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

Abstract

This paper is concerned with global well-posedness to the following fully parabolic kinetic systemin a smooth bounded domain Ω ⊂ R n , n ≥ 1 with no-flux boundary conditions. This model was recently proposed in [8,20] to describe the process of stripe pattern formations via the so-called self-trapping mechanism. The system features a signal-dependent motility function γ(·) which is decreasing in v and will vanish as v tends to infinity. The major difficulty in analysis comes from the possible degeneracy as v ր +∞. In this work we develop a new comparison method different from the conventional energy method in literature which reveals a striking fact that there is no finite-time degenercay in this system. More precisely, we use comparison principles for elliptic and parabolic equations to prove that degeneracy cannot take place in finite time in any spatial dimensions for all smooth motility functions satisfying γ(s) > 0, γ ′ (s) ≤ 0 when s ≥ 0 and lim s→+∞ γ(s) = 0.Then we investigate global existence of classical solutions to (0.1) when n ≤ 3 and discuss the uniform-in-time boundedness under certain growth conditions on 1/γ.In particular, we consider system (0.1) with γ(v) = e −v , which shares the same set of equilibria as well as the Lyapunov functional as the classical Keller-Segel model. In the two-dimensional setting, we observe a critical-mass phenomenon which is distinct from the well-known fact for the classical Keller-Segel model. We prove that classical solution always exists globally which is uniformly-in-time bounded with arbitrary initial data of sub-critical mass. On the contrary, with certain initial data of super-critical mass, the solution will become unbounded at time infinity which differs from the finite-time blowup behavior of the Keller-Segel model.

Delayed blow‐up for chemotaxis models with local sensing

“…A crucial step is to define an appropriate approximate system that conserves both the entropy and the duality structures, which are of very different nature. Note that we can connect our weak solutions with the strong solutions obtained in [13,20] thanks to our regularity and uniqueness results.…”

“…Both works focus on smooth solutions and the question of the possibility of blow-up. They show the existence of a critical mass such that, in the subcritical case, there exist uniformly bounded smooth solutions, and convergence to a single stationary solution is further shown in [13]. In the supercritical case, they obtain the blow-up in L ∞ , which is furthermore proved to happen only in infinite time in [13].…”

“…Both works [9,33] rely on duality and energy techniques (but not on an entropy in the strict sense of the term). Shortly afterwards, in [13], smooth solutions are obtained in dimension d = 2 for a very generic motility (positive and decreasing to zero). Furthermore, the authors obtain uniformly bounded smooth solutions if the motility γ(v) decays at most algebraically in dimension d = 2, and at most in 1/v in dimension d = 3.…”

“…When it comes to the case where the motility has precisely the exponential decay (1.1e), it has been studied specifically in two-dimensional domains in [13,20]. Both works focus on smooth solutions and the question of the possibility of blow-up.…”

Traveling wave solutions for a bacteria system with density-suppressed motility

“…For example, if γ(v) = e −χv , by constructing a Lyapunov functional, it was proved in [9] that there exists a critical mass m * = 4π χ such that the solution of (1.1) with τ = 1 exists globally with uniform-in-time bound if Ω u 0 dx < m * while blows up if Ω u 0 dx > m * in two dimensions, where u 0 is the initial value of u. The result of [9] was further refined in [4] by showing that the blow-up time is infinite. When γ(v) has both positive lower and upper bounds, the global existence of classical solutions in two dimensions was proved in [35].…”