Analysis of traveling fronts for chemotaxis model with the nonlinear degenerate viscosity

: In this paper, we are interested in chemotaxis model with nonlinear degenerate viscosity under the assumptions of β = 0 (without the e ﬀ ect of growth rate) and u + = 0. We need the weighted function deﬁned in Remark 1 to handle the singularity problem. The higher-order terms of this paper are signiﬁcant due to the nonlinear degenerate viscosity. Therefore, the following higher-order estimate is introduced to handle the energy estimate:

When m 1, the chemotaxis system (1.1) represents the reinforced movement of cells (or bacteria) in porous media, where u, c and β > 0 are the population density of cells, concentration of chemical signals (e.g., nutrients), and growth rate, respectively.Moreover, the diffusion rate of cells and the chemotactic coefficient are denoted by D > 0 and χ, respectively.The chemotaxis is said to be attractive if χ > 0 and repulsive if χ < 0. The logarithmic sensitivity ln c was derived from Weber-Fechner law [13] and has been verified by the experimental data [11].The above PDE-ODE system is the special case of the following Keller-Segel model with porous media type diffusion: where f (u, c) is a function characterizing the chemical growth and degradation defined as f (u, c) = βc − ug(c).This system describes the chemotactic dynamics, where cells move up the chemical concentration gradient and consume (or degrade) the chemical along the path.As stated in [26], the function g(c) is called the consumption rate function in the form constant rate, p = 0, sublinear rate, 0 < p < 1, linear rate, p = 1, superlinear rate, p > 1.
The problems of chemotaxis model in porous media are extensively studied for both the experiments and mathematical modeling.The experiments of bacterial chemotaxis in porous media were investigated in [21,25], and the nonlinear diffusion to a chemotaxis model in order to avoid overcrowding was instroduced in [1,8].Tao and Winkler [24] established the global existence and boundedness of solutions to a chemotaxis model of self-aggregation with arbitrary porous medium diffusion.However, few results are available to the chemotaxis model (1.1) except for the existence of compactly supported traveling waves in [2].
When m = 1, the system (1.1) is exactly the chemotaxis model proposed in [22] to describe the reinforced random walks.There are many other interesting analytical works with reinforced random walks.Othmer and Stevens [22] studied the model from random walk and presented the numerical simulations of the formation of spikes and blowup.The analytic results that support some numerical results in [22] were established in [23].The global existence and blowup of classical solutions on a bounded domain with no-flux boundary conditions were studied in [29,30].Moreover, the further study of global existence of smooth solutions to system (1.1) was investigated by Li et al. [16].Zhang and Zhu [31] presented the weak solutions to system (1.1) with the Robin boundary condition.Other references for global dynamics including well-posedness and large time behaviors of solutions in the whole space were presented in [3,14,18,28].The spike solution and blowup solution, traveling wave is another biological pattern observed in chemotaxis [13].The existence of traveling fronts to (1.1) was firstly established in [27].The stability problem of such a traveling front in the case of u + > 0 was obtained in [17].Moreover, when u + = 0, the energy estimate has the singular term, which is extremely difficult to overcome.This singular term was presented in [10] by employing it as the weighted function in the energy estimate.Recently, the half-space problem of (1.1) under the nonzero flux boundary condition was considered in [15].The authors showed that the system still admits traveling wave profiles on the half-space by introducing a wave selection mechanism.For other related work on traveling waves of chemotaxis models and Burger's equations, we refer the readers to these references [4,5,7,9,26].
By ignoring the effect of growth rate (β = 0), then one can derive the chemotaxis model for m > 0 and the initial data The major problem of this paper is concerned with the nonlinear diffusion and singularity problem.Under small perturbations and large wave amplitude, we prove the existence and stability of traveling waves to system (1.2) with m > 0 and u + = 0.The logarithmic singularity for the first equation of (1.2) is very difficult to study.Therefore, to handle these barriers, we employ the following Cole-Hopf transformation as in [10,17]: which presents the chemotaxis system as follows: and the initial data (u, v)(x, 0) = (u 0 , v 0 )(x) → (u ± , v ± ) as x → ±∞. (1.6) We organize this paper as follows: In Section 2, we present the theorems of existence and stability of the transformed system (1.5) and the original system (1.2).The proofs of weighted energy estimates and the stability of transformed system (1.5) are provided in Section 3.Then, we transfer the obtained results to prove the existence and stability of the main results for original system (1.2).
Notation 1.The norms in the Sobolev space of H r (R) are stated as q r := r k=0 ∂ k x q and q := q L 2 (R) .
Moreover, the weighted norms in the Sobolev space of H r w (R) are given by q r,w := r k=0 √ w(x)∂ k x q and q w := q L 2 w (R) .

Main results
We first establish the solutions of traveling wave (U, V)(x − st) of the parabolic-hyperbolic system (1.5).Substituting the following traveling wave ansatz into (1.5),where s and z are the traveling wave speed and moving coordinate, respectively.Then, we have where := d dz , and the boundary conditions are given as follows: (U, V)(z) → (u ± , v ± ) as z → ±∞. (2.3) 2) in z over (−∞, z) and (z, +∞), then one has and Moreover, we can rewrite the above results as follows: and lim Employing (2.3), the fact u + = 0, and U (z) → 0 as z → ±∞, then one has the following Rankine-Hugoniot conditions which presents s 2 + sχv + − χu − = 0. (2.5) In this paper, we only consider s > 0 and

.6)
Remark 1.To provide the weighted energy estimate, the weighted function is defined as follows: According to the weighted function in (2.7), the following condition is given where C 2 > C 1 > 0.
Lemma 1.Let (2.4) and u + = 0 hold.Then, the system (2.2) has a monotone traveling wave solution (U, V)(x − st), which is unique up to a translation satisfying U < 0, V > 0.Moreover, (U, V) has the following monotonicity behavior: (2.9) Proof.To show (2.9), we first assume that Then, one has ) (2.11) Let H s be the solution of Eq (2.11), then one can derive By employing the first equation of Eqs (2.10), (2.12) and L'Hospital's rule, one has lim Moreover, we assume that H s is the solution of Eq (2.11), then one has Similarly, by employing the first equation of Eq (2.10), L'Hospital's rule and Eq (2.13), one gets lim Therefore, we have Remark 2. Based on the second equation of Eq (2.10), one can derive where F = sv − + u − = sv + + u + and the wave speed s is given by (2.6).
For the parabolic-hyperbolic system (1.5), we define where the zero perturbation is obtained (see [12,19]).Then the stability result can be stated as follows: and By applying the change of variables (x, t) → (z = x − st, t), the system (1.5) becomes The solutions (u, v) of (2.15) are decomposed as follows: (2.17) Substituting (2.16) into (2.15) and integrating the results with respect to z, one has where , which is one of the barriers for the nonlinear diffusion.Moreover, the initial perturbation of (π, ρ) is given by with (π 0 , ρ 0 )(±∞) = 0. We present the solution of reformulated problem (2.18) and (2.19) in the space where 0 < T ≤ +∞ and w is the weighted function defined in (2.7). Let Then for any t > 0.Moreover, it holds that According to the classical works (see [17]), the global smooth solution can be constructed by the local well-posedness, the a priori estimate and an extension procedure.By the standard ways, the local well-posedness can be inferred (e.g., see [20]).Proposition 1.Let (π, ρ) ∈ X(0, T ) be a solution of (2.18) and (2.19) for several times T > 0. Then a constant ε 1 > 0 is presented, which is independent on T , such that if N(T ) < ε 1 , then (π, ρ) satisfies (2.20) for any 0 ≤ t ≤ T .
We further transfer the results of the transformed system (1.5) to the original chemotaxis model (1.2).Finally, our main results on the existence and stability of traveling waves (1.2) are stated in the following theorems: Theorem 3 (Existence).Let χ ∈ R ( 0).Then, the chemotaxis model with the nonlinear degenerate viscosity (1.2) does not have a traveling wave solution if χ < 0. If χ > 0 then the chemotaxis model with the nonlinear degenerate viscosity (1.2) has a unique monotone traveling wave solution (U, C)(z) such that U z < 0, C z > 0 for any given 0 = u + < u − and 0 = c − < c + .
Proof.The existence of U is from Lemma 1.We further show the existence of C. It follows from the second equation of (1.2), which implies that sC = CU, (2.22) which can be further calculated as follows: Since U converges to u ± exponentially as z → ±∞, and s > 0, then C(z) is bounded for any z ∈ R. It has the consequence C(z) > 0 for any z ∈ R, since otherwise c + ≡ 0 and hence C(z) = 0, which is not desired.
Noting that s > 0 and the Eq (2.22) at z = ±∞ yields Since C(z) > 0 for all z ∈ R, then it causes 0 ≤ c − < c + .It follows from the fact c + > 0 and (2.24), then it leads to u + = 0, which is only possible when U z < 0 and hence 0 = u + < u − .Finally, the proof of Theorem 3 is completed.

Weighted energy estimates
We further present the a priori estimates for solutions (π, ρ) of (2.18) and (2.19), and hence prove Proposition 1.In this paper, we deal with the case u + = 0 as z → +∞ which provides the singularity of 1  U .Therefore, we modify the idea of [17] by considering the singular term of 1 U as the weighted function of w in the energy estimates.
From Eq (2.14) and u + = 0, one has We further approximate (U + π z ) m in (2.18) through the following estimation: where P m l = m! (m−l)! .By dealing with N(t) 1, one has π z (•, t) L ∞ ≤ 1, and then (3.4) becomes Note that 0 < U ≤ u − , π z (•, t) L ∞ ≤ N(t) 1, and the term (π z + U) m−1 consists of two conditions: where the weighted function w(z) is given in Remark 1. Then we can derive By employing Young's inequality, we have where π(•, t) L ∞ ≤ N(t) has been employed.Similarly, Moreover, the higher-order estimate U m−2 has two possibilities as follows: where C = max {K, L} = max a m−a , (m + a) m for a > 0 and m > a.Then, (3.9) becomes 1 2 Conducting N(t) ≤ Dm(w(z) + u − ) and 1/U(z) ≤ Cw(z) for all z ∈ R, and further calculation of the integration of (3.10) with respect to t, the proof of estimate (π, ρ) in L 2 is completed. (3.11) Multiplying (3.12) 1 by π z U and (3.12) 2 by χρ z , we have 1 2 , where γ is a small enough constant.Substituting this inequality into (3.13)leads to We further multiply the first equation of (2.18) by ρ z to present the estimate t 0 Uρ 2 z , and one yields By the second equation of (3.12), we have By Young's inequality, noting 0 < U ≤ u − , we have where and some constant C > 0. By Young's inequality again, Thus, by employing π 2 z ≤ Cπ 2 z U , we have (3.17) } for 0 < m < 1, substituting (3.17) into (3.14), and through Lemma 2, when N(t) 1, one gets Substituting (3.18) into (3.17)gives which is combined with (3.18), In view of (3.6), by Young's inequality, π z (•, t) L ∞ ≤ N(t), and Lemma 2, we get Substituting this inequality into (3.20) and employing the higher-order estimate U m−2 as in L 2 , then one can obtain By the assumption 1/U(z) ≤ Cw(z) for all z ∈ R, we show the term t 0 wρ 2 z .Multiplying the second equation of (3.12) by wρ z , and integrating the results in z, one provides 1 2 which is combined with (3.23) Since w = 1+e ηz , one has 1 < w < 2 and 0 ≤ w z = ηe ηz ≤ 2ηw in (−∞, 0).We further integrate (3.24) with respect to z over (−∞, 0) to obtain 1 2 Integrating (3.24) in z over (0, +∞), and using the fact w z = ηe ηz ≥ ηw 2 in (0, +∞) we get 1 2 Combining (3.25) and (3.26), and then integrating the results in t, one has Next, we combine (3.27) with (3.21), and we have (3.28) Applying N(t) = min {Dm(w(z) + u − ), 1}, we complete the proof of Lemma 3. Proof.We differentiate (3.12) in z to present (3.31)By Young's inequality, where γ is a small enough constant.Noting where By the second equation of (3.30), By Young's inequality, In view of (3.33), since Thus, integrating (3.36) and using π (3.37) Substituting (3.37) into (3.35),choosing γ 1 and N(t) 1, since 0 < U ≤ u − , by Lemmas 2 and 3, we have which is combined with (3.38), further gives Since w = 1+e ηz , one has 1 < w < 2 and 0 ≤ w z = ηe ηz ≤ 2ηw in (−∞, 0).We further integrate (3.43) in z over (−∞, 0) to obtain 1 2 Under the influence of nonlinear diffusion, one needs to establish the third order derivative of (π, ρ) in order to make sense with the energy estimates stated in Theorem 2. In similar ways to Lemmas 3 and 4, the estimate of (π, ρ) in H 3 can be inferred, where the details are omitted here.Proposition 1 follows from Lemma 2 to Lemma 4.
Proof of Theorem 2. The a priori estimate (2.20) states that small enough N(0) gives small N(t).Thus, applying the procedure in a standard way, the global well-posedness of (2.18) and (2.Hence, the proof is finished.

Use of AI tools declaration
The author declare no use of Artificial Intelligence (AI) tools in the creation of this paper.