1 µ 2 , and the large time behavior has been addressed in [53,54] (see also [56,57]).…”

“…Theorem 1.1 provides some explicit conditions involving the parameters χ i , µ i and a i (i = 1, 2), which partially improves the result by Zhang et al [53] under the condition that χ 1 χ 2 < µ 1 µ 2 and Tu et al [34] under the condition that χ1 µ1 < a1n n−2 and χ2 µ2 < a2n n−2 . Next, motivated by the ideas in [1,21,48] (see also [20,38,39]), we will discuss the large time behavior and convergence rates of (1.1) in the weak competition case a 1 , a 2 ∈ (0, 1). Theorem 1.2.…”

“…Stabilization. In this section, we will derive the asymptotic behavior of the solutions to model (1.1), the ideas mainly come from [1,21,48] (see also [20,38,39]). To achieve our goals, we first recall the following lemma which is important for the proof of Theorems 1.…”

On a quasilinear fully parabolic two-species chemotaxis system with two chemicals

“…In case τ = 1, the boundedness was obtained if the all parameters were positive, and the large time behavior of solution to (1.7) under the conditions that µ1 were large enough by Black [3]; Wang et al [41] further studied this system in n ≤ 3, under the condition that µ i (i = 1, 2) were large enough, the global boundedness and the large time behavior were obtained; recently, the global boundedness of n = 3 was further studied by Pan et al [28]. There are also some another results about system (1.7), we refer to [27,53,54,55,56].…”

Boundedness in a three‐dimensional two‐species and two‐stimuli chemotaxis system with chemical signalling loop

“…The ideas mainly come from previous studies. 14,29,37,47 Lemma 3.1. Suppose that (u 0 , v 0 , w 0 , z 0 ) satisfies (4), then there exists 1 ∈ (0, 1 2 ) such that the solution of (3) satisfy…”