Global Existence of a Chemotactic Movement with Singular Sensitivity by Two Stimuli

and In this study, we deal with the chemotaxis system with singular sensitivity by two stimuli under homogeneous Neumann boundary conditions in a bounded domain with smooth boundary. Under appropriate regularity assumptions on the initial data, we show that the system possesses global classical solution. Our results generalize and improve previously known ones.


Introduction
Chemotaxis is a well-known biological phenomenon describing the collective motion of cells or the evolution of density of bacteria driven by chemicals, including embryo development, skin wound healing, cancer invasion, and metastasis. e pioneering works of the chemotaxis model was introduced by Keller and Segel in [1], describing the aggregation of cellular slime mold toward a higher concentration of a chemical signal, which reads where Ω ⊂ R N (N ≥ 1) is a bounded domain with smooth boundary. e mathematical analysis of (1) and the variants thereof mainly concentrate on the boundedness and blowup of the solutions (refer to [2][3][4][5][6] and the references therein), that is, it is well-known that for all suitably regular initial data (u 0 , v 0 ), an associated Neumann initial boundary value problem, posed in a smooth N-dimensional domain Ω, Osaki and Yagi [4] proved that system (1) always possesses a global bounded classical solution in one-dimensional bounded domain, Nagai et al. [3] showed that system (1) also admits a global bounded classical solution in twodimensional bounded domain if ‖u 0 ‖ L 1 (Ω) is small, as N ≥ 3, Winkler [5] proved that for each q > (N/2), p > N, one can find ε 0 > 0, such that if the initial data (u 0 , v 0 ) fulfill ‖u 0 ‖ L q (Ω) < ε and ‖v 0 ‖ L p (Ω) < ε for some ε < ε 0 , then the solution is global in time and bounded; this is complemented by corresponding findings on the occurrence of finite-time blow-up of some solutions emanating from smooth but appropriately large initial data [2,6]. In the past few decades, system (1) has attracted extensive attentions. Keller and Segel [7] introduced a phenomenological model of the wave-like solution behavior without any type of cell kinetics, a prototypical version of which is given by where u represents the density of bacteria and v denotes the concentration of the nutrient. e second equation models consumption of the signal. In the first equation, the chemotactic sensitivity is determined according to the Weber-Fechner law, which says that the chemotactic sensitivity is proportional to the reciprocal of signal density. Winkler [8] proved that if initial data satisfy appropriate regularity assumptions, system (2) possesses at least one global generalized solution in two-dimensional bounded domains. Moreover, he took into account asymptotic behavior of solutions to system (2) and proved that v(·, t)⇀ ⇀ 0 in L ∞ (Ω) and v(·, t) ⟶ 0 in L p (Ω) as t ⟶ ∞ provided positive constants. When uv is replaced by g(u)v, g ∈ C 1 (R), and 0 ≤ g(u) ≤ u β , β ∈ (0, 1), χ ∈ (0, 1), and any sufficiently regular initial data, Lankeit and Viglialoro [9] showed that system (2) has a global classical solution. Moreover, if additionally m � ‖u 0 ‖ L 1 (Ω) is sufficiently small, then also their boundedness is achieved. When system (2) has a logistic source f(u), Lankeit and Lankeit [10] showed that system (2) possesses a global generalized solution for any Zhao and Zheng [11] proved that system (2) possesses a unique positive global classical solution provided k > 1 with N � 1 or k > 1 + (N/2) with N ≥ 2. When 1/v is replaced by 1/v α , α ≥ 1 and Δv turned into εΔv, Zhao et al. [12] obtained the global existence of classical solutions with N � 1. Moreover, for any global classical solution (u, v) to the case of N, it is shown that v converges to 0 in the L ∞ -norm as t ⟶ ∞ with decay rate established whenever ε ∈ (ε 0 , 1) with ε 0 � max 0, 1 − (χ/(α‖v 0 ‖ α− 1 L ∞ (Ω) )) . When u is replaced by f(u) and 0 < f(u) ≤ K(u + 1) α , Liu [13] showed that for any sufficiently smooth initial data, system (2) admits a global classical solution when either N � 1 and α < 2 or N ≥ 2 and α < 1 − (N/4). When Δu is replaced by Δu m (m ≥ 1), Lankeit [14] proved that if m > 1 + (N/4), system (2) admits a global classical solution or global locally bounded weak solution.
To the best of our knowledge, Dong et al. [26] first put forward the following chemotactic model with general rotational sensitivity caused by two stimuli: (7) admits at least one global generalized solution. roughout above analysis, compared with system (6), the theory of system (3) is so fragmentary. To the best of our knowledge, the global classical solution of model (3) in N ≥ 2 has never been touched. No matter biological relevance or mathematical meaning, we find it is worth addressing the basic solvability theory of the model (3). Inspired by the arguments in previous studies [8,13,14,26,27], we mainly investigate the global classical solution in a chemotactic movement with singular sensitivity by two stimuli. eorem 1 partially generalizes and improves previously known ones.
In this study, we use symbols C i and c i (i � 1, 2, . . .) as some generic positive constants which may vary in the context. For simplicity, u(x, t) is written as u, the integral e rest of this study is organized as follows. In Section 2, we summarize some useful lemmata in order to prove the main result. In Section 3, we give some fundamental estimates for the solution to system (3) and proof of eorem 1.

Preliminaries and Main Result
In this section, we give the main theorem and the local existence of the classical solution to (3) and also summarize some useful lemmata in order to prove the main result. Noting the singular chemotaxis term, we let en, we can rewrite (3) as Discrete Dynamics in Nature and Society At first, we give the main result of global existence of the classical solution to (3). (4) and en, for any choice of the initial data
In the sequel, we will consider system (9) to obtain the local boundedness of v, w and then come back to system (3) to prove the main theorem. Under the framework of fixed point theorem, we will prove the local existence of classical solution to system (3) in the following lemma. e proof is quite standard, and a more detailed display of a similar reasoning in a related circumstance can be found in [14].

Lemma 4 (See
where g ∈ L 1 loc (R) has the property that with some τ > 0 and b > 0. en,

Proof of Theorem 1
In this section, we establish some priori estimates for solutions to system (9); we first establish a bound for n 1 (x, t) and n 2 (x, t) in the one-dimensional case, which differs from that in the multidimensional settings. Discrete Dynamics in Nature and Society Proof.
Proof. By the Gagliardo-Nirenberg inequality, there exist p > 1 and q > 1, such that 6 Discrete Dynamics in Nature and Society for all t ∈ (0, T max ) with some c 5 , c 6 > 0 and as well as In accordance with Lemma 5, there exist some c 10 , c 11 > 0, such that for all t ∈ 0, T max .
(59) erefore, using the Young inequality, we derive for all t ∈ (0, T) with some constant C > 0 determined by T. us, the global existence is a consequence of (70) and the extensibility criterion provided by Lemma 1.

Data Availability
No data were used to support this study.

Conflicts of Interest
e author declares that there are no conflicts of interest.