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].

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.

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This paper deals with the parabolic–elliptic Keller–Segel system with signal‐dependent chemotactic sensitivity function, {alignedrightutleft=Δu−∇⋅MathClass-open(uχMathClass-open(vMathClass-close)∇vMathClass-close),rightleftx∈Ω,t>0,rightright0left=Δv−v+u,rightleftx∈Ω,t>0,under homogeneous Neumann boundary conditions in a smooth bounded domain ΩMathClass-rel⊂double-struckRnMathClass-punc,1emnbspnMathClass-rel≥2, with initial data u0MathClass-rel∈C0(trueΩ̄) satisfying u0 ≥ 0 and MathClass-op∫Ωu0MathClass-rel>0. The chemotactic sensitivity function χ(v) is assumed to satisfy 0MathClass-rel<χ(v)MathClass-rel≤χ0vkMathClass-punc,1emquadkMathClass-rel≥1MathClass-punc,1emnbspχ0MathClass-rel>0MathClass-punc.The global existence of weak solutions in the special case χ(v)MathClass-rel=χ0v is shown by Biler (Adv. Math. Sci. Appl. 1999; 9:347–359). Uniform boundedness and blow‐up of radial solutions are studied by Nagai and Senba (Adv. Math. Sci. Appl. 1998; 8:145–156). However, the global existence and uniform boundedness of classical nonradial solutions are left as an open problem. This paper gives an answer to the problem. Namely, it is shown that the system possesses a unique global classical solution that is uniformly bounded if χ0MathClass-rel<2n1emnbsp(kMathClass-rel=1)MathClass-punc;1emnbspχ0MathClass-rel<2nMathClass-bin⋅kk(kMathClass-bin−1)kMathClass-bin−1γkMathClass-bin−11emnbsp(kMathClass-rel>1), where γ > 0 is a constant depending on Ω and u0. Copyright © 2014 John Wiley & Sons, Ltd.

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