Absence of collapse in a parabolic chemotaxis system with signal‐dependent sensitivity

Abstract

Key words Chemotaxis, global existence, boundedness MSC (2000) 35B35, 35B45, 35K55, 92C17We consider the chemotaxis systemunder homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ R n . The chemotactic sensitivity function is assumed to generalize the prototypeIt is proved that no chemotactic collapse occurs in the sense that for any choice of nonnegative initial data u(·, 0) ∈ C 0 (Ω) and v(·, 0) ∈ W 1,r (Ω) (with some r > n), the corresponding initial-boundary value problem possesses a unique global solution that is uniformly bounded.

Global solutions in a fully parabolic chemotaxis system with singular sensitivity

“…Then known results state that if n = 2, >0, 0 and (1+ ) , then all reasonable solutions of (1) are global in time [6,Theorem 3 and Remark 4]. Moreover, in the case when both and are positive, the result in [7] goes beyond this and states that for arbitrary n 1 and >0, all solutions will be global and even bounded. Hence, either of the sets of conditions n 1, >0, >0, >0, and n = 2, >0, = 0, 1 is sufficient to prevent a chemotactic collapse in the sense of finite-time blow-up such as it may occur in the so-called minimal Keller-Segel model formally obtained when =−1 [8].…”

“…Therefore, we may invoke Theorem 2.2 in [7] to obtain global existence of classical solutions to (2):…”

On a class of Keller–Segel chemotaxis systems with cross-diffusion

“…However, one may argue that the chemotactic flux ∇v/v is too strong when v is close to zero. For this reason, sensitivity functions of the type φ 2 (v) = (v + c) −β , which saturate for very small signal concentrations v, have been chosen in the literature, where the cases β = 1 [41] and 0 < β < 1 [45] have been investigated with no cross diffusion in the v-equation.…”

Global existence and boundedness in a reaction–diffusion–taxis system with three species

“…Proof We use semigroup arguments (see [13,16]) to get the L ∞ -bound of v 1 , v 2 . As pointed out in Theorem 3.4, by using the variation of constants formula, we have…”