1. Introduction
The model and its motivation: In this paper, we mainly consider the following equation:
with the initial distribution
where $\lambda \gt 0$ is the growth rate, $\kappa \gt 0$ is the carrying capacity, $\chi \gt 0$ is the dispersion coefficient, $\sigma \gt 0$ is a sensing coefficient, $x \to u(t,x)$ is the density of population, and $p(t, x)$ is an external pressure.
Here, the term density of population means that
is the number of individuals between $x_1$ and $x_2$ (when $x_1\lt x_2$ ).
In the model the term, $\chi \partial _{x}\left (u(t, x) \partial _{x} p(t, x)\right )$ describes the cell-cell repulsion, and a logistic term $\lambda u(t, x)(1- u(t,x)/\kappa )$ corresponds the cell division, cell mortality, and the quadratic term $u(t,x)^2/\kappa$ corresponds to growth limitations due to quorum sensing (for short slow down the process of cell division) and due to competition for resources.
Replacing $u(t,x)$ and $p(t,x)$ by $\widehat{u}(t,x)=u(t/\lambda,x)/\kappa$ and $\widehat{p}(t,x)=p(t/\lambda,x)/ \kappa$ , we obtain (dropping the hat notation)
Therefore, through the paper we will assume that
Our original motivation comes from the description of cell motion in a Petri dish. In a previous paper [Reference Fu, Griette and Magal8], we derived a two-dimensional version of (1.3) to model the cell-cell repulsion in a Petri dish. We considered that cells grow in a circular domain (the Petri dish) and generate a repulsive gradient that pushes back neighbouring cells. We built a numerical simulation framework to study the solutions of the partial differential equation and compared the results to some real experiments realized by Pasquier and collaborators [Reference Pasquier, Magal, Boulangé-Lecomte, Webb and Le Foll17]. When starting from an isolated disk-like islet, the solution of the PDE looks like an expanding disk whose radius seems to be growing at a constant speed. We can study the shape of an expanding islet by considering travelling waves for (1.3). Previously, we studied the well-posedness of the problem (1.3) in [Reference Fu, Griette and Magal9] and proved the existence of an asymptotic propagation discontinuous profile – a travelling wave – in [Reference Fu, Griette and Magal10], corresponding to an initial data that is equal to 0 outside of some bounded region. In other words, in [Reference Fu, Griette and Magal10], we considered the case of an initial population of cells with compact support: no cell exists initially outside of the islet. The travelling waves constructed in [Reference Fu, Griette and Magal10] are called sharp because the transition between the occupied space (the area where $u(t, x)\gt 0$ ) and the empty space (when $u(t, x)=0$ ) occurs at some finite position. We also proved in [Reference Fu, Griette and Magal10] that sharp travelling waves are necessarily discontinuous. Our model is related to the study of Ducrot et al. [Reference Ducrot, Foll, Magal, Murakawa, Pasquier and Webb5] who introduced a complete model of in vitro cell dynamics with many different behaviours at the cellular level. Other features of closely related models have been investigated in [Reference Ducrot, Fu and Magal4, Reference Ducrot and Magal6, Reference Ducrot and Manceau7, Reference Hamel and Henderson12, Reference Henderson13].
In the previous paper, we proved the existence of sharp travelling waves for (1.3). Our goal here is to complete the description of existing travelling waves that are not sharp. Formally, our work relates to the result of de Pablo and Vazquez [Reference de Pablo and Vazquez18], who studied the existence of sharp and not sharp travelling waves for a porous medium equation. The porous medium equation corresponds (formally) to the case $\sigma \to 0$ . The convergence of the travelling waves when $\sigma \to 0$ has been observed only numerically in [Reference Fu, Griette and Magal10] and proved in [Reference Griette, Henderson and Turanova11], where the authors also propose an alternate method for the construction of discontinuous fronts by vanishing viscosity among other results.
The notion of solution: Throughout this paper, we impose that $p\in{\textrm{L}}^\infty (\mathbb{R})$ so the second line of (1.3) has a unique solution for a given $u(t, \cdot )\in{\textrm{L}}^\infty (\mathbb{R})$ . In order to give a sense of the solution (1.3), we first assume that $x \to p(t,x)$ is regular enough. Then, the nonlinear diffusion can be understood as
and by using the second equation of (1.3), we obtain
Therefore, the system (1.3) is understood for $t \geq 0$ and $x \in{\mathbb{R}}$ as
with the initial distribution
The existence and uniqueness of solutions of (1.4) in ${\textrm{L}}^\infty \left ({\mathbb{R}}\right )$ have been considered as a subset of the weighted space $L^1_\eta \left ({\mathbb{R}}\right )$ (with $\eta \gt 0$ ) with the norm
The existence and uniqueness of solutions for (1.4) have been studied by Fu, Griette, and Magal [Reference Fu, Griette and Magal9, Theorem 2.2].
1.1. Notion of travelling wave:
Definition 1.1. A travelling wave is a special solution of (1.3) such that $u(t, x)$ has the specific form
where the profile $U$ has the following behaviour at $\pm \infty$ :
A travelling wave is sharp if there exists $x_0 \in{\mathbb{R}},$ such that
A travelling wave is not sharp if
We will say that system (1.3) has a travelling wave with continuous profile if we can find a bounded, continuous, and decreasing continuous function $U\;:\;{\mathbb{R}} \to{\mathbb{R}}$ that is the profile of a travelling wave.
In Fu, Griette, and Magal [Reference Fu, Griette and Magal10, Proposition 2.4], we proved that the sharp travelling waves must be discontinuous. That is to say that $x \to U(x)$ , the travelling wave profile of (1.3) can be either continuous or discontinuous. We illustrated both situations in Figure 1.
Estimations on the travelling speed for the discontinuous profile: Under a technical assumption on $\hat \chi =\frac{\chi }{\sigma ^2}$ , we can prove the existence of sharp travelling waves which present a jump at the vanishing point.
Assumption 1.2 (Bounds on $\hat \chi$ ). Let $\chi \gt 0$ and $\sigma \gt 0$ be given and define $\hat \chi \;:\!=\;\frac{\chi }{\sigma ^2}$ . We assume that $0\lt \hat \chi \lt \bar \chi$ , where $\bar \chi$ is the positive unique root of the function
The existence of travelling waves with discontinuous profile has been studied by Fu, Griette, and Magal [Reference Fu, Griette and Magal10, Theorem 2.4].
Theorem 1.3 (Existence of a sharp discontinuous travelling wave). Let Assumption 1.2 be satisfied. There exists a travelling wave $u(t,x)=U(x-ct)$ travelling at speed
where
Moreover, the profile $U$ satisfies the following properties (up to a shift in space):
-
(i) $U$ is sharp in the sense that $U(x)=0$ for all $x\geq 0$ ; moreover, $U$ has a discontinuity at $x=0$ with $ U(0^-)\geq \frac{2}{2+\hat \chi }$ .
-
(ii) $U$ is continuously differentiable and strictly decreasing on $(\!-\!\infty, 0]$ and satisfies
\begin{equation*} -c\,U'-\chi (UP')' =U(1-U) \text { on } (\!-\!\infty, 0), \end{equation*}and\begin{equation*} U=0 \text { on } (0, \infty ), \end{equation*}and\begin{equation*} P-\sigma ^2 P''=U \text { on }{\mathbb {R}}. \end{equation*}
In this article, we focus on the existence of travelling waves with continuous profiles. The main result of this paper is the following theorem.
Theorem 1.4 (Existence of a continuous travelling wave). We assume that
There exists a travelling wave $u(t, x)=U(x-ct)$ with a continuous profile $x \to U(x)$ that is continuously differentiable and strictly decreasing, and
and satisfies travelling wave problem
where
Estimations on the travelling speed: We obtain the following condition for the existence of a travelling wave with a continuous profile for all-speed
For the Fisher-KPP equation [Reference Aronson and Weinberger2, Reference Kolmogorov, Petrovsky and Piskunov15, Reference Weinberger21], travelling waves only exist for half-line of positive travelling speeds. Moreover, there is a minimum speed $c_*\gt 0$ below which no travelling wave exists. Moreover, we can construct travelling waves for any values $c$ above $c_*$ . The existence of minimum speed is also true for porous medium equations with logistic dynamics [Reference de Pablo and Vazquez18]. By analogy with the porous medium equations, we expect that the minimal speed of the travelling waves corresponds to the sharp travelling wave constructed in [Reference Fu, Griette and Magal10]. In contrast, the continuous travelling waves constructed in the present paper correspond to higher velocities. Recall from [Reference Fu, Griette and Magal10, Theorem 2.4] that the sharp travelling wave is expected to travel at a speed $c_{\text{sharp}}\in \left (\frac{\chi/\sigma }{2+\chi/(\sigma ^2)}, \frac{\chi }{2\sigma }\right )$ , and indeed, we have that
Further analysis will be necessary to connect the gap between $c_{\text{cont}}^*$ and $c_{\text{sharp}}$ and to possibly prove the non-existence of travelling waves slower than sharp waves. Understanding the relationships between the profiles and the travelling speeds is still an open problem.
The paper is organized as follows. Section 2 is devoted to preliminary results. Section 3 presents the fixed point problem and its properties. Section 4 is devoted to the proof of Theorem 1.4. In Section 5, we present some numerical simulations. In Section 6, we present an application to wound healing.
2. Preliminary
We are interested in travelling waves of the system (1.3). In Figure 2, we illustrate the continuous travelling wave profiles.
Lemma 2.1. Assume that system (1.4) has a travelling wave $u(t,x)=U(x-ct)$ . Then, we must have
where $P\;:\;{\mathbb{R}} \to{\mathbb{R}}$ is the unique bounded continuous function satisfying the elliptic equation
Proof.
therefore, $p(t,x)=P(x-ct)$ where
The following proposition was proved by Fu, Griette and Magal [Reference Fu, Griette and Magal10, Proposition 2.4].
Proposition 2.2. Assume that $U$ is a continuous profile of travelling wave. Then, $U\;:\;{\mathbb{R}} \to{\mathbb{R}}$ is continuously differentiable, and
Transforming $U(x)$ into $\widehat{U}(x)=U(\!-\!x)$ : In order to work with increasing functions rather than decreasing functions, we reverse the space variable. By Definition 1.1 and Lemma 2.1, we get the following travelling wave problem
where
Equation (2.3) has the following behaviour at $\pm \infty$ :
Now, let us perform the change of variables to reverse the space direction. Setting $\widehat{U}(x) = U(\!-\!x)$ , and $\widehat{P}(x) = P(\!-\!x)$ , then equations (2.3) and (2.4) become
where
Assume that $c+\chi P'(x) \gt 0, \forall x \in{\mathbb{R}},$ then we have $c-\chi \widehat{P}'(x) \gt 0, \forall x \in{\mathbb{R}}$ by using $\widehat{P}(x) = P(\!-\!x)$ .
For convenience, we drop the hat notation, and system (2.5) becomes a logistic equation
where
and
with $P(x)$ is the unique solution of the elliptic equation
System (2.6) has the following behaviour at $\pm \infty$
Lemma 2.3. Assume that $U\;:\;{\mathbb{R}} \to{\mathbb{R}}$ is an increasing $C^1$ function. Then, the map $x \to P(x)$ solving the elliptic equation
is an increasing $C^3$ function, and we have the following estimation of the first derivative of $P(x)$
Proof. The result follows the following inequality
Lemma 2.4. Assume that $c\geq \sqrt{\chi \left (1+\frac{\chi }{\sigma ^2}\right )}$ ,
and
Then, $U\;:\;{\mathbb{R}} \to{\mathbb{R}}$ is an increasing $C^1$ function satisfying (2.6), (2.10) and
which is given by the following formula
where $\lambda (x)$ , $\kappa (x)$ , and $P(x)$ are given by equations (2.7), (2.8), and (2.9) above.
Proof. Let us prove that the formula
is well defined for all $x \in{\mathbb{R}}$ . So let us prove that if $0\lt u_0\lt \frac{\sigma ^2}{2 \left ( \sigma ^2+\chi \right )}{\left (1-\sqrt{1-\frac{\chi }{c^2}\left (1+\frac{\chi }{\sigma ^2}\right )}\right )}$ , then we have
Indeed, since by assumption
and Lemma 2.3, we deduce that
hence,
and since $P'(x)\geq 0$ , we deduce that
Now by combining (2.17), and $0\leq P(x)\leq 1,$ for any $x\in{\mathbb{R}}$ , we deduce that
Define
Using (2.18), we have that
by using the assumption
3. The relationship between the fixed point and travelling waves
Definition 3.1. Let $\mathcal{A}$ be the set of all admissible function $U\;:\;{\mathbb{R}}\rightarrow [0, 1]$ satisfying
-
(i) $U \in C^1({\mathbb{R}})$ ;
-
(ii) $0 \leq U(x) \leq 1, \forall x \in \mathbb{R}$ ;
-
(iii) $0\leq U'(x)\leq{C_U=\dfrac{c+\sqrt{c^2-\chi \left (1+\frac{\chi }{\sigma ^2}\right )}}{2\chi }}, \forall x \in \mathbb{R}$ .
Note that the upper bound in (iii) is the same as in the statement of Lemma 2.4 (2.13).
For each $U\in \mathcal{A}$ , we define
where
with
and
and
and $P(x)$ is the unique solution of the elliptic equation
Assumption 3.2. We assume that
Lemma 3.3 (Invariance of $\mathcal{A}$ by $\mathcal{T}$ ). Let Assumption 3.2 be satisfied. Let $\mathcal{T}$ be the map defined by (3.1). Then,
Proof. We divide the proof into three steps.
Step 1. We prove that $V=\mathcal{T}(U)\in C^1({\mathbb{R}})$ . Indeed, $V$ is continuously differentiable and
hence,
It follows from the definitions of $\lambda (x)$ and $\kappa (x)$ (see (3.4) and (3.5)) that $\lambda (x)$ and $\kappa (x)$ are continuously differentiable. Therefore, we have
Step 2. We prove that $0\lt V(x)\leq 1$ , $\forall x\in{\mathbb{R}}$ . By (2.11) and $U\in \mathcal{A}$ , we have that
Therefore, we have $c-\chi P'(x)\gt 0$ . Recall that
By using (3.3), we know that
Therefore, by definition of $V(x)$ , we have that $V(x)\gt 0, \forall x\in{\mathbb{R}}$ . On the other hand, by using (3.8), we have that
and
Since $0\leq U(x)\leq 1,$ $\forall x\in{\mathbb{R}},$ we have
We deduce that for $x \in \mathbb{R}$
By (3.12) and the comparison principle, we have that
Step 3. Let us prove that
Since
then we have
By (3.14), we have that
Therefore, we deduce
Since $0\leq U(x)\leq 1$ , and
we deduce that
Therefore, by using the fact that $0\leq V(x)\leq 1$ , $\forall x\in{\mathbb{R}}$ , (3.10), (3.15), and (3.16), we have $V(x)(1-V(x))\leq 1/4$ and we deduce that for $x\in{\mathbb{R}}$
Using the definition of $C_U$ and $c^2\geq \chi \left (1+\frac{\chi }{\sigma ^2}\right )$ , we have
and we finally reach
On the other hand, by using (3.7), we have
where
From (3.18), to prove $V'(x)\gt 0$ for $x\in{\mathbb{R}}$ , we only need to prove that
Indeed, we have
By using the definitions $\lambda (x)$ and $\kappa (x)$ in (3.4) and (3.5), and the formula (3.19), we deduce that
Since by assumption $U$ is increasing, it follows from (3.14) that $P$ is increasing. Then, for any $s\lt x$ , we have $P(s)\leq P(x)$ . We deduce that
therefore by combining (3.20) and the above inequality, we obtain
By using (3.21) and the definitions of $\lambda (x)$ and $\kappa (x)$ for $x\in{\mathbb{R}}$ , we have that
To conclude, it remains to recall that by assumption we have
therefore since $P(x)\geq 0$ and $c-\chi P'(x)\gt 0$ , we have that
which implies
The conclusion of the Step 3 now follows from (3.17) and (3.23). The proof is completed.
Let $\eta \gt 0$ . Let ${\textrm{BUC}}\left ({\mathbb{R}} \right )$ be the space of bounded and uniformly continuous maps from $\mathbb{R}$ to itself. Define the weighted space of continuous functions
and the weighted space $n$ times continuous differentiable functions
which is a Banach space endowed with the norm
Here we will use the above weighted space of ${\textrm{BUC}}^1_\eta \left ({\mathbb{R}} \right )$ maps to ensure that
is a closed subset of ${\textrm{BUC}}^1_\eta \left ({\mathbb{R}} \right )$ . As a consequence, the subset $\mathcal{A}$ equipped with the distance
is a complete metric space.
Lemma 3.4 (Compactness of $\mathcal{T}$ ). Let Assumption 3.2 be satisfied. Then, the set $\overline{ \mathcal{T} \left (\mathcal{A} \right )}$ is a compact subset of the metric space $\mathcal{A}$ equipped with the distance $ d_\eta$ .
Proof. Let $ \left \{ U_n \right \}_{n \geq 0} \subset \mathcal{A}$ be a sequence and define the corresponding sequence $ \left \{P_n\right \}_{n \geq 0}$ solution of equation (3.6) where $U$ is replaced by $U_n$ . Define the corresponding sequences $\left \{ \lambda _n \right \}_{n \geq 0}$ and $\left \{ \kappa _n \right \}_{n \geq 0}$ by using (3.4) and (3.5) where $P(x)$ is replaced by $P_n(x)$ . Denote $V_n=\mathcal{T}(U_n), \forall n\in \mathbb{N}$ . By Lemma 3.3, we know that $\mathcal{T}(\mathcal{A})\subset \mathcal{A}$ . Therefore, we have
Similarly to equation (3.8) in the proof of Lemma 3.3, we have that
where
and
and $P_n(x)$ is the unique solution of the elliptic equation
It follows from Lemma 2.3 that the map $x \to P_n(x)$ solving the above equation is an increasing $C^3$ function. By (3.14), we have that $c-\chi P'_{\!\!n}(x)\gt 0$ . Since $U_n(x)\in C^1({\mathbb{R}})$ , we have that $\lambda _n(x), \kappa _n(x)\in C^2({\mathbb{R}})$ and $V'_{\!\!n}(x)\in C^1({\mathbb{R}})$ . Then, we obtain that $V_n(x)\in C^2({\mathbb{R}}).$ Therefore by using (3.25) and (3.26), we deduce that the families $V'_{\!\!n}|_{[\!-\!k, k]}$ and $V''_{\!\!\!n}|_{[\!-\!k, k]}$ are uniformly Lipschitz continuous on $[\!-\!k,k]$ for each $k\in \mathbb{N}$ . Applying Ascoli-Arzelà theorem, we have that the sets $\{V_n|_{[\!-\!k, k]}\}_{n\gt 0}$ and $\{V'_{\!\!n}|_{[\!-\!k, k]}\}_{n\gt 0}$ are relatively compact on $[\!-\!k, k]$ for each $k\in \mathbb{N}$ .
Using a diagonal extraction process, there exists a sub-sequence $n_p$ and a bounded continuous function $V$ such that $V_{n_p}\rightarrow V$ uniformly on every compact subset of $\mathbb{R}$ as $p\rightarrow \infty$ . Indeed, recall that $0\lt V_n(x)\leq 1$ and $0\lt V'_{\!\!n}(x)\leq C_U$ , $ \forall x\in{\mathbb{R}}$ . By the Ascoli-Arzelà theorem, there exists a sub-sequence $\{V_{m_{p}^1}\}_{p\geq 0}$ of $V_{n}$ and a function $V_1 \in C^1([\!-\!1,1])$ such that
Now we can extract $\{V_{m_{p}^{2}}\}_{p\geq 0}$ a sub-sequence of $\{V_{m_{p}^1}\}_{p\geq 0}$ , and function $V_2 \in C^1([-2,2])$
By construction, we will have
Replacing eventually $m_{1}^{2}$ by $m_{1}^{1}$ , we can assume that $m_{p}^{2}=\{m_{1}^{1}, m_{2}^{2},\cdots,m_{p}^{2},\cdots \}$ . Proceeding by induction, we can find a $\{V_{m_{p}^{k}}\}_{p\geq 0}$ a sub-sequence $\{V_{m_{p}^{k-1}}\}_{p\geq 0}$ , such that
and a function $V_k \in C^1([\!-\!k,k])$ such that
Set
and
Then, the sub-sequence $\{V_{n_{p}}\}_{p\geq 0}$ converges locally uniformly with respect to the $C^1$ norm, so we can define
and
By construction, we will have
Now, we are ready to show that $\|V_{n_p}-V\|_{1,\eta }\rightarrow 0$ as $p\rightarrow +\infty$ . Let $\varepsilon \gt 0$ be given. Let $k$ be large enough to satisfy
For all $k$ large enough, since $0\lt V_{n_p}(x)\leq 1$ , $0\lt V'_{\!\!n_p}(x)\leq C_U$ , $\forall x\in{\mathbb{R}}$ and (3.27), we deduce that
and
Moreover, since $V_{n_p}$ converges locally uniformly to $V$ and $V'_{\!\!n_p}$ converges locally uniformly to $V'$ , for any fixed $x\in [\!-\!k, k]$ , there exists an integer $p_0\gt 0$ such that
and
It follows from (3.29), (3.30), (3.31), and (3.32) that for $p \geq p_0$ ,
Since the above inequality is true for any $\varepsilon \gt 0$ , this completes the proof of lemma.
The most difficult part of the proof of existence of travelling waves is the continuity of the map $\mathcal{T}\;:\; \mathcal{A} \to \mathcal{A}$ . To consider this problem, we decompose the real line into several intervals $(\!-\!\infty, -K]$ , $[\!-\!K,K]$ , and $[K,\infty )$ . Before proving the continuity of $\mathcal T$ , we establish the continuity of its components separately.
Lemma 3.5 (Continuity of $P$ , $P'$ , $\lambda$ and $\kappa$ ). Let Assumption 3.2 be satisfied. Assume that $0\lt \eta \lt \frac{1}{\sigma }$ . Let $U_1, U_2\in \mathcal{A}$ and define, for $i=1,2,$
There exist continuous functions of $x\in \mathbb R$ , $C_P(x)$ , $C_\lambda (x)$ , and $C_\kappa (x)$ such that, for all $x\in \mathbb R$ ,
The functions $C_P(x)$ , $C_\lambda (x)$ , and $C_\kappa (x)$ do not depend on the particular choice of $U_1\in \mathcal{A}$ and $U_2\in \mathcal{A}$ but only on $\eta$ , $\sigma$ , and $\chi$ .
Proof. Step 1: We show (3.33). We have, for $x\gt 0$ :
and similarly for $x\lt 0$ :
Rearranging the terms in (3.37) and (3.38), we have
and (3.33) is proved.
Step 2: We show (3.34). We have
so that the exact computations leading to (3.39) can be reproduced, and we have
(3.34) is proved.
Step 3: We show (3.35). It follows from the definitions of $\lambda _1(x)$ and $\lambda _2(x)$ that, for all $x\in \mathbb R$ ,
Since by Definition 3.1, we have $U_i'(x)\leq C_U$ ( $i=1,2$ ), then $P_i'(x)=\int _{\mathbb R} \frac{1}{2\sigma }e^{-\frac{|x-y|}{\sigma }} U'(y)\mathrm{d} y\leq C_U$ ( $i=1,2$ ), therefore
It follows from (3.40) and (3.41) that
Using the fact that $0\leq P_1(x)\leq 1$ and $0\leq P'_1(x)\leq C_U$ for $x\in{\mathbb{R}}$ , then (3.35) is a consequence of (3.33) and (3.34).
Step 4: We show (3.36). We have:
Thus, (3.36) is a consequence of (3.34). Lemma 3.5 is proved.
Lemma 3.6 (Continuity of $\mathcal{T}$ ). Let Assumption 3.2 be satisfied. Assume that $0\lt \eta \lt \frac{1}{\sigma }$ . Then, the map $\mathcal{T}: \mathcal{A} \to \mathcal{A}$ is continuous on $\mathcal{A}$ endowed with distance $d(U_1,U_2)=\Vert U_1-U_2\Vert _{1,\eta }$ .
Proof. Let $U_0 \in \mathcal{A}$ be fixed, and $U \in \mathcal{A}$ , and define
Part A: We prove that for each admissible profile $U_0 \in \mathcal A$ and $\varepsilon \gt 0$ , there is a $\delta _1\gt 0$ such that
whenever
Let $K\gt 0$ be such that
Then since $V\in \mathcal{A}$ by Lemma 3.3, we have $0\leq V(x)\leq 1$ and $0\leq V_0(x)\leq 1$ for all $x\in \mathbb R$ , therefore
Thus, there remains only to establish that
if $\Vert U-U_0\Vert _{1, \eta }\leq \delta _1$ , for $\delta _1\gt 0$ sufficiently small. Recall that
wherein
and
and $P(x)$ is the unique solution of the elliptic equation
By using the definitions of $V_0(x)$ and $V(x)$ for $x\in{\mathbb{R}}$ , we find that
Since
and similarly
we have that
hence,
where
and
We divide the rest of the proof of Part A into two steps, to estimate $H(x)$ and $I(x)$ .
Step 1: We show that
for some continuous function $C_H(x)$ independent of $\varepsilon$ , $U$ , $U_0$ .
By Taylor’s theorem, we have that
Now we use (3.49) to estimate $H(x)$ defined in (3.46).
Recall from (3.35) in Lemma 3.5 that there is a continuous function $C_\lambda (x)$ such that
Thus, we can rewrite (3.50) as
Next recall the definition of $\lambda (x)$ :
Since by Definition 3.1, we have $U'(x)\leq C_U$ , then $P'(x)=\int _{\mathbb R} \frac{1}{2\sigma }e^{-\frac{|x-y|}{\sigma }} U'(y)\mathrm{d} y\leq C_U$ ; therefore,
and therefore,
Clearly, we have the same upper bound for $\lambda _0(x)$ and $\lambda (x)$ , and (3.51) becomes
where $C_H(x)$ is a continuous function. Therefore, (3.48) is proved.
Step 2: We show that
for some continuous function $C_I(x)$ independent from $\varepsilon$ , $U$ , $U_0$ .
Indeed, we have
where
and
Using (3.49), we rewrite (3.54) as
Since by Definition 3.1, we have $U'(x)\leq C_U$ , then $P'(x)=\int _{\mathbb R} \frac{1}{2\sigma }e^{-\frac{|x-y|}{\sigma }} U'(y)\mathrm{d} y\leq C_U$ ; therefore,
and finally
By using (3.56), (3.57), and (3.35) in Lemma 3.5, we rewrite as
Thus, there exists a continuous function $C_{I_1}(x)$ such that
Next we estimate $I_2(x)$ in (3.55). By using (3.35) and (3.57), we have
thus there exists a continuous function $C_{I_2}(x)$ such that
Combining (3.53), (3.58) and (3.59), there exists a continuous function $C_I(x)\;:\!=\;C_{I_1}(x)+C_{I_2}(x)$ such that (3.52) holds. Step 2 is completed.
Conclusion of Part A: By choosing $\delta _1$ such that
we conclude from (3.38), (3.48), and (3.52) that indeed
whenever $\Vert U-U_0\Vert _{0, \eta }\leq \delta _1$ . Thus, (3.43) holds, and this concludes Part A.
Part B: We prove that for each admissible profile $U_0\in \mathcal{A}$ and $\varepsilon \gt 0$ , there is $\delta \gt 0$ such that whenever
we have
By Lemma 3.3, we know that $V=\mathcal{T}(U)\in \mathcal{A}$ and $V_0=\mathcal{T}(U_0)\in \mathcal{A}$ . Therefore,
Let $K\gt 0$ be such that
We have
Thus, there remains only to establish that
We note that $V$ and $V_0$ satisfy (3.2); therefore,
Then, we have that
By using the fact that $0\leq V(x)\leq 1$ and $0\leq V_0(x)\leq 1$ for $x\in{\mathbb{R}}$ , we have that
It follows from (3.45), (3.57), (3.52), (3.48), (3.35), and (3.36) that
Let
then whenever $\Vert U-U_0\Vert _{0, \eta }\leq \delta _2$ , we have
therefore, recalling (3.61),
Conclusion of Part B: Let $\delta \;:\!=\;\min (\delta _1, \delta _2)$ where $\delta _1$ is defined in (3.60) and $\delta _2$ is defined in (3.66). Then if $\Vert U-U_0\Vert _{0, \eta }\leq \delta$ , we know from Part A (3.42) that
and from (3.67) that
So finally,
Part B is proved. Since we always have $\Vert U-U_0\Vert _{0, \eta }\leq \Vert U-U_0\Vert _{1, \eta }$ , the continuity holds for the norm $\Vert \cdot \Vert _{1, \eta }$ . Lemma 3.5 is proved.
4. Proof of Theorem 1.4
From the definition of admissible functions $\mathcal{A}$ , it is a nonempty, closed, convex, bounded subset of the Banach space $BUC^1_\eta \left ({\mathbb{R}} \right )$ . By Lemmas 3.4 and 3.6, we obtain that $\mathcal{T}$ is a continuous compact operator on $\mathcal{A}$ . Therefore, by the Schauder fixed point theorem, there exists $U$ in $\mathcal{A}$ such that
Applying Lemma 3.3, we have that $U\in C^1({\mathbb{R}})$ and $0\leq U'(x)\leq C_U$ for any $x\in{\mathbb{R}}$ . Therefore, we have that
wherein
and
and $P(x)$ is the unique solution of the elliptic equation
Namely, we have that
Therefore, we have that
By using (4.1), we have that
We prove that
Indeed, since $U'(x)\geq 0$ and $0\lt U(x)\leq 1$ for any $x\in{\mathbb{R}}$ , then $U(\infty )$ exists. By using $P$ equation (1.8), the function $x \to P(x)$ is increasing and bounded, and by Lebesgue’s dominated convergence theorem, we have
Therefore,
It follows from (4.3) and (4.4), we have that
and since $x \to U(x)$ increasing and $U(0)=u_0\gt 0$ , this implies that
This completes the proof of the Theorem 1.4.
5. Numerical simulations
We choose a bounded interval $[\!-\!K,K]$ and an initial distribution $u_0\in C([\!-\!K,K])$ as follows:
In the following numerical simulations, we solve the PDE numerically using the upwind scheme, and we refer to Leveque [Reference Leveque16] and Toro [Reference Toro19] for more results on this subject. The numerical method used for the simulations is presented in Section A of the Appendix.
In this section, we set the parameters of the system (A.1) all equal to one. That is,
In Figure 3, we plot $x \to u_0(x)$ with the parameter values $\beta =1$ , and $K=20$ , and the corresponding travelling wave profile which coincides with $x \to u(20,x)$ the solution of system (A.1) at $t=20$ days.
In Figure 4, we run a simulation from $t=0$ until $t=20$ of the model (A.1). We observe that the travelling wave appears almost immediately after the starting time $t=0$ .
Next, we use the following initial value
In Figure 5, we plot $x \to u_0(x)$ the initial distribution of system (A.1) (on the left-hand side) and the corresponding travelling wave profile which coincides with $x \to u(20,x)$ the solution of system (A.1) at $t=20$ days.
In Figure 6, we run a simulation from $t=0$ until $t=20$ of the model (A.1). We observe that the travelling wave appears almost immediately after the starting time $t=0$ .
On the one hand, our numerical simulations show that continuous travelling waves can be observed from an initial distribution decaying exponentially (slowly enough). On the other hand, sharp travelling waves can also be observed when starting the PDE with initial distributions equal to zero on the half-plane. So in practice, both types of travelling waves can be observed numerically.
Now concerning the travelling speed, we observe numerically that sharp travelling waves are slower than continuous travelling waves. In this aspect, the situation is somehow similar to what is observed with reaction-diffusion equations (like the Fisher-KPP equation), in that the ‘slowest’ wave is caught by starting from compactly supported initial data. The question of the minimal speed is quite intricate given the nonlinear nature of the equation, and we leave it for future works.
6. Application to wound healing
The wound healing assay is used in a range of disciplines to study the coordinated movement of a cell population (Figure 7). We refer to the paper of Jonkman et al. [Reference Jonkman, Cathcart, Xu, Bartolini, Amon, Stevens and Colarusso14] for a review on this topic. In this paper, we consider the cell-cell repulsion described by nonlinear diffusion, but cell-cell attraction also occurs and this problem was recently considered by Webb [Reference Webb20] (see also the references therein for more results).
In this section, we set the parameters of the system (A.1) as follows
Initial distribution for imperfect wound: We choose a bounded interval $[\!-\!K,K]$ and an initial distribution $u_0\in C([\!-\!K,K])$ as follows
In Figure 8, we plot $x \to u_0(x)$ with the parameter values $\beta =0.5$ , and $K=20$ , and $x \to u(7,x)$ the solution of system (A.1) at $t=7$ days.
In Figure 9, we run a simulation from $t=0$ until $t=7$ of the model (A.1). We observe that two travelling waves moving in opposite directions appear almost immediately after the starting time $t=0$ . They merge together to give a flat distribution approximately on day $2$ .
Initial distribution for perfect wound: We choose a bounded interval $[\!-\!K,K]$ and an initial distribution $u_0\in C([\!-\!K,K])$ as follows
In Figure 10, we plot $x \to u_0(x)$ with the parameter values $\beta =0.07$ , and $K=20$ , and $x \to u(7,x)$ the solution of system (A.1) at $t=7$ days.
In Figure 11, we run a simulation from $t=0$ until $t=7$ of the model (A.1) for the parameter values $\sigma =1$ and $\chi =1$ . We observe that two travelling waves moving in opposite directions appear almost immediately after the starting time $t=0$ . They merge together to give a flat distribution approximately on day $5$ .
It is observed that the speed of healing depends strongly on the imperfection of the wound. If we compare the two simulations, we see that the wound seems much larger in Figure 8 than in Figure 10. But the time required for healing is about $2$ days in Figure 9 whereas it is about $5$ days in Figure 11. Therefore, the imperfection of the wound has a strong influence on the healing time.
Financial support
Q.G. acknowledges support from ANR via the project Indyana under grant agreement ANR-21-CE40-0008. M.Z is supported by Natural Science Foundation of Tianjin (No. 23JCQNJC01010) and China Scholarship Council.
Competing interests
The authors declare none.
Appendix
An Upwind method applied to the numerical scheme
In Section 5, we use the following system of PDE to run the numerical simulations
with
Now we use the finite volume method to consider equation (A.1). Our numerical scheme reads as follows:
where the flux $\phi (u_{i+1}^{n},u_{i}^{n})$ for $i=0, \ldots, M$ is defined as
where
and
where
Moreover, the vector $P^{n}$ is defined by
where
Indeed, we have
and since we use the Neumann boundary condition, we must impose
Since the Neumann boundary condition corresponds to a no flux boundary condition, we have
which corresponds to $p_0^n=p_1^n$ and $p_{M+1}^n=p_M^n$ . Therefore, the numerical scheme at the boundary becomes
Due to the boundary condition, we have the conservation of mass for equation (A.1) when the reaction term equals zero.