The Keller-Segel system is used to model chemotactic movement in biology . The mathematical study of the system has attracted great interest in recent years . In this paper, we consider the Neumann initial-boundary value problem for the chemotaxis system with singular sensitivity
in a bounded domain with smooth boundary, where and are diffusion coefficients of cell density and chemical stimulus, respectively. The Keller-Segel systems were introduced to describe the aggregation of cellular slime molds, u represents the density of the cells and v represents the concentration of a chemical substance secreted by themselves. The chemical substance is an attractant, they sense a gradient of the chemical substances and move towards higher concentrations. The function is called a sensitivity function, and expresses the relation between the chemical concentration and the cells response, the symbol denotes differentiation with respects to the outward normal on and the initial data and are sufficiently smooth functions. For system (1.2) with , the global existence and boundedness of classical solution is proved under the assumption see   . Lankeit  extended the range of in the two-dimensional case. Also the generalized solutions with large are constructed in   . More results on the related model with general sensitivity can be found in    . In this present paper, we prove the existence of global bounded classical solutions for (1.2) without assumptions on the space dimensions or the smallness assumption on the initial data in the case . Our main result reads as follows.
Lemma 1.2. (Poincaré inequality)  Let be a bounded domain, then there is exists a constant , such that for all
Theorem 1.1. Let be a bounded domain with smooth boundary and let the parameters and satisfy . Then for any nonnegative function and positive function , the problem (1.2) has a unique global classical solution which is bounded in .
3. Proof of Theorem 1.1
As a preparation to the proof, we first state one result concerning local-in-time classical solution of the problem (1.2), which can be proved by standard contraction mapping arguments and parabolic regularity theory (see ( , Proposition 2.2) and the references therein).
Lemma 3.1. Suppose that is a nonnegative function and that is a positive function in . Then there exist the maximal existence time and a uniquely determined pair of positive functions
that solves (1.2) classically in . In additions, for the second component v of the solution one can find such that
Furthermore, if , Then
The following lemma is a generalization of the maximum principle, which plays a major role in the proof of the main result.
Lemma 3.2. Suppose that is a bounded domain with smooth boundary, is a positive constant and is a positive continuous function satisfying . Let , in . If
then z is bounded in .
By simple calculations we can show that y is the solution of
and it is bounded in by our supposition. Therefore, by the comparison principle, we see that z is bounded in .
We are now in the position to prove global boundedness of solutions for (1.2).
4. Proof of the Main Result
Motivated by , let us introduce the function . by using this assumption , we shall transform the system (1.2) into
and then, by the comparison principle we will obtain
where . Hence, the second equation in (1.5) implies that
If , we deduce that
by using the maximum principle. For , Let . Through direct computation we establish that
We shall also use the maximum principle for the second time, it follows that
which implies that
Along with this, the Lemma 3.1 guarantees that is global in time. Then the integral
we apply the Lemma 3.2 to (1.6), it follows that v is bounded in , and hence is bounded in with smooth boundary, , Thus we complete the proof.
5. Conclusion and Remarks
In the paper, we presented that the Neumann initial-boundary value problem for the chemotaxis system with singular sensitivity in problem (0.1) is bounded in with smooth boundary, . Then we established that the problem (1.2) has a unique global classical solution which is bounded in . And we showed that is a bounded domain with smooth boundary, is a positive constant and a is a positive continuous function.
The authors would like to thank the anonymous referees for their helpful comments. Referees’ comments led to improvements of this paper.
 Bellomo, N., Bellouquid, A., Tao, Y. and Winkler, M. (2015) Toward a Mathematical Theory of Keller-Segel Models of Pattern Formation in Biological Tissues. Mathematical Models and Methods in Applied Sciences, 25, 1663-1763.
 Fujie, K. (2015) Boundedness in a Fully Parabolic Chemotaxis System with Singular Sensitivity. Journal of Mathematical Analysis and Applications, 424, 675-684.
 Lankeit, J. (2016) A New Approach toward Boundedness in a Two-Dimensional Parabolic Chemotaxis System with Singular Sensitivity. Mathematical Methods in the Applied Sciences, 39, 394-404.
 Stinner, C. and Winkler, M. (2011) Global Weak Solutions in a Chemotaxis System with Large Singular Sensitivity. Nonlinear Analysis: Real World Applications, 12, 3727-3740.
 Lankeit, J. and Winkler, M. (2017) A Generalized Solution Concept for the Keller-Segel System with Logarithmic Sensitivity: Global Solvability for Large Nonradial Data. Nonlinear Differential Equations and Applications NoDEA, 24, Article number: 49.
 Mizukami, M. and Yokota, T. (2017) A Unified Method for Boundedness in Fully Parabolic Chemotaxis Systems with Signal-Dependent Sensitivity. Mathematische Nachrichten, 290, 2648-2660.
 Fujie, K. and Senba, T. (2016) Global Existence and Boundedness of Radial Solutions to a Two Dimensional Fully Parabolic Chemotaxis System with General Sensitivity. Nonlinearity, 29, 2417-2450.
 Fujie, K. and Senba, T. (2018) A Sufficient Condition of Sensitivity Functions for Boundedness of Solutions to a Parabolic-Parabolic Chemotaxis System. Nonlinearity, 31, 1639-1672.
 Yang, Y., Chen, H. and Liu, W. (2001) On Existence of Global Solutions and Blow-Up to a System of the Reaction-Diffusion Equations Modelling Chemotaxis. SIAM Journal on Mathematical Analysis, 33, 763-785.