Boundedness, Stabilization, and Pattern Formation Driven by Density-Suppressed Motility

Abstract

We are concerned with the following density-suppressed motility model: ut = Δ(γ(v)u) + μu(1 − u); vt = Δv + u − v, in a bounded smooth domain Ω ⊂ R 2 with homogeneous Neumann boundary conditions, where the motility function γ(v) ∈ C 3 ([0, ∞)), γ(v) > 0, γ (v) < 0 for all v ≥ 0, limv→∞ γ(v) = 0, and limv→∞ γ (v) γ(v) exists. The model is proposed to advocate a new possible mechanism: density-suppressed motility can induce spatio-temporal pattern formation through self-trapping. The major technical difficulty in the analysis of above density-suppressed motility model is the possible degeneracy of diffusion from the condition limv→∞ γ(v) = 0. In this paper, by treating the motility function γ(v) as a weight function and employing the method of weighted energy estimates, we derive the a priori L ∞-bound of v to rule out the degeneracy and establish the global existence of classical solutions of the above problem with a uniform-in-time bound. Furthermore, we show if μ > K 0 16 with K 0 = max 0≤v≤∞ |γ (v)| 2 γ(v) , the constant steady state (1, 1) is globally asymptotically stable and, hence, pattern formation does not exist. For small μ > 0, we perform numerical simulations to illustrate aggregation patterns and wave propagation formed by the model.

Global existence for a kinetic model of pattern formation with density-suppressed motilities

“…Once we establish (4.2), we can argue as before to obtain uniform-in-time boundedness of u L ∞ (Ω) in the 2D case [3]. In addition, one can argue in the same way as in [18] to obtain the stability of the classical solutions. Thus, we have Proposition 4.2.…”

“…, v 0 ) ≤ 16πa log λ − 32πa 2 log λ + 16πa 2 log λ + C = −16πa(a − 1) log λ + C ≤ −2Λ Λ 8π − 1 log λ + C → −∞ as λ → ∞,(6 18). …”

Boundedness and large time behavior in a two-dimensional Keller-Segel-Navier-Stokes system with signal-dependent diffusion and sensitivity

“…Moreover, one can show the map Φ is compact in X. Using the Schauder fixed point, we can conclude that there exists a Z ∈ S T such that Φ(Z) = Z, the uniqueness can be proved by using the similar arguments as in [24,46]. (ii) Regularity and non-negativity.…”

“…The results in Theorem 1.1 are new even in the fluid-free system (4) with u = 0. Indeed, in our results, we do not require the special structure χ(c) = −d (c), which plays an important role in [24,42,51].…”

“…2 are two constants). Furthermore, if the first equation of system (3) was replaced by n t = ∆(d(c)n) + µn(1 − n) (i.e., cell has growth), Jin et al [24] studied the global dynamics and patterns ( like aggregation and wavefronts) without small assumption as imposed in [51] and the lower-upper bound assumption in [42].…”