Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with the sensitivity $v^{-1}S(u)$

This paper is concerned with global existence and boundedness of classical solutions to the quasilinear fully parabolic 
Keller-Segel system 
$u_t 
 = \nabla \cdot(D(u)\nabla u) 
 -\nabla \cdot (v^{-1}S(u)\nabla v)$, 
$v_t= \Delta v-v+u$. 
In [7,4], global existence and boundedness were established in the system without $v^{-1}$. 
In this paper the signal-dependent sensitivity $v^{-1}$ is taken into account via the Weber-Fechner law. 
A uniform-in-time estimate for $v$ obtained in [2] defeats the singularity of $v^{-1}$.


1.
Introduction. The Keller-Segel system, which was proposed by Keller and Segel [5] in 1970, describes a motion of cellular slime molds with chemotaxis. The system has been widely studied (see e.g., Hillen and Painter [3]).
In the present paper we consider the following quasilinear fully parabolic Keller-Segel system: where Ω is a bounded domain in R N with smooth boundary, N ∈ N and ∂ ∂ν denotes differentiation with respect to the outward normal of ∂Ω. The initial data (u 0 , v 0 ) is assumed to be a pair of functions fulfilling u 0 ≥ 0, u 0 ∈ C 2 (Ω) and v 0 > 0, v 0 ∈ C 1 (Ω). (1.2) Moreover we suppose that D and S satisfy the following conditions: From a mathematical point of view it is important to study whether solutions remain bounded or blow up. As to the problem (1.1) without 1 v , i.e., in the case that the chemotaxis term in the first equation in (1.1) is replaced with −∇·(S(u)∇v), Tao and Winkler [7] proved boundedness of solutions, provided that D and S satisfy (1.3), (1.4), (1.5) and (1.6) and Ω is convex. Recently this convexity condition of Ω was removed in [4]. As to blow-up of solutions to the problem (1.1) without 1 v , Winkler [10,8] and Ciéslak and Stinner [1] established that the solutions blow up in finite time under the conditions that S(u) D(u) ≥ Ku 2 N +η for u > 1 with K > 0, η > 0 and that S(u) ≥ cu for some c > 0. Therefore the optimal exponent is known as 2 N . In the last decade, a growing literature has been concerned with signal-dependent sensitivity. The case that the chemotaxis term is −χ 0 ∇ · ( u v ∇v) was already proposed in the original model by Keller and Segel from a biological point of view such as the Weber-Fechner law. In [9,2] it has been shown that (1.1) with χ 0 > 0 small enough has a globally bounded solution, provided that the first equation has the linear diffusion ∆u, i.e., D(u) ≡ 1. However, to the best of our knowledge, no results are available for the system with both nonlinear diffusion and signal-dependent sensitivity. As opposed to the case without 1 v , we find that all solutions of (1.1) are global and bounded in the case D(u) ≡ 1 and S(u) ≡ χ 0 u with sufficiently small χ 0 > 0 in [9,2]. This means that the case α = 1 and sufficiently small K > 0 admits global existence and boundedness. As 1 > 2 N for N ≥ 3, this fact indicates that the constant 2 N is not optimal in the condition (1.6). The question of optimality of (1.6) remains an open problem.
The purpose of the present paper is to establish a globally bounded solution of the Keller-Segel system with not only the nonlinear diffusion ∇ · (D(u)∇u) but also the singular sensitivity function S(u) v . Our main result reads as follows.
The difficulty in the proof of Theorem 1.1 lies in the singularity of 1 v . In the present paper, a uniform-in-time lower bound for v ([2]) builds a "bridge" between the regular case ( [7,4]) and the singular case. We will consider approximate problems in Section 2 and prepare some estimates. Section 3 is devoted to discussing convergence of approximate solutions and completing the proof of Theorem 1.1.
furthermore, u ε has the following mass conservation: The following lemma is a cornerstone of this work, which was essentially established in [2, Lemma 2.2]. Mass conservation property plays a key role in the proof of the lemma. In view of the lemma we can ensure a uniform-in-time estimate for v ε .
where δ does not depend on ε and T .
As a preparation for the passage to the limit, we present three lemmas.
where C p and C 2q do not depend on ε and T .
Proof. Proceeding similarly as in [7, Lemma 3.3] and [4, Proposition 3.2], we define φ as Thus we can calculate Now in virtue of Lemma 2.2 we have the following independent-in-ε bound: where C ∞ and C ∞ do not depend on ε and T .
As to (2.4), using the representation formula for v ε and standard smoothing estimates, we see that with constants c > 0, η > 0 and θ > 1. Now we can choose θ > 1 large enough satisfying 3) ensures boundedness of the right-hand side of the above inequality which leads to the conclusion.
where C ∞ does not depend on ε and T .
Proof. We can calculate the first equation in (2.1) as From (2.2) we have the following upper estimates: By noting that u 0 ∈ C 2 (Ω), these estimates allow us to apply standard parabolic theory [6, Theorem V.7.2] and to complete the proof.
3. Proof of the main theorem. We start by showing that {u ε } and {v ε } satisfy the Cauchy condition.
We are now in a position to prove Theorem 1.1.
In the same fashion as before (3.8) implies As to the second equation in (1.1), we can similarly deduce the following identity: Thus we conclude that (u, v) is a weak solution of (1.1).