EXISTENCE AND BOUNDEDNESS OF SOLUTIONS FOR A KELLER-SEGEL SYSTEM WITH GRADIENT DEPENDENT CHEMOTACTIC SENSITIVITY

. We consider the Keller-Segel system with gradient dependent chemo-tactic sensitivity


Introduction
In this article, we consider the chemotaxis system with gradient dependent chemotactic sensitivity u t = ∆u − ∇ • (u|∇v| p−2 ∇v), x ∈ Ω, t > 0, where Ω ⊂ R n (n ≥ 2) is a bounded domain with smooth boundary and 1 < p < n/(n − 1).Keller and Segel [9] introduced a mathematical model to describe chemotactic aggregation of cellular slime molds.The classical Keller-Segel system is where u denotes the cell density and v describes the concentration of the chemical signal secreted by cells.This parabolic-parabolic Keller-Segel system has been studied extensively in literature, see the review paper [2,6,7] for details.Here we point out that the authors in [11] proved that (1.2) has global bounded solutions under the condition Ω u 0 (x) < 4π in R 2 or under the condition Ω u 0 (x) < 8π for radial solutions on a disk.Winkler [20] proved that finite-time blow-up occurs for radially symmetric initial data when Ω u 0 is arbitrary prescribed number.The chemotactic sensitivity can depend nonlinearly on the cell density.Some authors studied the system u t = ∇(D(u)∇u) − ∇(S(u)∇v), in the past decades.Horstmann and Winkler [8] determined the critical blow-up exponent for (1.3), where D(u) = 1 and the chemotactic sensitivity equals some nonlinear function of the particle density.In [18], it is proved that if S(u)/D(u) grows faster than u 2/n as u → ∞ and D(u) satisfies some technical conditions, then there exist solutions that blow up in either finite or infinite time.In [14], Tao and Winkler showed that if S(u)/D(u) ≤ cu α with α < 2/n and D(u) satisfies algebraic upper and lower growth, then the classical solutions to (1.3) are uniformly bounded.By the Weber-Fechner law, the classical Keller-Segel system has been modified to the Keller-Segel system with a singular sensitivity Winkler [19] proved that if 0 < χ < 2/n, (1.4) has a global-in-time classical solution.Furthermore, relaxing the solution concept, the global existence of weak solutions is established whenever 0 < χ < (n + 2)/(3n − 4).In [13], Stinner and Winkler introduced a generalized solution concept, and then proved that such generalized solution for any χ > 0. In [10], the authors introduced another generalized solution concept, which exists for the some range of χ.
Recently, Bellomo and Winkler posed a model where the chemotactic sensitivity depends on ∇v.In [3] the authors deduced the existence of a unique radial classical solution to the system where M = 1 |Ω| Ω u 0 (x)dx, n ≥ 2 and χ < 1.In [4], it is showed that for some T > 0, (1.5) possesses a uniquely determined classical solution blowing up at time T .[22] concerns the null controllability of a control system governed by coupled degenerate parabolic equations with lower order terms.
In this article, we study the global existence and boundedness of (1.1), the parabolic-parabolic version of (1.6).Now we state the main results of this article.We assume that the initial data u 0 and v 0 satisfy (1.7) Our main results read as follows. Theorem The rest of this article is organized as follows.In Section 2, we introduce the conception of the weak solution.Section 3 is devoted to showing the existence of the weak solution.Finally, we give the proof of the boundedness in Section 4.

A weak solution concept and approximate problems
Let us firstly introduce a natural concept of weak solutions to (1.1).Definition 2.1.Assume that u 0 and v 0 satisfy (1.7).For all T > 0, a pair (u, v) of functions with u ≥ 0 a.e. in Ω × (0, T ) and v ≥ 0 a.e. in Ω × (0, T ), ( and will be called a weak solution of (1.4) if u has the mass conservation property and the following two identities and We intend to construct a solution of (1.1) as the limit of a sequence of solutions to the approximate problems where ε ∈ (0, 1) is a positive parameter.We construct a suitable fixed point framework to prove the existence of classical solutions to (2.7).
Lemma 2.2.Assume that (1.7) holds, and let ε ∈ (0, 1).Then there exists as well as Proof.Let us prove the existence of solutions by a standard contraction argument referring to [8].For T ∈ (0, 1), we define a Banach space Consider the closed set We claim that for T sufficiently small, the map where we have used the estimate (2.11) From (2.10) and (2.11), it follows that ΨS ⊂ S if we choose T small.For all (u ε , v ε ), (ū ε , vε ) ∈ S, we have so Ψ is shown to be a contraction if T is sufficiently small.By the Banach's fixed point theorem, we obtain that the existence of (u, v) ∈ X satisfies (u, v) = Ψ(u, v).Properties (2.8) and (2.9) follow by integrating the PDEs in (2.7) in space.

Existence of the weak solutions
The construction of a global weak solution is based on a limit procedure of solutions to suitably regularized problems.The Aubin-Lions lemma is very helpful.We collect some ε-independent a priori estimates of the solutions to (2.7).For the second equation in (2.7), using the parabolic theory, we obtain the following lemma.
Proof.For convenience, we give the proof.
(ii) Applying ∇ to both sides in (3.3) and invoking corresponding smoothing properties involving gradient [16], we similarly find that with a certain C > 0. So we conclude using the similar method of proving (i).
Next, we prove the almost everywhere convergence of u ε k by referring to the method in [21].Lemma 3.3.Let 1 < p < n/(n − 1).For all T > 0, there exists C > 0 such that for any ε ∈ (0, 1), we have Proof.We multiply the first equation in (2.7) by 1 uε+1 , and integrate by parts to obtain By the Cauchy-Schwarz inequality, we obtain Then, we have By integrating with respect to time we obtain where m := Ω u 0 .From 2(p − 1) < n/(n − 1), we obtain (3.8) by Lemma 3.1.
Lemma 3.4.Let 1 < p < n/(n − 1).For all T > 0, there exists C > 0 such that for any ε ∈ (0, 1), Proof.Testing the first equation in (2.7) by ψ uε+1 for fixed t > 0 and arbitrary ψ ∈ C ∞ ( Ω), we obtain By the Cauchy-Schwarz inequality and Young's inequality, we have Since in view of the fact that for any such ψ, this entails After an integration with respect to time, by Lemmas 3.1 and 3.3, this implies (3.9).

Boundedness
In this section, our goal is to prove Theorem 1.2.Firstly, by means of a Moser-Alikakos iteration, we can achieve the following boundedness results.Lemma 4.1.Let 1 < p < n/(n − 1).For all t > 0, there exists C > 0 such that for any ε ∈ (0, 1), Proof.We multiply the first equation in (2.7) by u q−1 ε (for q > 1), and integrate by parts to find that By the Cauchy-Schwarz inequality, we have We can find a positive constant µ satisfying 2(p−1) < µ < n/(n−1).Using Lemma 3.1 and Hölder's inequality, we have By the Gagliardo-Nirenberg inequality, we can find a positive constant C > 0 such that where Since 1 < p < n/(n − 1), we have a ∈ (0, 1).We apply inequality (4.3) to (4.2) and use Young's inequality to obtain By the Poincaré-Wirtinger inequality, we obtain By the maximum principle, we have Using the Moser-Alikakos iteration [1] and assuming, without loss of generality, that δ k ≥ 1, we have Finally by taking the 1/2 k power of both sides of (4.4) and by passing to the limit as k → ∞ we obtain sup Next, to obtain the limit function u, we need a regularity estimate for ∂ t u ε .
Finally, we give the proof of Theorem 1.2 by referring to the method in [17].