Global boundedness in the higher-dimensional chemotaxis system with indirect signal production and rotational flux
Introduction
Chemotaxis system. Chemotaxis is a well-known biological phenomenon that describes the evolution of density of bacteria driven by chemical stimulus. Mathematically, the following chemotaxis system describing such a phenomenon has been widely studied, which consists of the dynamics of cell density and the concentration of chemical attractant substance , where the proliferation term may represent a logistic-type source. The classical Keller–Segel system (i.e. in (1)) was proposed by Keller and Segel in 1970s [1], [2] and have been extensively studied. For more general chemotaxis system with or measure initial data, we refer to [3], [4] and references therein.
The chemotactic signal generation undergoes intermediate instead of direct in realistic biological processes [5], [6], [7]. In particular, the study of the following chemotaxis system with indirect signal production is initially inspired by the chemotaxis model of tumor invasion. When in (2), Fujie and Senba [8] established global existence and boundedness of solutions when either , or and , while for and , they established the existence of blowup solutions in [9]. When and with , Zhang et al. [10] proved that the global classical solution exists and is bounded in a smooth bounded domain for .
Chemotaxis with rotational flux. More experimental results and corresponding modeling approaches find that chemotactic movements of cells need not necessarily be directed along the gradient of the chemical substance, but can rather have rotational components. Accordingly, the chemotactic sensitivity can be represented by a tensor [11], [12], and this tensor need not be symmetric due to external influences [13]. For instance, it was showed that bacteria are subject to a net rotational force to their right when they swim close to a surface in [14], [15]. In contract with scalar-valued chemotactic sensitivity, the rotational chemotactic sensitivity case is more difficult due to the fact that chemotaxis system with such rotational fluxes lose some energy structure. Li et al. [16] first considered the following chemotaxis system with rational sensitivity under no-flux boundary conditions in a smooth bounded planar convex domain, where and are given parameter functions which are assumed to satisfy for some nondecreasing functions and on , respectively. In particular, the existence of globally defined bounded classical solutions were proved for smallness condition on . Winkler [17] further established the existence of global generalized solution for general sufficiently regular nonnegative initial data. For more recent chemotaxis system coupled with fluid, we refer to [18], [19] and references therein.
Main results. Motivated by above works, we will investigate the interaction of the indirect signal production, logistic source and rotational sensitivity in the chemotaxis system of the form in a smooth bounded domain with boundary conditions and the initial conditions We suppose that is a smooth function representing a logistic-type source and satisfies with , , , and the rotational sensitivity function satisfies for all , with constant . Our main results read as follows.
Theorem 1.1 Assume that is a bounded domain with smooth boundary and that satisfies (7). For all and , if satisfies (6) with , then for any nonnegative , and fulfilling the inclusions , and , there exists a unique global nonnegative solution to system (3)–(5). Moreover, the solution is bounded in .
Remark 1.1 Theorem 1.1 implies that the rotational flux in indirect signal production mechanism remain the regularity of the system. This is the main novelty of the paper.
The rest of this paper is organized as follows. In Section 2, we show the global boundedness of classical solution for the special case on . For the general matrix case, we will establish the global bounded classical solution by an approximate procedure in Section 3.
Section snippets
Global classical solutions for special case: on
In this section, we shall consider the special case that on . The nonlinear boundary condition for in (4) therefore reduces to the homogeneous Neumann boundary condition, i.e., (4) becomes
Global classical solution for general
In this section, we will devote ourselves to the case of general rotational sensitivity and establish the global existence theory of system (3)–(5).
Let be a family of standard cut-off functions fulfilling in for all and in pointwisely as . In order to reach our goal, we define for and consider the following approximating model
Acknowledgments
The authors are deeply grateful to Professor Z. Xiang for suggesting this problem and his key advice, and to the anonymous reviewers for their fruitful comments. This work is supported by the Applied Fundamental Research Program of Sichuan Province, China (no. 2020YJ0264).
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