Global boundedness in higher dimensions for a fully parabolic chemotaxis system with singular sensitivity

In this paper we study the global boundedness of solutions to the fully parabolic chemotaxis system with singular sensitivity: \begin{document}$u_t=\Delta u-\chi\nabla·(\frac{u}{v}\nabla v)$\end{document} , \begin{document}$v_t=k\Delta v-v+u$\end{document} , subject to homogeneous Neumann boundary conditions in a bounded and smooth domain \begin{document}$\Omega\subset\mathbb{R}^{n}$\end{document} ( \begin{document}$n\ge 2$\end{document} ), where \begin{document}$\chi, \, k>0$\end{document} . It is shown that the solution is globally bounded provided \begin{document}$0 . This result removes the additional restriction of \begin{document}$n \le 8 $\end{document} in Zhao, Zheng [ 15 ] for the global boundedness of solutions.

1. The main result and its proof. This paper is concerned with the global boundedness of solutions to the following chemotaxis system with singular sensitivity in a bounded and smooth domain Ω ⊂ R n (n ≥ 2), where χ, k > 0 are constants, ν represents the outer normal vector on ∂Ω, and the initial data (u 0 , v 0 ) ∈ C(Ω) × W 1,∞ (Ω) satisfies u 0 ≥ 0 with u 0 ≡ 0 and v 0 > 0 onΩ. The problem (1) with the sensitivity function χ/v was first proposed in [9] due to the Weber-Fechner law of stimulus perception in the process of chemotactic response, where the sensitivity function χ/v is singular near v = 0 and reflects an inhibition of chemotactic migration at high signal concentrations.
Some rich understanding to this model with k = 1 is available. • For the parabolic-elliptic analogue of (1) with the second equation replaced by 0 = ∆v − v + u, it was shown in [11] that the radial solutions are globally bounded with χ > 0, n = 2, or χ < 2/(n − 2), n ≥ 3, and there exist radial blow-up solutions if χ > 2n/(n − 2), n ≥ 3. Without the requirement for symmetry, Biler [2] obtained the global existence of solutions for χ ≤ 1, n = 2, or χ < 2/n, n ≥ 3. Furthermore, the global boundedness of solutions was established with χ < 2/n, n ≥ 2 [7]. Lately, Fujie and Senba [4] proved that the solutions are globally bounded for arbitrary χ > 0 under the 2-dimensional case, for which it is important to determine the behavior of solutions for large χ > 0.
As to the problem (1) with the second equation replaced by τ v t = ∆v − v + u in a ball Ω ⊂ R 2 , it was shown in [5] that if τ ∈ (0, 1] is properly small, then the classical radially symmetric solutions are globally bounded for all χ > 0. But whether the smallness of τ > 0 and the radial symmetry of solutions can be removed has been left as an open problem. Refer to e.g. [4,5,7,8,14] for general singular sensitivity, and [1,6] for the chemotaxis model with singular sensitivity and logistic source.
Recently, Zhao and Zheng [15] showed for general k > 0 that if then (1) has a global classical solution. Moreover, the solution is globally bounded under n ≤ 8. However, the global boundedness of solutions for n > 8 is not discussed therein. So, it is natural to ask whether the solution is globally bounded when n > 8. In this paper, we give a positive answer to this question. As the main result, it can be formulated as follows.
Theorem 1. Assume that χ satisfies (2) with n ≥ 2 and k > 0. Then the global classical solution of (1) is bounded in the sense that there exists C > 0 such that To prove this main result, we first give several well-understood facts described as the coming three propositions. Throughout this paper, let (u, v) be the global classical solution of (1).
for each t > 0.
(ii) Assume that 1 2 + n for all t > 0 with some C > 0.
with some c > 0, then there exists C > 0 such that To obtain the global boundedness of solutions to (1), as performed in [3], Proposition 2 (i) and Proposition 3 with the proper choice of r allow one to elevate the L 1 -conservation of u to the boundedness of u(·, t) L q (Ω) in (0, ∞) for some q > n/2 by an iterative argument. In [15], r is directly taken as satisfying r ∈ (r − (p), r + (p)) spontaneously. But such complicated choice for r makes it possible to carry out the iteration only under the case n ≤ 8. To remove the restriction of n ≤ 8, we alternatively pick r proportional to p − 1. Moreover, the following lemma guarantees that such selected r with a suitable ratio to p − 1 does lie in (r − (p), r + (p)), and hence plays a key role in the proof of Theorem 1 although it is elementary.
Again by (2), we have Therefore, the continuity results in that there exists ε 2 ∈ (0, 1) small enough such that for all 0 < ε ≤ ε 2 , Take ε * = min{ε 1 , ε 2 , k χ(χ−1+k)+ − n 2 }. By (3), we know that the quadratic equation admits two positive roots with the smaller one, written by a * , satisfying With the constants ε * and a * determined above, we have for any p ∈ (1, n 2 + ε * ) that The proof is complete. Now we are in a position to prove Theorem 1.
Proof of Theorem 1. The argument, similar to that in [3], can be performed via two steps.
Step 1. We prove that there is q > n/2 such that u(·, t) L q (Ω) ≤ C for all t > 0 with some C > 0. Let ε * > 0 and a * ∈ (0, 1) be the constants as in Lemma 2. To this end, we first claim the following facts.