Existence of a Nontrivial Steady-State Solution to a Parabolic-Parabolic Chemotaxis System with Singular Sensitivity

Chemotaxis is the biased movement of cells toward the concentration gradient of a chemical. It plays a critical role in a wide range of biological phenomena. For example, cells migrate toward resources of food and stay away fromharmful substances. The first mathematical model of chemotaxis was introduced by Patlak in [1] and Keller and Segel in [2]. There are numerous works dedicated to the analysis of chemotaxis models. For example, Othmer and Stevens in [3] modeled myxobacteria as individual random walkers and proposed a microscopic model based on a velocity jump process. By taking the parabolic limit of the microscopic model, they obtain the macroscopic chemotaxis model, which is the wellknown Keller-Segel system


Introduction
Chemotaxis is the biased movement of cells toward the concentration gradient of a chemical.It plays a critical role in a wide range of biological phenomena.For example, cells migrate toward resources of food and stay away from harmful substances.The first mathematical model of chemotaxis was introduced by Patlak in [1] and Keller and Segel in [2].There are numerous works dedicated to the analysis of chemotaxis models.For example, Othmer and Stevens in [3] modeled myxobacteria as individual random walkers and proposed a microscopic model based on a velocity jump process.By taking the parabolic limit of the microscopic model, they obtain the macroscopic chemotaxis model, which is the wellknown Keller-Segel system   = ∇ ⋅ (∇) − ∇ ⋅ (∇Θ (V)) +  (, V) ,  ∈ Ω,  > 0, V  = ΔV +  (, V) ,  ∈ Ω,  > 0, (1) where Ω ⊂ R  is a bounded connected domain with a smooth boundary Ω.The function  = (,) denotes the cell density and V = V(, ) represents the chemical concentration, for example, oxygen.The constant  is called the chemotactic coefficient, and the sign of  corresponds to chemoattraction if  > 0 and chemorepulsion if  < 0. Parameters  and  are the diffusion coefficients of the cells and the chemical, respectively.The function () represents the kinetic function describing production and degradation of cells, and Θ(V) is commonly referred to as the chemotactic potential function.Function (, V) describes the production and degradation of the chemical.
The existence of global solutions, blow-up, and traveling wave solutions to the chemotactic system (1) were extensively studied during the past four decades (see, e.g., [4][5][6][7][8][9][10][11][12] and references therein).The authors studied the roles of growth, death and random in promoting population persistence through band popation in [4], and the work was related to the significance of cell motility and chemotaxis in microbial ecology.
X.F.Wang addressed the trivial and nontrivial steady states with small  and  of the quasi-linear system (1) in [13].In the absence of population dynamics (i.e., () ≡ 0), there have been extensive studies.The main feature of solutions to the Keller-Segel model is the possibility of blowup in finite time in [9,14,15].Moreover, recent results in [6,10,11,16,17] proved the global existence of solutions under some conditions.Moreover, the global existence, asymptotic behavior, and steady states of classical solutions were studied in [18] for the one-dimensional case.
The purpose of this paper is to study the existence of the nontrivial steady-state solution to a parabolic-parabolic coupled system arising from chemotaxis with singular sensitivity.We consider the following system with the initial-boundary value conditions where Ω = [0, 1] and "" = /.The function  = (, ) denotes the density of the cells and V = V(, ) denotes the concentration of the chemical.The parameters , , , and  are all positive constants, with  being the diffusion coefficient of cells and  the chemotactic sensitivity coefficient as above.
In this paper, we establish the existence of nontrivial steady-state solutions to the parabolic-parabolic coupled chemotactic system (2)-(3) with singular sensitivity.Here we assume that the function (V) satisfies Obviously,  ≡ 0 and V ≡ 1 are the trivial solutions to system (4).
Our main results in this paper are presented in the following theorem.
The rest of this paper is organized as follows.In Section 2, we give some preliminary lemmas.In Section 3, we complete the proof of Theorem 1.

Lemmas
To prove Theorem 1, we need to establish boundary estimates of solutions to system (4).Since the state variables represent densities, we only consider nonnegative solutions, that is, () ≥ 0 and V() ≥ 0 on [0, 1].Firstly, we state one result concerning the estimate of a solution (, V).
(ii) Integrating the first equation of (4) yields Lemma 3. Let V() solve system (2) with the initial-boundary value condition of (3).Then we have the lower-bound estimate and  is defined in (11) below.
Proof.From the second equation in (4), we obtain Integrating the above equation once, we have Integrating by part and applying the condition V  (0) = 0, we rearrange the above equation From the boundary value condition of From the definition of the function , we easily get Applying the implicit function theorem, we have the estimate (7).
Lemma 4.There are two positive constants  1 = / and  2 = / such that a nonnegative steady state of system ( 4) satisfies Proof.By the steady-state system (4), we know V  > 0, and for every  ∈ (0, 1), we have Applying the condition V  (1) + V(1) = 1, we obtain the following equation Similarly, we also get On the other hand, integrating once the first equation of (4), we get which is equal to Then, applying the boundary condition (  −(ln V)  )| =1 = 0 and rewriting (18), we have the following estimate Next, we review the condition (V  + V)| =1 = 1, which implies that V  (1) ≤ 1.Combined with the condition of 1/V ≤  in Lemma 3, inequality (19) can be rewritten as Integrating (20) once, we have Here  1 = / and  2 = /.Hence, we obtain By Gronwall's inequality, we get Substituting the definition of () into inequality (25), we obtain Combining this result with   () ≥ 0 in Lemma 2 (), we obtain estimate (13).
Proof.If not, there exists some  0 ∈ [, +∞) such that () > N. According to Lemma 2 and the definition of , we have From ( 13), we can easily get However, V( 0 ) < .
In fact, if not we have which contradicts the fact that V() ≤ 1.
Next, we consider 0 = ∫ (i) For each () ∈  0 [0, 1], system (27) has a unique solution Proof.For any given () ∈  0 [0, 1], it is obvious that V() = 1 and V() = 0 are a pair of sup-sub-solutions.According to the standard comparison theorem, we easily obtain the estimate of the solution to system (27).If V 1 () and V 2 () are solutions to system (27), then we have The difference between the first and second equation of ( 34) is with the notation  V fl /V.By the maximum principle, we obtain V 1 ≡ V 2 .Moreover, A is uniquely defined and 0 ≤ A ≤ 1.
Next we give the proof of (ii) and (iii) in Lemma 6, respectively.
(i) Assume that there exists a sequence {  ()} ∞ =1 , which converges to ().Then we only point out that there exists a subsequence A   of A  such that A   → () on  0 [0, 1].By regularity theory, we know that () is the solution to system (34) and () ̸ = A, which contradicts the uniqueness of A.(ii) We can directly prove (iii) from the maximum principle.
Define () to be the unique solution to the system −  + ( −  (1))  = 0,  ∈ (0, 1) ,       =0 = 0, where  is a constant satisfying  >  (1).For any given () ∈  0 [0, 1], we define  to be the unique solution to the following system Next, we introduce some notation: It is easy to see that the operator  :  →  is compact and linear.Define the operator L as the map Similar to the proof of Lemma 5(i), we can show that the operator L is continuous and bounded from  to .We also have Hence,  ∘ L is compact operator from  to  with the norm ‖∘L‖  = (‖‖  ).Moreover, the operator  is also compact from  to  and ‖( ∘ )()‖  = (‖‖  ).
From the above lemmas, we can show the proof of Theorem 1.
(2) Assume that there is a constant  such that  > (1).
We firstly prove  ⊆  × .Otherwise, there exists a point (49) However, ũ ∈  which means that ũ is equal to zero at some point.We can directly have ũ ≡ 0 by the maximum principle.That is, (0, ) is a bifurcation point of (40).