Boundedness in a fully parabolic chemotaxis system with singular sensitivity

Abstract

Please cite this article in press as: K. Fujie, Boundedness in a fully parabolic chemotaxis system with singular sensitivity, AbstractThis paper deals with a fully parabolic chemotaxis system u t = Δu−χ∇·( u v ∇v), v t = Δv−v+u with singular sensitivity χ v (χ > 0) on a bounded domain Ω ⊂ R n , n ≥ 2. The main result solves the open problem of uniform-in-time boundedness of solutions for χ < 2 n , which was conjectured by Winkler [16].

Singular sensitivity in a Keller–Segel-fluid system

“…As in related situations (see [38,9], but also [22]), the key for establishing estimates significantly going beyond those of Lemma 2.2 lies in the following:…”

“…Its characteristics are shaped by the chemotactic effects becoming very strong at fast-varying small signal concentrations -and, indeed, for sufficiently large values of the coefficient χ (namely, χ > 2N N −2 ), radial solutions undergoing blow-up within finite time have been found in the corresponding parabolic-elliptic fluid-free setting ( [27]). On the other hand, for small values of χ and in the absence of fluid, classical solutions are known to exist globally in bounded domains of dimension two ( [2,28]) or arbitrary dimension ( [38]) and to be bounded ( [9]; for a generalization involving different diffusion coefficients see also [48]), where the precise condition χ < 2 N imposed on the chemotactic coefficient in these works is known to be not strict: For two-dimensional domains, global existence and boundedness of classical solutions were shown for any slightly larger χ in [21]. There is still a range of values for χ where it is unknown whether blow-up can occur.…”

“…Now, however, the source term is not n anymore, but n − u · ∇c, and a priori little is known about bounds for u or even ∇c, so that the reasoning of [9,Lemma 2.4] or [38,Lemma 2.4] cannot be used here. We will hence resort to a differential inequality for Ω c q for q ∈ [1, ∞) (Lemma 2.4) and estimate the production term arising therein by means of…”

“…Having derived (time-local) L ∞ (Ω)-bounds for the fluid velocity field in the Stokes-and Navier-Stokes settings in Sections 3.1 and 3.2, respectively, in Section 4 we can, mainly leaning on semigroup estimates, conclude global existence of solutions. The second regard in which the fluid poses an obstacle concerns the crucial uniform-in-time lower bound for c constituting the core of the boundedness proof in [9]. Again viewing the equation for c as inhomogeneous heat equation with source term n, in [9, Lemma 2.2] estimates of the heat kernel provide this uniform positive lower bound on c. These estimates rely on the nonnegativity of n, whereas no information about the sign of u · ∇c (and hence of n − u · ∇c) seems available.…”

Locally bounded global solutions to a chemotaxis consumption model with singular sensitivity and nonlinear diffusion

“…(For this system, global solutions are known to exist if χ is sufficiently small, where the precise condition depends on the dimension as well as on whether classical ( [17,40,4]) or weak solutions ( [34,40]) are considered and on radial symmetry of initial data ( [4,28]); but for large χ also blow-up may occur in the corresponding parabolic-elliptic system ( [28]). ) The proof of boundedness of solutions for χ < 2 N in [6] even relies on the second equation actually ensuring a positive pointwise lower bound for v. In (1), we cannot hope for such a convenient bound and thus have to deal with the influence of the actual singularity in the sensitivity function.…”