Delayed blow‐up for chemotaxis models with local sensing

Abstract

The aim of this paper is to analyze a model for chemotaxis based on a local sensing mechanism instead of the gradient sensing mechanism used in the celebrated minimal Keller–Segel model. The model we study has the same entropy as the minimal Keller–Segel model, but a different dynamics to minimize this entropy. Consequently, the conditions on the mass for the existence of stationary solutions or blow‐up are the same; however, we make the interesting observation that with the local sensing mechanism the blow‐up in the case of supercritical mass is delayed to infinite time. Our observation is made rigorous from a mathematical point of view via a proof of global existence of weak solutions for arbitrary large masses and space dimension. The key difference of our model to the minimal Keller–Segel model is that the structure of the equation allows for a duality estimate that implies a bound on the false(H1false)′‐norm of the solutions, which can only grow with a square‐root law in time. This additional false(H1false)′‐bound implies a lower bound on the entropy, which contrasts markedly with the minimal Keller‐Segel model for which it is unbounded from below in the supercritical case. Besides, regularity and uniqueness of solutions are also studied.

Global existence and uniform boundedness in a chemotaxis model with signal-dependent motility

“…In [33], the motility γ is assumed to range in a closed interval of (0, ∞), thereby excluding the possibility of degeneracy and global classical solutions in two-space dimension, as well as global weak solutions in higher dimensions, were obtained. In [14] and [13], global weak solutions were constructed for γ(s) = 1/(c + s k ), c ≥ 0, when N ≤ 3, and for γ(s) = e −s when N ≥ 1, respectively.…”

“…More recently, through an adaptation of the comparison argument together with a standard compactness argument, the existence of weak solutions to system (1.1) in any space dimension with a generic motility is also studied in [29]. Still in arbitrary space dimension but with a completely different method, global existence of weak solutions to (1.4) is also shown in [13] for the particular choice γ(s) = e −s , exploiting the gradient structure of (1.4) available in that case.…”

“…Moreover, for the specific case γ(s) = e −χs (φ(s) = χs, correspondingly) with χ > 0, a new critical mass phenomenon in two space dimensions is observed: there is a critical value of the total mass u in L 1 (Ω) below which the global solution to (1.4) is uniformly-in-time bounded while the trajectory {u(t) : t ≥ 0} may be unbounded in L ∞ (Ω) when u in L 1 (Ω) exceeds the threshold value [17]. The threshold value turns out to be the same as for the minimal Keller-Segel model (φ(s) = χs in (1.5)), but the dynamical behavior is different in the two models, blowup being delayed to infinite time for (1.4) [13,17,24]). In a subsequent work [16], global existence of classical solutions to (1.4) when N ≥ 3 is established for γ(s) = s −k provided that 0 < k < ( √ 2N + 2)/(N − 2) and boundedness is verified for 0 < k < 2/(N − 2), again by a different method.…”

On the parabolic‐elliptic Keller–Segel system with signal‐dependent motilities: A paradigm for global boundedness and steady states

“…This result was further refined in previous work 17 showing that the blowup occurs at the infinity time. Recently, the existence and uniqueness of global weak solutions in all dimensions as well as blow‐up of solution to () with were discussed in 18 . For the system () with logistic growth (i.e., σ > 0), the blowup in two dimensions was ruled out for a large class of motility function γ ( v ).…”

Logistic damping effect in chemotaxis models with density-suppressed motility

“…Meanwhile, it was proved in [8,39,41] that global bounded classical solutions exist in all dimensions with some additional conditions on γ(v). Recently global weak solutions of (1.3) in all dimensions and the blow-up of solutions of (1.3) in two dimensions were investigated in [5].…”