Traveling wave solutions for a one dimensional model Of cell-to-cell adhesion and diffusion with monostable reaction term

This work is concerned with the properties of the traveling wave solutions of a one dimensional model of cell diffusion and aggregation, incorporating volume filling and cell-to-cell adhesion with net birth term \begin{document}$\begin{equation*} ρ_t = [ D(ρ)ρ_x]_x + g(ρ)\ \ \ t≥0,\ \ \ x∈ \mathbb{R},\end{equation*}$ \end{document} where \begin{document}$ D(ρ) $\end{document} may take positive or negative values with different population density \begin{document}$ ρ $\end{document} and adhesion coefficient \begin{document}$ α ∈ [0,1] $\end{document} , and the negative one will lead to the ill-posedness of the equation. In all these cases we prove the existence of infinitely many traveling wave solutions, where these solutions are parameterized by their wave speed and monotonically connect the stationary states \begin{document}$ ρ\equiv0 $\end{document} and \begin{document}$ ρ\equiv 1 $\end{document} .

1. Introduction. Reaction diffusion equations have been applied extensively to population dynamics. By using a random walk approach and assuming that the individuals of a population have the same probability of moving from one point to another, Skellam [26] derived the standard reaction diffusion equation (Fisher-KPP equation) ρ t = D∆ρ + g(ρ) where ρ(r, t) is the population density, D > 0 is a constant and g(ρ) is the net rate of growth. However, it is now clear that in a number of biology contexts, motility varies with population density, requiring nonlinear diffusion terms. This was first realized in ecology [15,16,25] and density-dependent dispersal is now a common feature of spatial modeling in biology. This includes models that are of degenerate reaction diffusion equations [8,9,12,13,14,22], as well as reaction diffusion aggregation equations [1,5,10,20,21].
1.1. The derivation of the model. Here we consider one species living in a onedimensional habitat. To derive the model we follow a biased random walk approach plus a diffusion approximation. First we discretize space in a regular manner. Let h be the distance between two successive points of the mesh, and let ρ(x, t) be the population density at the point x, time t where 0 ≤ ρ(x, t) ≤ 1 with scaled density. During a time period τ an individual which is at the position x and time t can either: is analogous. We can see that 0 ≤ R(x, t), L(x, t), R(x, t) + L(x, t) ≤ 1, which is different from the form given in [3]. In their case, L(x, t) = (1 − ρ(x − h, t))(1 − αρ(x + h, t))/h 2 may go to infinity as h → 0, which can not be explained from a probabilistic standpoint. Using the notations above, we can rewrite the density ρ(x, t) as follows: ρ(x, t+τ ) = N (x, t)ρ(x, t)+R(x−h, t)ρ(x−h, t)+L(x+h, t)ρ(x+h, t)+τ g(ρ(x, t)), where g(ρ(x, t)) is the net birth rate at (x, t). By using Taylor series, we obtain the following approximation, we obtain Now substituting β and ν in the above equation, we obtain Using the same diffusion approximation as in [5], [18], [24], [26], and assuming that h 2 /τ → C > 0 as τ, h → 0, we obtain the following, Then, letting C = 2 for simplicity, we obtain the equation (1). We recall from [3] that the equation with quadratic coefficient D(ρ) = 3αρ 2 −4αρ+1, is globally well posed if 0 ≤ α < 3 4 . When α = 3/4, the equation (4) degenerates at ρ = 2/3, and when 1 ≥ α > 3 4 , the equation (4) is ill posed if and only if the initial density profile protrudes into the unstable interval Equations (1)-(2) can be seen as an extension of the standard Fisher-KPP equation where D > 0 is constant. Traveling wave solutions (t.w.s) are very important in reaction diffusion equation, which in ecology correspond to invasions, and in cell biology correspond to the advancing edge of an expanding cell population, such as a growing tumor. We recall that a traveling wave solution is a solution ρ(x, t) having a constant profile, that is ρ(x, t) = ρ(x − ct) = ρ(τ ) for some function ρ(τ ), and the constant c is the wave speed. In particular, a traveling wave solution connecting the steady states ρ = 0 and ρ = 1 satisfies the boundary value problem ρ(−∞) = 1, ρ(+∞) = 0.
The first systematic analysis on the existence of traveling wave solution of the standard Fisher-KPP equation appeared in two separate works due to Fisher [11] and Kolmogorov et al. [19]. The main ideas of the methodology introduced by Kolmogorov et al. are still used today.
For the non-linear diffusion equation where D(ρ) is a strictly positive function on [0,1] and the kinetic part g(ρ) is as in the classic Fisher-KPP equation, Hadeler [17] gave the lower bound on c for the existence of traveling wave solution of front type. In the degenerate case where D(0) = 0 with D(ρ) > 0 ∀ρ ∈ (0, 1], Sánchez-Garduño and Maini [12], [13] used a dynamical systems approach to prove the existence and nonexistence of the traveling wave and the monotone decreasing property of the traveling wave solution. In the doubly degenerate Fisher-KPP equations when D(0) = D(1) = 0 with D(ρ) > 0 elsewhere, under less regularity conditions on g(ρ) and D(ρ), Malaguti and Marcelli [22] obtained a continuum of traveling wave solutions having wave speed c greater than a threshold value c * and showed the appearance of a sharptype or finite type profile when c = c * .
For the diffusion aggregation equation where D(ρ) changes its sign once, from positive to negative values, in the interval ρ ∈ [0, 1], and g(ρ) is a monostable nonlinear term. Maini et al. [21] proved the existence of infinitely many C 1 traveling wave solutions. These solutions are parameterized by their wave speed and monotonically connect to the stationary states ρ = 0 and ρ = 1. In the degenerate case, i.e, when D(0) = 0 and/or D(1) = 0, sharp profiles appear, corresponding to the minimum wave speed. In 2010 Sánchez-Garduño et al. [14] investigated a family of degenerate negative diffusion equation (D(ρ) ≤ 0) with mono-stable reaction term g(ρ) and studied the one dimensional traveling wave solutions for these equations.
In 2011 Kuzmin and Ruggerini [20] considered a modified aggregation diffusion model with the diffusivity D(ρ) satisfying the following and g(ρ) satisfying the bi-stable condition. They gave necessary and sufficient conditions under which traveling wave solutions of the equation exist and provided an estimate for the minimal speed c * . In their situation, D (γ) = 0 is a very important condition in proving the existence of C 1 traveling waves. However, in this paper we will see if D (γ) > 0, traveling wave solutions may not differentiable at γ with minimal wave speed, and that is the motivation of the weak traveling wave solution which improves results in [10].
DiCarlo et al. [6] obtained traveling wave solutions to a nonlinear diffusion equation where D(ρ) contains a negative region and the nonmonotonic traveling wave is driven by a boundary condition on a semi-infinite domain, whereas in our model the traveling wave is driven by the source term on an infinite domain and has the property of monotonicity.
The structure of our paper is as follows: In section 2, we define the weak traveling wave solution and prove the traveling wave speed is positive and the traveling wave solution is monotone decreasing. This helps us define the inverse function of the traveling solution. In section 3, we prove the existence of C 1 traveling wave solutions when 0 ≤ α < 3/4. In the complicated case when 3/4 ≤ α ≤ 1, we prove the existence of weak and C 1 traveling wave solutions in Theorem 1.1. For the precise definitions of C 1 traveling wave and weak traveling wave solutions, see Definitions 2.1 and 2.2.
Theorem 1.1. Let D(ρ) and g(ρ) given functions respectively satisfy (2) and (3) and 3/4 ≤ α ≤ 1. There exists a value c * > 0, satisfying such that the equation (1) has i) no weak traveling wave solution for c < c * , ii) a unique (up to space shifts) weak traveling wave solution for c = c * , ii) a unique (up to space shifts) C 1 traveling wave solution for c > c * , Where ρ (α) and ρ (α) are given in (5).
Remark 1. When 0 ≤ α < 3/4, we can see the diffusion coefficient D(ρ) > 0 and the existence of the traveling wave solution is already known.
2. Preliminaries and necessary conditions. We recall that a traveling wave solution for the equation (1) is a solution of the form ρ(x, t) = ρ(x − ct) for some constant traveling speed c. The equation we have to deal with is changed to where stands for differentiation with respect to the wave variable ξ = x − ct.
Following [21], we have the definition of the classical traveling wave solution for , satisfying the equation (9) in (a, b) and the boundary conditions Condition (11) added to the classical boundary condition (10) is motivated by the possible occurrence of sharp type profiles, that is, solutions that reach the equilibria at a finite value a or b. However, when the existence interval is the whole real line, then the condition (11) is automatically satisfied and then Definition 2.1 is reduced to the classical one which is the front type traveling wave solution.
However ρ may not be C 1 because of the singularity of the equation at ρ = ρ (α), ρ (α), and we have to give a definition of weak traveling wave solution.

LIANZHANG BAO AND ZHENGFANG ZHOU
When 3 4 ≤ α ≤ 1, the diffusivity D(ρ) is negative whenever and D(ρ) ≥ 0 otherwise. In the following, we split the proof into four steps.
We can see that η < +∞, and η is a local maximum point.
), by using the equation (12), we obtain which is a contradiction. In the case ρ(η) ≥ which is a contradiction.
) and ρ (ξ 0 ) = 0. We only , 1] and the case for ] is analogical. By using the equation (12), we have This is a contradiction.

3.
Main results of traveling wave solutions. The first part of this section will deal with the case 0 ≤ α < 3 4 , then we will use these results in the second part to prove the existence and nonexistence of the traveling wave solution when 3 4 ≤ α ≤ 1. 3.1. Results for 0 ≤ α < 3 4 . In this case, we obtain that the diffusivity coefficient D(ρ) > 1 − 4 3 α > 0. Engler [7] and Hadeler [17] obtained the following result. In [21] the result is extended to the possible degenerate case. such that the boundary value problem: is solvable if and only if c ≥ c * . Moreover, for every c ≥ c * , the solution is unique.  (1), with wave speed c, satisfying boundary conditions (10) and (11), is equivalent to the solvability of the equation (16), with the same c. 3 4 ≤ α ≤ 1. In the model when 3 4 ≤ α ≤ 1, the diffusivity coefficient will be negative which will make the proof of the existence of the traveling wave complicated, especially at the point

Results for
. The derivative of z(ρ) may have possibly two different values which will lead to two possible derivative values of ρ at by L'Hôpital's rule). To overcome this difficulty, we will use the idea of weak traveling wave solution from [5] and consider the traveling waves in different intervals: [0, , 1]. Under certain conditions, we can glue these traveling waves together which will give the existence of the weak traveling wave solution in the whole interval.
Before proving Theorem 1.1, we introduce the following Lemma, which will be used in the proof of Theorem 1.1.