3.1. Preliminaries

Let $L\gt 0$ and let $1 \le p\lt \infty$ . We denote by $L^p(0,L)$ and $W^{1,p}(0,L)$ the usual Lebesgue and Sobolev spaces. For $p= 2$ , we write $H^1({0,L})$ instead of $W^{1,2}({0,L})$ . Furthermore, $H^1({0,L})'$ is the dual space of $H^1({0,L})$ .

It is well known (see e.g., [Reference Adams1]) that there exists a unique linear, continuous map $\Gamma \colon W^{1,p}({0,L}) \to{\mathbb{R}}$ known as the trace map such that $\Gamma (u)= u(0)$ for all $u \in W^{1,p}({0,L}) \cap C([0,L])$ . In addition, let us recall the following trace estimate [Reference Brenner and Scott3, Theorem 1.6.6]:

(3.1) \begin{equation} |u(0)| \le C_{\text{e}}\,\|u\|_{L^2({0,L})}^{1/2}\,\|u\|_{H^1({0,L})}^{1/2}. \end{equation}

Let $T\gt 0$ and let $(B, \|\cdot \|_B)$ be a Banach space. For every $1 \le r\lt \infty$ , we denote by $L^r(0,T; B)$ the Bochner space of all measurable functions $u\colon [0,T] \to B$ such that $ \|u\|_{L^r(0,T; B)}^r\;:\!=\; \int _0^T \|u(t)\|_B^r \; \,\textrm{d}t \lt \infty .$ For $r= \infty$ , the norm of the corresponding space $L^\infty (0,T; B)$ is given by $ \|u\|_{L^\infty (0,T; B)}\;:\!=\; \mathrm{ess \,sup}_{0 \le t \le T} \|u(t)\|_B.$ Finally, $C([0, T]; B)$ contains all continuous functions $u\colon [0, T] \to B$ such that

\begin{equation*} \|u\|_{C([0, T]; B)}\;:\!=\; \max _{0 \le t \le T} \|u(t)\|_B\lt \infty . \end{equation*}

We refer to [Reference Evans14] as a reference for Bochner spaces. For every $a \in{\mathbb{R}}$ , we set $a^{\pm }\;:\!=\; \max \{\pm a, 0\}$ and for $u \in W^{1,p}({0,L})$ we define $u^\pm (\cdot )\;:\!=\; u(\cdot )^\pm$ and will use the fact that $u^\pm \in W^{1,p}(0,L)$ .

3.2. Transformation for a fixed reference domain

Before we give our definition of weak solutions, state the necessary assumptions and our main theorem, we transform (3.6) into an equivalent system set on a fixed reference domain. This facilitates the proofs and also the spaces that we need to work in. To this end, we make the following change of variables:

\begin{equation*} y= y(t, x)=: \frac {x}{L(t)} \quad \longleftrightarrow \quad x= L(t) y. \end{equation*}

Then we define the functions $ \bar{f}_i(t, y)= f_i(t, x)= f_i(t, L(t) y)$ and observe that

(3.2) \begin{equation} \begin{split} \partial _x f_i= \frac{1}{L(t)}\, \partial _y \bar{f}_i, \quad \partial _t f_i= \partial _t \bar{f}_i- L'(t)\,y\, \partial _x f_i= \partial _t \bar{f}_i- \frac{L'(t)}{L(t)}\, y\, \partial _y \bar{f}_i, \quad \,\textrm{d}x= L(t) \,\textrm{d}y. \end{split} \end{equation}

Using (3.2), taking into account that, by the product rule, $y\, \partial _y \bar{f}_+= \partial _y(y\, \bar{f}_+)- \bar{f}_+$ and rearranging, the first equation of (2.2) reads as:

(3.3) \begin{align} \partial _t \bar{f}_+&= -\frac{1}{L^2(t)}\, \partial _y \big (L(t)\, v_0\, \bar{f}_+\,(1- \bar{\rho })- D_T\, \partial _y \bar{f}_+ - L'(t)\,L(t)\, y\, \bar{f}_+\big ) - \frac{L'(t)}{L(t)}\, \bar{f}_++ \lambda \,(\bar{f}_- - \bar{f}_+) \nonumber \\[5pt] &= -\frac{1}{L^2(t)}\, \partial _y \bar{J}_+- \frac{L'(t)}{L(t)}\, \bar{f}_++ \lambda \,(\bar{f}_- - \bar{f}_+), \end{align}

and with similar arguments:

\begin{equation*} \partial _t \bar {f}_- = -\frac 1{L^2(t)}\,\partial _y \bar {J}_- - \frac {L'(t)}{L(t)}\, \bar {f}_- +\lambda \,(\bar {f}_+ - \bar {f}_-), \end{equation*}

where the fluxes are defined by:

\begin{align*} \bar{J}_+(t,y) &= \phantom{-}L(t)\,v_0\,\bar{f}_+(t,y)\,(1- \bar{\rho }(t,y))- D_T\, \partial _y \bar{f}_+(t,y) - L'(t)\,L(t)\,y\,\bar{f}_+(t,y), \\[5pt] \bar{J}_-(t,y) &= -L(t)\,v_0\,\bar{f}_-(t,y)\,(1- \bar{\rho }(t,y))- D_T\, \partial _y \bar{f}_-(t,y) - L'(t)\,L(t)\,y\,\bar{f}_-(t,y). \end{align*}

Note that the fluxes $J_+$ and $\bar{J}_+$ are related to each other via

\begin{align*} & J_+(t, y\,L(t))- L'(t)\, y\, f_+(t, y\,L(t)) \\[5pt] &\qquad = v_0 \,f_+(t, y\,L(t))\, (1- \rho (t, y\,L(t)))- D_T\, \partial _x f_+(t, y\,L(t))- L'(t)\,y\,f_+(t, y\,L(t)) \\[5pt] &\qquad = v_0 \,\bar{f}_+(t,y)\,(1- \bar{\rho }(t, y))- \frac{D_T}{L(t)}\, \partial _y \bar{f}_+(t,y)- L'(t) \,y\,\bar{f}_+(t,y) \\[5pt] &\qquad =\frac{1}{L(t)}\, \big (L(t)\, v_0\, \bar{f}_+(t,y)\,(1- \bar{\rho }(t,y)) - D_T\, \partial _y \bar{f}_+(t,y)- L'(t)\,y\,\bar{f}_+(t,y)\big ) \\[5pt] &\qquad = \frac{1}{L(t)} \,\bar{J}_+(t,y), \end{align*}

and a similar relation can be deduced for $J_-$ and $\bar{J}_-$ . The boundary conditions (2.3) in the reference configuration then read

(3.4) \begin{equation} \begin{aligned} \bar{J}_+(t,0) &= L(t)\,\alpha _+(\Lambda _{\text{som}}(t))\,g_+(\bar{\boldsymbol{f}}(t,0)), \\[5pt] \bar{J}_+(t,1) &= L(t)\,\beta _+(\Lambda (t))\,\bar{f}_+(t, 1), \\[5pt] -\bar{J}_-(t,0) &= L(t)\,\alpha _-(\Lambda _{\text{som}}(t))\,g_-(\bar{\boldsymbol{f}}(t,0)), \\[5pt] -\bar{J}_-(t,1) &= L(t)\,\beta _-(\Lambda (t))\,\bar{f}_+(t, 1). \end{aligned} \end{equation}

The ODE (2.4) remains unchanged, while for (2.5) quantities are evaluated at $y=1$ instead of $x=L_j(t)$ , which results in

(3.5) \begin{equation} \begin{aligned} \Lambda '_{\text{som}}(t) &= \sum _{j=1,2} \bigl ( \beta _{-,j}(\Lambda _{\text{som}}(t))\,\bar{f}_{-,j}(t,0) - \alpha _{+,j}(\Lambda _{\text{som}}(t))\,g_{+,j}(\bar{\boldsymbol{f}}_{j}(t,0)) \bigr ) + \gamma _{\text{prod}}(t), \\[5pt] \Lambda '_{1}(t) &= \beta _{+,1}(\Lambda _1(t))\,\bar{f}_{+,1}(t, 1)- \alpha _{-,1}(\Lambda _1(t))\, g_{-,1}(\bar{\boldsymbol{f}}_{1}(t, 1)) -\chi \, h_1(\Lambda _1(t), L_1(t)), \\[5pt] \Lambda '_{2}(t) &= \beta _{+,2}(\Lambda _2(t))\,\bar{f}_{+,2}(t, 1)- \alpha _{-,2}(\Lambda _2(t))\,g_{-,2}(\bar{\boldsymbol{f}}_{2}(t, 1)) -\chi \, h_2(\Lambda _2(t), L_2(t)), \end{aligned} \end{equation}

for $t\in (0,T)$ .

3.3. Notion of weak solution and existence result

We now define the notion of weak solution to our problem. Whenever not differently specified, we assume $i \in \{+,-\}$ as well as $j \in \{1,2\}$ , $k \in \{1,2, \text{som}\}$ , while $C\gt 0$ denotes a constant that may change from line to line but always depends only on the data. From now on, we always write $f_{i,j}$ instead of $\bar{f}_{i,j}$ as we always work with the equations on the reference interval.

We next state the assumptions on the data and non-linearities which read as follows:

1. (H₀) $\Lambda _k^0\gt 0$ and $L_j^0 \ge L_{\text{min}, j}$ , where $L_{\text{min}, j}\gt 0$ is given.

2. (H₁) For $f_{+,j}^0, f_{-,j}^0 \in L^2(0,1)$ it holds $ f_{+,j}^0, f_{-,j}^0 \ge 0$ and $0 \le \rho _j^0 \le 1$ a.e. in $(0,1)$ , where $\rho _j^0\;:\!=\; f_{+,j}^0+ f_{-,j}^0$ .

3. (H₂) The non-linearities $g_{i,j}\colon{\mathbb{R}}^2 \to{\mathbb{R}}_+$ , are Lipschitz continuous and such that $g_{i,j}(s,t)= 0$ whenever $s+t = 1$ as well as $g_{-,j}(s,0) = g_{+,j}(0,s) = 0$ for all $0 \le s \le 1$ .

4. (H₃) The functions $h_j\colon{\mathbb{R}}_+ \times [ L_{\text{min}, j},+\infty ) \to{\mathbb{R}}$ are such that

   1. (i) there exist non-negative functions $K_{h_j} \in L^\infty ((0,\infty ))$ such that

      \begin{equation*} |h_j(t,a) - h_j(t,b)| \le K_{h_j}(t)|a-b|, \end{equation*}
      for all $a,\,b \in [ L_{\text{min}, j},+\infty )$ ;

   2. (ii) $h_j(s, L_{\text{min}, j})\ge 0$ for every $s\ge 0$ .

5. (H₄) The functions $\alpha _{i,j}\colon{\mathbb{R}}_+ \to{\mathbb{R}}_+$ are increasing and Lipschitz continuous. Moreover, $\alpha _{-,j}(t) \ge 0$ for all $t\gt 0$ and $\alpha _{-,j}(0)= 0$ .

6. (H₅) The functions $\beta _{i,j}\colon{\mathbb{R}}_+ \to{\mathbb{R}}_+$ are nonnegative and Lipschitz continuous. Moreover, there exists $\Lambda _{j, \text{max}}\gt 0$ such that $\beta _{+,j}(\Lambda _{j, \text{max}})= 0$ .

7. (H₆) The parameters satisfy $ v_0, D_T, \chi \gt 0$ and $ \lambda \ge 0$ .

8. (H₇) The function $\gamma _{\text{prod}} \colon{\mathbb{R}}_+ \to{\mathbb{R}}_+$ is such that $\lim _{t\to \infty } \gamma _{\text{prod}}(t) =0$ .

Then we can state our existence result.

3.4. Proof of Theorem 3.3

The proof of Theorem3.3 is based on a fixed point argument applied to an operator obtained by concatenating linearised versions of (3.6), (2.4) and (3.5). Let us briefly sketch our strategy before we provide the corresponding rigorous results. We work in the Banach space

\begin{equation*} X= \prod _{j=1,2} L^2(0,T; H^1(0, 1))^2 \end{equation*}

endowed with the norm

\begin{equation*} \|(\boldsymbol{f}_1, \boldsymbol{f}_2 )\|^2_X= \sum _{j=1,2} \sum _{i=+,-} \|f_{i,j}\|^2_{L^2(0,T; H^1(0, 1))}. \end{equation*}

Given $(\widehat{\boldsymbol{f}}_1, \widehat{\boldsymbol{f}}_2 ) \in X$ , let $\boldsymbol{\Lambda }\;:\!=\;(\Lambda _{\text{som}}, \Lambda _1, \Lambda _2) \in C^1([0,T])^3$ be the unique solution to the ODE system

(3.7) \begin{equation} \begin{aligned} \Lambda _{\text{som}}'(t) &= \sum _{j=1,2} \bigl ( \beta _{-,j}(\Lambda _{\text{som}}(t)) \widehat{f}_{-,j}(t,0) - \alpha _{+,j}(\Lambda _{\text{som}}(t))\, g_{+,j}(\widehat{\boldsymbol{f}}_{j}(t,0)) \bigr ) + \gamma _{\text{prod}}(t), \\[5pt] \Lambda _{1}'(t) &= \beta _{+,1}(\Lambda _1(t))\,\widehat{f}_{+,1}(t, 1)- \alpha _{-,1}(\Lambda _1(t))\,g_{-,1}(\widehat{\boldsymbol{f}}_{1}(t, 1)) - \chi \, h_1(\Lambda _1(t), L_1(t)), \\[5pt] \Lambda _{2}'(t) &= \beta _{+,2}(\Lambda _2(t))\,\widehat{f}_{+,2}(t, 1)- \alpha _{-,2}(\Lambda _2(t))\,g_{-,2}(\widehat{\boldsymbol{f}}_{2}(t, 1)) - \chi \,h_2(\Lambda _2(t), L_2(t)). \end{aligned} \end{equation}

We denote the mapping $(\widehat{\boldsymbol{f}}_1, \widehat{\boldsymbol{f}}_2) \mapsto \boldsymbol{\Lambda }$ by $\mathcal{B}_1$ . This $\boldsymbol{\Lambda }$ is now substituted into (2.4), that is, we are looking for the unique solution $\boldsymbol{L}=(L_1, L_2) \in C^1([0,T])^2$ to the ODE problems

(3.8) \begin{equation} L_j'(t)= h_j(\Lambda _j(t), L_j(t)), \end{equation}

and the corresponding solution operator is denoted by $\mathcal{B}_2$ . Finally, these solutions $\boldsymbol{\Lambda }$ and $\boldsymbol{L}$ are substituted into (3.6), and we look for the unique solution $\boldsymbol{f}_j \in L^2(0, T; H^1(0, 1))^2$ , with $\partial _t \boldsymbol{f}_j \in L^2(0, T; H^1(0, 1)^{\prime })^2$ , to the (still non-linear) PDE problem

(3.9) \begin{align} & \int _0^{1} \partial _t f_{+,j} \,\varphi _+ \,\textrm{d}y = \int _0^{1} \frac{1}{L(t)^2}\left [L(t)\,v_0\,f_{+,j}\,(1- \rho _j)- D_T\,\partial _y f_{+,j} - L'(t)\,L(t)\,y\,f_+ \right ] \partial _y \varphi _+ \\[5pt] &\quad +\left (\lambda \, (f_{-,j}- f_{+,j}) - \frac{L'(t)}{L(t)}\,f_{+,k}\right )\, \varphi _+ \,\textrm{d}y \nonumber \\[5pt] & + \frac{1}{L(t)}\left [\alpha _{+,j}(\Lambda _{\text{som}})\,g_{+,j}(\boldsymbol{f}_{j}(t, 0))\,\varphi _+(0) - \beta _{+,j}(\Lambda _j)\,f_{+,j}(t, 1)\,\varphi _+(1)\right ], \nonumber \end{align}

(3.10) \begin{align} & \int _0^{ 1} \partial _t f_{-,j}\,\varphi _- \,\textrm{d}y = \int _0^{ 1} - \frac{1}{L(t)^2}\left [L(t)\,v_0\,f_{-,j}(1- \rho _j)+ D_T\,\partial _y f_{-,j} - L'(t)\,L(t)\,y\,f_-\right ]\, \partial _y \varphi _-\\[5pt] &\quad +\left (\lambda \, (f_{+,j}- f_{-,j}) - \frac{L'(t)}{L(t)}\,f_{-,k}\right )\, \varphi _- \,\textrm{d}y \nonumber \\[5pt] &-\frac{1}{L(t)}\left [\beta _{-,j}(\Lambda _{\text{som}}(t)) \,f_{-,j}(t, 0)\,\varphi _-(0) - \alpha _{-,j}(\Lambda _j)\,g_{-,j}(\boldsymbol{f}_{j}(t, 1))\,\varphi _-(1)\right ],\nonumber \end{align}

for every $\varphi _+, \varphi _- \in H^1(0, 1)$ . We call the resulting solution operator $(\boldsymbol{\Lambda }, \boldsymbol{L}) \mapsto (\boldsymbol{f}_1, \boldsymbol{f}_2 )$ $\mathcal{B}_3$ . Then, given an appropriate subset $\mathcal{K} \subset X$ , we define the (fixed point) operator $\mathcal{B} \colon \mathcal{K} \to X$ as

\begin{equation*} \mathcal {B}(\widehat {\boldsymbol{f}}_1, \widehat {\boldsymbol{f}}_2 )= \mathcal {B}_3 \bigl (\mathcal {B}_1(\widehat {\boldsymbol{f}}_1, \widehat {\boldsymbol{f}}_2), \mathcal {B}_2(\mathcal {B}_1(\widehat {\boldsymbol{f}}_1, \widehat {\boldsymbol{f}}_2 )\bigr )= (\boldsymbol{f}_1, \boldsymbol{f}_2 ). \end{equation*}

We show that $\mathcal{B}$ is self-mapping and contractive, so that existence is a consequence of the Banach’s fixed point theorem.

Let us begin with system (3.7).

Lemma 3.4. Let $(\widehat{\boldsymbol{f}}_1, \widehat{\boldsymbol{f}}_2) \in X$ , then, there exists a unique $\boldsymbol{\Lambda }= (\Lambda _{\textit{som}}, \Lambda _1, \Lambda _2) \in C^1([0, T])^3$ that solves (3.7) with initial conditions

(3.11) \begin{equation} \Lambda _k(0)= \Lambda _k^{0}, \quad k= \textit{som}, 1, 2. \end{equation}

Proof. From Lemma3.4, $\mathcal{B}_1$ is well defined. Let now $(\widehat{\boldsymbol{f}}_1^{(1)},\widehat{\boldsymbol{f}}_2^{(1)}),$ $(\widehat{\boldsymbol{f}}_1^{(2)}, \widehat{\boldsymbol{f}}_2^{(2)} ) \in X$ and let $\boldsymbol{\Lambda }^{(1)}=: \mathcal{B}_1(\widehat{\boldsymbol{f}}_1^{(1)}, \widehat{\boldsymbol{f}}_2^{(1)})$ and $\boldsymbol{\Lambda }^{(2)}=: \mathcal{B}_1(\widehat{\boldsymbol{f}}_1^{(2)}, \widehat{\boldsymbol{f}}_2^{(2)})$ be solutions to (3.7) satisfying the same initial condition (3.11). We fix $t \in [0, T]$ and consider

\begin{equation*} \begin{split} (\Lambda _{j}^{(a)})'(t)&= \beta _{+,j}(\Lambda _j^{(a)}(t)) \,\widehat{f}_{+,j}^{(a)}(t, 1)- \alpha _{-,j}(\Lambda _j^{(a)}(t)) \,g_{-,j}(\widehat{\boldsymbol{f}}_{j}^{(a)}(t, 1)) \\[5pt] & \qquad - \chi \, h_j(\Lambda _j^{(a)}(t), 1), \end{split} \end{equation*}

$a=1,2$ . Taking the difference of the two equations, setting $\delta \Lambda _j\;:\!=\; \Lambda _j^{(1)}- \Lambda _j^{(2)}$ , exploiting hypotheses (H $_2$ )–(H $_5$ ) and summarising the constants give

\begin{equation*} \begin{split} & |(\delta \Lambda _j)'(t)| \le C |\delta \Lambda _j(t)| + C \left (|(\widehat{f}_{+,j}^{(1)}- \widehat{f}_{+,j}^{(2)})(t, 1)|+ |(\widehat{f}_{-,j}^{(1)}- \widehat{f}_{-,j}^{(2)})(t, 1)| \right ), \end{split} \end{equation*}

while the trace inequality (3.1) and a Gronwall argument imply

\begin{align*} |\Lambda _j^{(1)}(t)- \Lambda _j^{(2)}(t)| & \le C\, \|\widehat{\boldsymbol{f}}_j^{(1)}- \widehat{\boldsymbol{f}}_j^{(2)} \|_{L^2(0, T; H^1(0, 1))^2} . \end{align*}

A similar argument holds for the equation for $\Lambda _{\text{som}}$ , and we eventually have

(3.12) \begin{equation} \|\boldsymbol{\Lambda }^{(1)}- \boldsymbol{\Lambda }^{(2)}\|_{C([0, T])^3} \le C \,\|(\widehat{\boldsymbol{f}}_1^{(1)}, \widehat{\boldsymbol{f}}_2^{(1)})- (\widehat{\boldsymbol{f}}_1^{(2)}, \widehat{\boldsymbol{f}}_2^{(2)})\|_X. \end{equation}

We next show the following existence result for equation (3.8).

Lemma 3.6. Let $\boldsymbol{\Lambda } \in C^1([0, T])^3$ be the unique solution to (3.7). Then, there exists a unique $\boldsymbol{L}= (L_1, L_2) \in C^1([0, T])^2$ that solves (3.8) with initial condition $ L_j(0)= L_j^{(0)}$ . Furthermore, $\text{for all } t \in (0, T)$ it holds

(3.13) \begin{align} & L_{\textit{min}, j} \le L_j(t) \le L_j^{(0)}+ T \,\|h_j\|_{L^{\infty }({\mathbb{R}}^2)}, \end{align}

(3.14) \begin{align} & \frac{|L_j'(t)|}{L_j(t)} \le \frac{\|h_j\|_{L^{\infty }({\mathbb{R}}^2)}}{L_{\textit{min}, j}}. \end{align}

Proof. The existence and uniqueness follow as before. The lower bound in (3.13) can be deduced by applying TheoremA.1 ( [Reference Deimling10, Theorem 5.1]) in the appendix with $X={\mathbb{R}}$ , $\Omega = [L_{\text{min}, j},\infty )$ and $f=h_j$ . Assumption (A1) in the theorem is satisfied as, due to (H $_3)$ , the choice $\omega = K_{h_j}$ fulfils the requirements. For (A2), we note that the unit outward normal of the set $[L_{\text{min}, j},\infty )$ at $L_{\text{min},j}$ is $-1$ and that $h_j(s, L_{\text{min}, j})\ge 0$ for every $s\ge 0$ (again by (H $_3$ )). This yields $(h_j(s, L_{\text{min}, j}),-1) = -h_j(s, L_{\text{min}, j}) \le 0$ as needed. In order to prove the upper bound in (3.13), we fix $t \in (0,T)$ , integrate (3.8) and use (H $_3$ )-(i) to have

\begin{equation*} L_j(t)= L_j^{(0)}+ \int _0^t h_j(\Lambda _j(s), L_j(s)) \,{\,\textrm {d}s} \le L_j^{(0)}+ T\, \|h_j\|_{L^{\infty }({\mathbb {R}}^2)}. \end{equation*}

In addition, we observe that (3.8) gives $|L_j'(t)| \le \|h_j\|_{L^{\infty }({\mathbb{R}}^2)}$ . Then, the fact that $L_j(t) \ge L_{\text{min}, j}$ allows us to conclude (3.14).

Lemma 3.7. The operator $\mathcal{B}_2 \colon C([0, T])^3 \to C([0, T])^2$ that maps $\boldsymbol{\Lambda }$ to the solution $\boldsymbol{L}$ to (3.8) is Lipschitz continuous in the sense of

(3.15) \begin{equation} \|\boldsymbol{L}^{(1)}- \boldsymbol{L}^{(2)}\|_{C([0, T])^2} \le T\, \max _{j= 1,2}\bigl \{L_{h_j}\, e^{2\,T\,L_{h_j}} \bigr \} \|\boldsymbol{\Lambda }^{(1)}- \boldsymbol{\Lambda }^{(2)}\|_{C([0, T])^3}, \end{equation}

where $L_{h_j}$ is the Lipschitz constant of $h_j$ . If $T \,\max _{j= 1,2}\bigl \{L_{h_j}\, e^{2\,T\,L_{h_j}} \bigr \}\lt 1$ , then $\mathcal{B}_2$ is contractive.

We next investigate the existence of solutions to system (3.9)–(3.10).

Proof. To simplify the notation, in this proof we will drop the use of the $j$ -index and, for the reader’s convenience, we split the proof in several steps.

Step 1: Approximation by truncation. Given a generic function $a$ we introduce the truncation

(3.16) \begin{equation} a_{\text{tr}}= \begin{cases} a & \text{if } 0 \le a \le 1, \\[5pt] 0 & \text{otherwise}. \end{cases} \end{equation}

We apply this to the non-linear transport terms $f_{\pm }\,(1-\rho )$ in (3.6) which yields (after summing up)

(3.17) \begin{equation} \begin{split} &\sum _{i= +,-}\int _0^1 \partial _t f_i \,\varphi _i \,\textrm{d}y+ \frac{D_T}{L^2(t)} \,\sum _{i= +,-} \int _0^ 1 \partial _y f_i \, \partial _y \varphi _i \,\textrm{d}y \\[5pt] &= \int _0^1 \frac{v_0}{L(t)} (f_+ \,(1-\rho ))_{\text{tr}} \, \partial _y \varphi _+ - \frac{v_0}{L(t)}\, (f_-\,(1-\rho ))_{\text{tr}} \, \partial _y \varphi _- \,\textrm{d}y \\[5pt] & \quad - \frac{L'(t)}{L(t)}\, \sum _{i= +,-} \int _0^1 \left (y \,f_i \,\partial _y \varphi _i + f_i\, \varphi _i \right )\,\textrm{d}y+ \int _0^1 \lambda \,[(f_- - f_+)\, \varphi _+ + (f_+- f_-)\,\varphi _-] \,\textrm{d}y \\[5pt] & \quad +\frac{1}{L(t)} \Big (\!-\! \beta _{+}(\Lambda (t)) \, f_{+}(t, 1)\,\varphi _+(1) + \alpha _{+}(\Lambda _{\text{som}}(t))\, g_{+}(\boldsymbol{f}(t, 0))\,\varphi _+(0) \\[5pt] & \quad \qquad +\alpha _{-}(\Lambda (t))\, g_{-}(\boldsymbol{f}(t, 1))\,\varphi _-(1) - \beta _{-}(\Lambda _{\text{som}}(t)) \, f_{-}(t, 0)\,\varphi _-(0)\Big ). \end{split} \end{equation}

We solve (3.17) by means of the Banach fixed point theorem. We follow [Reference Egger, Pietschmann and Schlottbom12], pointing out that a similar approach has been used also in [Reference Marino, Pietschmann and Pichler22], yet in a different context.

Let us set $Y\;:\!=\; (L^{\infty }((0,T); L^2(0,1)))^2$ and introduce the following nonempty, closed set

\begin{equation*} \mathcal {M}= \left \{{\boldsymbol{f}}= (f_+, f_-) \in Y: \, \|{\boldsymbol{f}}\|_{Y} \le C_{\mathcal {M}}\right \}, \end{equation*}

with $T, C_{\mathcal{M}}\gt 0$ to be specified. Then we define the mapping $\Phi \colon \mathcal{M} \to Y$ such that $\Phi (\widetilde{\boldsymbol{f}})={\boldsymbol{f}}$ where, for fixed $\widetilde{\boldsymbol{f} }\in \mathcal{M}$ , $\boldsymbol{f}$ solves the following linearised equation (cf. [Reference Lieberman21, Chapter III])

(3.18) \begin{equation} \begin{split} &\sum _{i= +,-}\int _0^1 \partial _t f_i \,\varphi _i \,\textrm{d}y+ \frac{D_T}{L^2(t)} \,\sum _{i= +,-} \int _0^ 1 \partial _y f_i \, \partial _y \varphi _i \,\textrm{d}y \\[5pt] &= \int _0^1 \frac{v_0}{L(t)} \,(\widetilde{f}_+\, (1-\widetilde{\rho }))_{\text{tr}} \, \partial _y \varphi _+ - \frac{v_0}{L(t)} \,(\widetilde{f}_-\,(1-\widetilde{\rho }))_{\text{tr}} \, \partial _y \varphi _- \,\textrm{d}y \\[5pt] & \quad - \frac{L'(t)}{L(t)} \, \sum _{i= +,-} \int _0^1 y\, f_i \,\partial _y \varphi _i + f_i \,\varphi _i \,\textrm{d}y+ \int _0^1 \lambda \, [(f_- - f_+)\, \varphi _+ +(f_+- f_-)\,\varphi _-] \,\textrm{d}y \\[5pt] & \quad +\frac{1}{L(t)}\, \Big (\!-\!\beta _{+}(\Lambda (t)) \, f_{+}(t, 1)\,\varphi _+(1) + \alpha _{+}(\Lambda _{\text{som}}(t))\, g_{+}(\boldsymbol{f}(t, 0))\,\varphi _+(0) \\[5pt] & \quad \qquad +\alpha _{-}(\Lambda (t))\, g_{-}(\boldsymbol{f}(t, 1))\,\varphi _-(1) - \beta _{-}(\Lambda _{\text{som}}(t)) \, f_{-}(t, 0)\,\varphi _-(0)\Big ). \end{split} \end{equation}

$\underline{\textrm{Step 2:}\; \Phi\; \textrm{is self-mapping.}}$ We show that

(3.19) \begin{equation} \|{\boldsymbol{f}}\|_Y \le C_{\mathcal{M}}. \end{equation}

We choose $\varphi _i= f_i$ in (3.18) and estimate the several terms appearing in the resulting equation separately. From (3.13), on the left-hand side we have

\begin{equation*} \begin{split} & \frac{1}{2} \frac{\textrm{d}}{\,\textrm{d}t} \sum _{i= +,-} \int _0^1 |f_i|^2 \,\textrm{d}y+ \frac{D_T}{L^2(t)} \sum _{i= +,-} \int _0^ 1 |\partial _y f_i|^2 \,\textrm{d}y \\[5pt] & \ge \frac{1}{2} \frac{\textrm{d}}{\,\textrm{d}t} \sum _{i= +,-} \int _0^1 |f_i|^2 \,\textrm{d}y+ \frac{D_T}{(L^{(0)}+ T\, \|h\|_{L^\infty ({\mathbb{R}})})^2} \sum _{i= +,-} \int _0^ 1 |\partial _y f_i|^2 \,\textrm{d}y. \end{split} \end{equation*}

On the right-hand side we first use equation (3.14) along with Young’s inequality for some $\varepsilon _1\gt 0$ and the fact that $y \in (0,1)$ to achieve

\begin{equation*} \sum _{i= +,-} \int _0^1 \frac {L'(t)}{L(t)} \,y\,f_i\,\partial _y f_i \,\textrm {d}y \le \sum _{i= +,-} \left (\varepsilon _1 \,\|\partial _y f_i\|_{L^2(0,1)}^2+ \frac {\|h\|_{L^\infty ({\mathbb {R}})}^2}{2\,\varepsilon _1 \,L_{\text {min}}^2} \|f_i\|_{L^2(0,1)}^2 \right ), \end{equation*}

while (3.14) once again gives

\begin{equation*} -\frac {L'(t)}{L(t)} \sum _{i=+,-} \int _0^1 f_i^2 \,\textrm {d}y \le \frac {\|h\|_{L^\infty ({\mathbb {R}}^2)}}{L_{\text {min}}} \sum _{i=+,-} \|f_i\|_{L^2(0,1)}^2. \end{equation*}

On the other hand, (3.13), Young’s inequality for some $\varepsilon _2\gt 0$ and (3.16) give

\begin{equation*} \pm \int _0^1 \frac {v_0}{L(t)} \bigl (\widetilde {f}_i \,(1- \widetilde {\rho })\bigr )_{\text {tr}}\, \partial _y f_i \,\textrm {d}y \le C + \varepsilon _2 \|\partial _y f_i\|_{L^2(0,1)}^2. \end{equation*}

We further observe that

\begin{equation*} \lambda \int _0^1 (f_- - f_+)\, f_+ + (f_+ - f_-)\,f_- \,\textrm {d}y= -\lambda \int _0^1 (f_+ + f_-)^2 \,\textrm {d}y \le 0. \end{equation*}

Finally we estimate the boundary terms. We use hypotheses (H $_2$ ), (H $_4$ ), (H $_5$ ) and equation (3.13) together with Young’s inequality with some $\varepsilon _3, \dots, \varepsilon _6\gt 0$ and the trace inequality (3.1) to achieve

\begin{equation*} \begin{split} \frac{1}{L(t)}\, \beta _+(\Lambda (t)) \,f_+^2(t, 1) & \le C \,\|f_+\|_{L^2(0,1)}^2+ \varepsilon _3 \, \|\partial _y f_+\|_{L^2(0,1)}^2, \\[5pt] \frac{1}{L(t)} \,\alpha _{+}(\Lambda _{\text{som}}(t))\,g_{+}(\widetilde{\boldsymbol{f}}_{+}(t, 0))\,f_+(t, 0) & \le C\,\|f_+\|_{L^2(0,1)}^2+ \varepsilon _4 \, \|\partial f_+\|^2_{L^2(0,1)}, \\[5pt] \frac{1}{L(t)} \,\alpha _{-}(\Lambda (t))\,g_{-}( \widetilde{\boldsymbol{f}}_{+}(t, 1))\,f_-(t,1) & \le C \, \|f_-\|_{L^2(0,1)}^2 + \varepsilon _5\,\|\partial _y f_-\|^2_{L^2(0,1)}, \\[5pt] \frac{1}{L(t)} \, \beta _{-}(\Lambda _{\text{som}}(t)) \,f_{-}^2(t, 0) & \le C \, \|f_-\|_{L^2(0,1)}^2 + \varepsilon _6\, \|\partial _y f_-\|_{L^2(0,1)}^2. \end{split} \end{equation*}

We choose $\varepsilon _{\kappa }$ , $\kappa =1, \dots, 6$ , in such a way that all the terms of the form $\|\partial _y f_i\|_{L^2(0,1)}$ can be absorbed on the left-hand side of (3.18), which simplifies to

\begin{equation*} \frac {\textrm {d}}{\,\textrm {d}t} \sum _{i= +,-} \int _0^1 |f_i|^2 \,\textrm {d}y \le C \sum _{i= +,-} \|f_i\|_{L^2(0,1)}^2+ C. \end{equation*}

We then use a Gronwall argument to infer

\begin{equation*} \sup _{t \in (0, T)} \|f_i(t, \cdot )\|_{L^2(0,1)}^2 \le C=: C_{\mathcal {M}}^2. \end{equation*}

This implies that (3.19) is satisfied and therefore $\Phi$ is self-mapping.

$\underline{\textrm{Step 3:}\; \Phi \textrm{is a contraction.}}$ Let $\widetilde{\boldsymbol{f}}_1, \widetilde{\boldsymbol{f}}_2 \in \mathcal{M}$ and let ${\boldsymbol{f}_1}=: \Phi (\widetilde{\boldsymbol{f}}_1)$ and ${\boldsymbol{f}_2}=: \Phi (\widetilde{\boldsymbol{f}}_2)$ be two solutions to (3.18) with the same initial datum $\boldsymbol{f}^0$ . We then consider the difference of the corresponding equations and choose $\varphi _i= f_{i, 1}- f_{i, 2}$ .

Reasoning as in Step 2 and exploiting the Lipschitz continuity of the functions

(3.20) \begin{equation} {\mathbb{R}}^2 \ni (a,b) \mapsto (a\,(1- a- b))_{\text{tr}} \quad \text{and} \quad{\mathbb{R}}^2 \ni (a,b) \mapsto (b\,(1- a- b))_{\text{tr}} \end{equation}

we get

\begin{equation*} \begin{split} & \frac{\textrm{d}}{\,\textrm{d}t} \sum _{i= +,-} \|f_{i, 1}- f_{i, 2}\|_{L^2(0,1)}^2 \le C \sum _{i= +,-} \Big (\|f_{i, 1}- f_{i, 2}\|_{L^2(0,1)}^2+\|\widetilde{f}_{i, 1}- \widetilde{f}_{i, 2}\|_{L^2(0,1)}^2\Big ). \end{split} \end{equation*}

Again by means of a Gronwall argument we have

\begin{equation*} \sum _{i= +,-} \|(f_{i, 1}- f_{i, 2})(t, \cdot )\|_{L^2(0,1)}^2 \le C\,T\, e^{CT} \sum _{i= +,-} \|\widetilde {f}_{i, 1}- \widetilde {f}_{i, 2}\|_{L^{\infty }(0,T; L^2(0,1))}^2, \end{equation*}

and then $\Phi$ is a contraction if $T\gt 0$ is small enough so that $ C\, T\, e^{C\, T}\lt 1$ . Then Banach’s fixed point theorem applies and we obtain a solution $\boldsymbol{f} \in (L^{\infty }(0,T; L^2(0,1)))^2$ to (3.17). A standard regularity theory then gives $\boldsymbol{f} \in (L^2(0,T; H^1(0,1)))^2$ , with $\partial _t \boldsymbol{f} \in (L^2(0,T; H^1(0,1)')^2$ .

Step 4: Box constraints. We show that such $\boldsymbol{f}$ obtained in Step 4 is actually a solution to (3.6), because it satisfies the box constraint $ f_+, f_- \ge 0$ and $\rho \le 1$ .

We start by showing that $f_+ \ge 0$ , and to this end we consider only the terms involving the $\varphi _+$ -functions in (3.17), that is,

(3.21) \begin{equation} \begin{split} &\int _0^1 \partial _t f_+\,\varphi _+ \,\textrm{d}y+ \frac{D_T}{L^2(t)}\, \int _0^1 \partial _y f_+ \, \partial _y \varphi _+ \,\textrm{d}y \\[5pt] &= \int _0^1 \Big ( \frac{v_0}{L(t)} \, \bigl (f_+ \,(1- \rho )\bigr )_{\text{tr}} - \frac{L'(t)}{L(t)}\, y\, f_+ \Big ) \partial _y \varphi _+ + \Big (\lambda \,( f_- - f_+)- \frac{L'(t)}{L(t)} \,f_+ \Big )\,\varphi _+ \,\textrm{d}y \\[5pt] & \qquad +\frac{1}{L(t)}\, \Big (\!-\! \beta _{+}(\Lambda (t))\,f_{+}(t, 1) \, \varphi _+(1)+ \alpha _{+}(\Lambda _{\text{som}}(t))\,g_{+}( \boldsymbol{f}(t, 0))\,\varphi _+(0)\Big ). \end{split} \end{equation}

For every $\varepsilon \gt 0$ we consider the function $\eta _\varepsilon \in W^{2, \infty }({\mathbb{R}})$ given by

(3.22) \begin{equation} \eta _\varepsilon (u)= \begin{cases} 0 & \text{if } u \le 0, \\[5pt] \displaystyle \frac{u^2}{4\varepsilon } & \text{if } 0\lt u \le 2\varepsilon, \\[5pt] u- \varepsilon & \text{if } u \gt 2\varepsilon, \end{cases} \qquad \text{with} \quad \eta ''_\varepsilon (u)= \begin{cases} 0 & \text{if } u \le 0, \\[5pt] \displaystyle \frac{1}{2\varepsilon } & \text{if } 0\lt u \le 2\varepsilon, \\[5pt] 0 & \text{if } u \gt 2\varepsilon . \end{cases} \end{equation}

Next we choose $\varphi _+= -\eta _\varepsilon '(\!-\!f_+)$ and observe that by the chain rule we have $\partial _y \varphi _+= \eta _\varepsilon ''(\!-\!f_+)\,\partial _y f_+$ . Using such $\varphi _+$ in (3.21) gives

\begin{equation*} \begin{split} & -\int _0^1 \partial _t f_+\, \eta _\varepsilon '(\!-\!f_+) \,\textrm{d}y+ \frac{D_T}{L(t)^2} \int _0^ 1 \eta _\varepsilon ''(\!-\!f_+)\, |\partial _y f_+|^2 \,\textrm{d}y \\[5pt] & = \int _0^1 \Big (\frac{v_0}{L(t)} \, \bigl (f_+\, (1- \rho )\bigr )_{\text{tr}} - \frac{L'(t)}{L(t)}\, y\, f_+ \Big ) \,\eta ''_\varepsilon (\!-\!f_+) \,\partial _y f_+ - \Big (\lambda \,(f_- - f_+) - \frac{L'(t)}{L(t)} \,f_+ \Big )\, \eta '_\varepsilon (\!-\!f_+) \,\textrm{d}y \\[5pt] & \qquad + \frac{1}{L(t)} \big ( \beta _+(\Lambda (t))\, f_+(t,1)\, \eta _\varepsilon '(\!-\!f_+(t, 1)) -\alpha _+(\Lambda _{\text{som}}(t)) \,g_+(\boldsymbol{f}(t,0))\,\eta _\varepsilon '(\!-\!f_+(t,0))\big ). \end{split} \end{equation*}

Thanks to Young’s inequality with a suitable $\kappa \gt 0$ and to (3.16) we have

\begin{equation*} \begin{split} &\int _0^1 \frac{v_0}{L(t)}\, \bigl (f_+\,(1- \rho )\bigr )_{\text{tr}}\, \eta _\varepsilon ''(\!-\!f_+)\,\partial _y f_+ \,\textrm{d}y \\[5pt] &\qquad \le \kappa \int _0^1 \eta _\varepsilon ''(\!-\!f_+)\,|\partial _y f_+|^2 \,\textrm{d}y+ \frac{1}{4\,\kappa } \int _0^1 \eta _\varepsilon ''(\!-\!f_+) \,\Big (\frac{v_0}{L(t)}\Big )^2 \,\bigl (f_+\, (1- \rho )\bigr )_{\text{tr}}^2 \,\textrm{d}y, \end{split} \end{equation*}

as well as

\begin{align*} & -\int _0^1 \frac{L'(t)}{L(t)}\, y\, f_+ \eta _\varepsilon ''(\!-\!f_+)\, \partial _y f_+ \,\textrm{d}y \\[5pt] & \qquad \le \kappa \int _0^1 \eta _\varepsilon ''(\!-\!f_+)\, |\partial _y f_+|^2 \,\textrm{d}y+ \frac{1}{\kappa } \,\Big (\frac{L'(t)}{L(t)}\Big )^2 \int _0^1 \eta _\varepsilon ''(\!-\!f_+)\, f_+^2 \,\textrm{d}y, \end{align*}

where $\kappa$ depends on the lower and upper bounds on $L(t)$ , see Lemma3.6. Choosing $\kappa$ sufficiently small and taking into account that the term involving $\alpha _+$ is non-negative we obtain

(3.23) \begin{equation} \begin{split} & \frac{\textrm{d}}{\,\textrm{d}t} \int _0^1 \eta _\varepsilon (\!-\!f_+) \,\textrm{d}y = -\int _0^1 \partial _t f_+ \,\eta _\varepsilon '(\!-\!f_+) \,\textrm{d}y \\[5pt] & \le C \int _0^1 \, \eta _\varepsilon ''(\!-\!f_+) \Big [\Big (\frac{v_0}{L(t)}\Big )^2\, \bigl (f_+\, (1- \rho )\bigr )_{\text{tr}}^2+ \Big (\frac{L'(t)}{L(t)}\Big )^2\, f_+^2\Big ] -\left (\lambda \,(f_- - f_+)- \frac{L'(t)}{L(t)}\, f_+ \right ) \eta '_\varepsilon (\!-\!f_+) \,\textrm{d}y \\[5pt] & \qquad + \frac{1}{L(t)}\, \beta _+(\Lambda (t))\,f_+(t,1)\,\eta _\varepsilon '(\!-\!f_+(t, 1)). \end{split} \end{equation}

To gain some sign information on the right-hand side of (3.23), we introduce the set

(3.24) \begin{equation} \Omega _\varepsilon \;:\!=\; \{y \in (0,1): \, 0 \lt f_+(t, y), f_-(t, y) \le 2\varepsilon \}. \end{equation}

We first use (3.13), (3.22), and the Lipschitz continuity of (3.20) to have

\begin{equation*} \begin{split} & \int _{\Omega _\varepsilon } \eta _\varepsilon ''(\!-\!f_+)\, \Big (\frac{v_0}{L(t)}\Big )^2 \,\bigl (f_+\,(1-\rho )\bigr )_{\text{tr}}^2\; \,\textrm{d}y \\[5pt] &= \int _{\Omega _\varepsilon } \eta _\varepsilon ''(\!-\!f_+)\, \Big (\frac{v_0}{L(t)}\Big )^2 \,\bigl (f_+\,(1-\rho )\bigr )_{\text{tr}} -(0 \cdot (1-\rho )\bigr )_{\text{tr}}\bigr ) ^2 \,\textrm{d}y \le \frac{v_0^2}{L_{\text{min}}^2}\,2 \varepsilon \, |\Omega _\varepsilon |=: \tilde{c}_1\,\varepsilon, \end{split} \end{equation*}

as well as

\begin{align*} \int _{\Omega _\varepsilon } \eta _\varepsilon ''(\!-\!f_+)\, \Big (\frac{L'(t)}{L(t)}\Big )^2\, f_+^2 \,\textrm{d}y \le \frac{\|h\|_{L^\infty ({\mathbb{R}}^2)}^2}{L_{\text{min}}^2}\, |\Omega _\varepsilon |\, 2 \varepsilon \;:\!=\; \tilde{c}_2\, \varepsilon, \end{align*}

and

\begin{align*} \frac{L'(t)}{L(t)} \int _{\Omega _\varepsilon } f_+ \,\eta _\varepsilon '(f_+) \,\textrm{d}y \le \frac{\|h\|_{L^\infty ({\mathbb{R}}^2)}^2}{L_{\text{min}}^2}\, |\Omega _\varepsilon |\, 2 \varepsilon = \tilde{c}_2\, \varepsilon . \end{align*}

On the other hand, from (3.24) it follows that $-2\varepsilon \lt f_+- f_- \le 2\varepsilon$ , which implies

\begin{equation*} -\int _{\Omega _\varepsilon } \lambda \,(f_- - f_+) \,\textrm {d}y \le 2\varepsilon \,\lambda \,|\Omega _\varepsilon |=: \tilde {c}_3\,\varepsilon . \end{equation*}

Concerning the boundary term, we note that by definition the function $\eta _\varepsilon '$ is nonzero only if its argument is nonnegative, which in any case gives

\begin{equation*} \frac {1}{L(t)}\, \beta _+(\Lambda )\,f_+(t,1)\, \eta _\varepsilon '(\!-\!f_+(t, 1)) \le 0. \end{equation*}

Summarising, from (3.23) we have

\begin{align*} \frac{\textrm{d}}{\textrm{d} t} \int _0^1 (f_+)_- \,\textrm{d}y= \lim _{\varepsilon \to 0} \frac{\textrm{d}}{\textrm{d} t} \int _{\Omega _\varepsilon } \eta _\varepsilon (\!-\!f_+) \,\textrm{d}y \le 0, \end{align*}

which implies

\begin{align*} \int _0^1 (f_+)_- \,\textrm{d}y \le \int _0^1 (f_+^0)_- \,\textrm{d}y= 0, \end{align*}

where the last equality holds thanks to assumption (H $_1$ ). It follows that $f_+ \ge 0$ . The proof that $f_- \ge 0$ works in a similar way, starting again from equation (3.17) and taking into account only the terms involving the $\varphi _-$ -functions.

We now show that $\rho \le 1$ and to this aim start from (3.17) with $\varphi _i\;:\!=\; \varphi$ , for $i=+,-$ . This gives

\begin{equation*} \begin{split} & \int _0^1 \partial _t \rho \,\varphi \,\textrm{d}y+ \frac{D_T}{L^2(t)} \int _0^ 1 \partial _y \rho \,\partial _y \varphi \,\textrm{d}y \\[5pt] &= -\frac{L'(t)}{L(t)} \int _0^1 y \,\rho \, \partial _y \varphi + \rho \, \varphi \,\textrm{d}y+ \frac{v_0}{L(t)} \int _0^1 \left (\bigl (f_+\,(1- \rho )\bigr )_{\text{tr}} - \bigl (f_-\,(1- \rho )\bigr )_{\text{tr}}\right ) \partial _y \varphi \,\textrm{d}y \\[5pt] &\quad - \frac{1}{L(t)}\, \Big [ \big (\beta _{+}(\Lambda (t)) \,f_{+}(t, 1)-\alpha _{-}(\Lambda (t))\,g_{-}(\boldsymbol{f}(t, 1))\big ) \,\varphi (1) \\[5pt] & \quad \quad +\big (\beta _{-}(\Lambda _{\textrm{som}}(t))\, f_{-}(t, 0)- \alpha _{+}(\Lambda _{\text{som}}(t))\, g_{+}(\boldsymbol{f}(t, 0))\big ) \, \varphi (0)\Big ]. \end{split} \end{equation*}

We choose $\varphi = \eta _\varepsilon '(\rho -1)$ , $\widetilde{\Omega }_\varepsilon = \{y \in (0,1): \, 1 \le \rho (t, y) \le 1+ 2\varepsilon \}$ , and reason as before exploiting hypothesis (H $_2$ ). Then the limit as $\varepsilon \to 0$ entails

\begin{equation*} 0 \le \int _0^1 (\rho -1)^+ \,\textrm {d}y \le \int _0^1 (\rho ^0-1)^+ \,\textrm {d}y= 0, \end{equation*}

thanks again to (H $_1$ ). It follows that $\rho \le 1$ , then the claim is proved.

Step 5: Conclusion. Using the original notation, we have shown the existence of a solution $\bar{\boldsymbol{f}}$ to (3.6) such that $\bar{f}_+, \bar{f}_- \ge 0$ and $\bar{\rho } \le 1$ for a.e. $y \in (0,1)$ , $t \in [0, T]$ . This implies that $\boldsymbol{f}$ is a weak solution to (3.9)–(3.10) such that $f_+, f_- \ge 0$ and $\rho \le 1$ for a.e. $x \in (0, L(t))$ , $t \in [0, T]$ . The system is completely solved.

Proof. Thanks to Theorem3.8, $\mathcal{B}_3$ is well defined. We choose $\boldsymbol{\Lambda }^{(a)} \in C([0,T])^3$ and $\boldsymbol{L}^{(a)} \in C([0, T])^2$ and set $(\boldsymbol{f}_1^{(a)}, \boldsymbol{f}_2^{(a)} )\;:\!=\; \mathcal{B}_3(\boldsymbol{\Lambda }^{(a)}, \boldsymbol{L}^{(a)})$ , $a=1,2$ . Then we start from equations (3.9), (3.10) for the respective copies, take the corresponding differences and use a Gronwall inequality to achieve

(3.25) \begin{align} &\|(\boldsymbol{f}_1^{(1)}, \boldsymbol{f}_2^{(1)})- (\boldsymbol{f}_1^{(2)}, \boldsymbol{f}_2^{(2)})\|_{\prod _{j=1,2} (L^2(0,T; H^1(0, 1)))^2} \nonumber \\[5pt] &\qquad \le \overline{C}\, e^{CT}\,\Big (\|\boldsymbol{\Lambda }^{(1)}-\boldsymbol{\Lambda }^{(2)}\|_{C([0,T])^3}^2 + \|\boldsymbol{L}^{(1)}- \boldsymbol{L}^{(2)}\|_{C([0, T])^2}^2\Big ), \end{align}

where the constant $\overline C$ is shown to be uniform.

Proof of Theorem 3.3. We use the Banach fixed point theorem to show the existence of a unique solution $(\boldsymbol{f}_1, \boldsymbol{f}_2, \boldsymbol{\Lambda }, \boldsymbol{L})$ to (2.1)–(2.5). We consider the set

\begin{equation*} \mathcal {K}= \{(\boldsymbol{f}_1, \boldsymbol{f}_2) \in X : \, 0 \le f_{i, j}(t,x) \le 1 \text { for a.e. $x \in (0, 1)$ and $t \in [0, T]$}\}. \end{equation*}

Let $\mathcal{B} \colon \mathcal{K} \to X$ be given by $\mathcal{B}(\boldsymbol{f}_1, \boldsymbol{f}_2) = \mathcal{B}_3(\mathcal{B}_1(\boldsymbol{f}_1, \boldsymbol{f}_2), \mathcal{B}_2(\mathcal{B}_1(\boldsymbol{f}_1, \boldsymbol{f}_2)))$ , where $\mathcal{B}_1, \mathcal{B}_2, \mathcal{B}_3$ are defined in Lemmas3.5, 3.7, and 3.9, respectively. Thus, $\mathcal{B}$ is well defined, while Theorem3.8 implies that $\mathcal{B}$ is self-mapping, that is, $\mathcal{B}\colon \mathcal{K} \to \mathcal{K}$ .

We next show that $\mathcal{B}$ is a contraction. Let $(\widehat{\boldsymbol{f}}_1^{(1)}, \widehat{\boldsymbol{f}}_2^{(1)}), (\widehat{\boldsymbol{f}}_1^{(2)}, \widehat{\boldsymbol{f}}_2^{(2)}) \in \mathcal{K}$ and set $( \boldsymbol{f}_1^{(1)}, \boldsymbol{f}_2^{(1)})\;:\!=\; \mathcal{B}(\widehat{\boldsymbol{f}}_1^{(1)}, \widehat{\boldsymbol{f}}_2^{(1)})$ as well as $(\boldsymbol{f}_1^{(2)}, \boldsymbol{f}_2^{(2)})\;:\!=\; \mathcal{B}(\widehat{\boldsymbol{f}}_1^{(2)}, \widehat{\boldsymbol{f}}_2^{(2)})$ . Using (3.25), (3.15) and (3.12) and summarising the not essential constants gives

\begin{equation*} \begin{split} & \|(\boldsymbol{f}_1^{(1)}, \boldsymbol{f}_2^{(1)} )- (\boldsymbol{f}_1^{(2)}, \boldsymbol{f}_2^{(2)})\|_X \\[5pt] & \le C T e^{CT} \big (T \max _{j=1,2} \big \{L_{h_j} e^{2T L_{h_j}}\big \}+1 \big ) \|(\widehat{\boldsymbol{f}}_1^{(1)}, \widehat{\boldsymbol{f}}_2^{(1)})- (\widehat{\boldsymbol{f}}_1^{(2)}, \widehat{\boldsymbol{f}}_2^{(2)})\|_X, \end{split} \end{equation*}

from which the contractivity follows if $T\gt 0$ is small enough that $C T e^{CT}\lt 1$ . Thanks to the Banach’s fixed point theorem, we then infer the existence of a unique $( \boldsymbol{f}_1, \boldsymbol{f}_2) \in \mathcal{K}$ such that $(\boldsymbol{f}_1, \boldsymbol{f}_2)= \mathcal{B}( \boldsymbol{f}_1, \boldsymbol{f}_2)$ . From (3.12), this implies the uniqueness of $\boldsymbol{\Lambda }$ , too. Due to the uniform $L^\infty$ -bounds on $\boldsymbol{f}_j$ , a concatenation argument yields existence on the complete interval $[0,T]$ . The proof is thus complete.

4.1. Constant stationary solutions

We assume a strictly positive reaction rate $\lambda \gt 0$ which requires $f_{+,j} = f_{-,j} =: f_j^\infty \in [0,1]$ . Assuming in addition $f_j^\infty = \textrm{const}.$ , (4.1) is automatically satisfied (note that $(L_j^\infty )'=0$ ). Thus, the actual constants are determined via the total mass, the stationary solutions to the ODEs and the boundary couplings, only, where the fluxes take the form:

(4.4) \begin{equation} J_{+,j}^{\infty }= -J_{-,j}^{\infty }= v_0\,f_{j}^{\infty }\,(1- \rho _j^{\infty })= v_0\,f_{j}^{\infty }\,(1- 2f_j^{\infty }). \end{equation}

Making the choice

(4.5) \begin{align} g_{+,j}(f_{+,j}, f_{-,j})= f_{+,j}\,(1- \rho _j) \quad \text{as well as} \quad g_{-,j}(f_{+,j}, f_{-,j})= f_{-,j}\,(1- \rho _j), \end{align}

the boundary conditions (4.2) become

(4.6) \begin{equation} \begin{split} J_{+,j}^{\infty }&={L_j^\infty }\alpha _{+,j}(\Lambda _{\text{som}}^\infty )\, f_j^{\infty }\,(1- \rho _j^{\infty })={L_j^\infty }\beta _{+,j}(\Lambda _j^{\infty })\, f_j^{\infty }, \\[5pt] -J_{-,j}^{\infty }&={L_j^\infty }\alpha _{-,j}(\Lambda _{j}^\infty )\,f_j^{\infty }\,(1- \rho _j^{\infty })={L_j^\infty }\beta _{-,j}(\Lambda _{\text{som}}^{\infty })\, f_j^{\infty }, \end{split} \end{equation}

which together with (4.4) yield

\begin{equation*} \alpha _{+,j}(\Lambda _{\text {som}}^{\infty })= \alpha _{-,j}(\Lambda _j^{\infty })= \frac {v_0}{L_j^\infty } \quad \text {as well as} \quad \beta _{-,j}(\Lambda _{\text {som}}^{\infty })= \beta _{+,j}(\Lambda _j^{\infty })= \frac {v_0(1- \rho _j^{\infty })}{L_j^\infty }. \end{equation*}

Thus, fixing the values of $f_j^\infty$ , we obtain $\Lambda _j^\infty$ , $\Lambda _{\text{som}}^\infty$ by inverting $\alpha _{+,j}$ , $\alpha _{-,j}$ , $\beta _{+,j}$ , and $\beta _{-,j}$ , respectively, together with the compatibility conditions:

\begin{equation*} \alpha _{-,j}(\Lambda _j^\infty )(1-\rho _j^\infty ) = \beta _{+,j}(\Lambda _j^\infty ) \quad \text {as well as} \quad \alpha _{+,j}(\Lambda _{\text {som}}^\infty )(1- \rho _j^\infty )= \beta _{-,j}(\Lambda _{\text {som}}^\infty ). \end{equation*}

We further observe that the equations in (4.3) are the differences of in- and outflow at the respective boundaries, which in the case of constant $f_j^\infty$ -s read as:

(4.7) \begin{equation} \begin{split} 0&= \sum _{j=1,2} \left (\beta _{-,j}(\Lambda _{\text{som}}^\infty )\,f_j^\infty - \alpha _{+,j}(\Lambda _{\text{som}}^\infty )\,f_j^\infty \,(1- \rho _j^\infty )\right ), \\[5pt] 0&= \beta _{+,j}(\Lambda _j^\infty )\,f_j^\infty - \alpha _{-,j}(\Lambda _j^\infty )\,f_j^\infty \,(1- \rho _j^\infty ) - \chi \,h_j(\Lambda _j^\infty, L_j^\infty ). \end{split} \end{equation}

It turns out that (4.7) will be automatically satisfied as soon as (4.6) and (4.3) hold. In particular, the last equation in (4.3) will determine the values of $L_j^\infty$ . We make the choices

(4.8) \begin{align} \beta _{+,j}(s)&= c_{\beta _{+,j}} \left (1- \frac{s}{\Lambda _{j, \text{max}}}\right ), \quad \beta _{-,j}(s)= c_{\beta _{-,j}} \left (1- \frac{s}{\Lambda _{\text{som}, \text{max}}}\right ), \end{align}

(4.9) \begin{align} \alpha _{+,j}(s)&= c_{\alpha _{+,j}} \frac{s}{\Lambda _{\text{som}, \text{max}}}, \quad \alpha _{-,j}(s)= c_{\alpha _{-,j}} \frac{s}{\Lambda _{j, \text{max}}}, \end{align}

for some $c_{\beta _{i,j}}, c_{\alpha _{i,j}} \ge 0$ , with $\Lambda _{\text{som}, \text{max}}$ and $\Lambda _{j,\text{max}}$ being the maximal capacity of soma and growth cones. Then, for given $f_j^\infty \in (0,\frac{1}{2}]$ to be a stationary solution, we need

\begin{align*} c_{\alpha _{+,j}}&={\frac{v_0}{L_j^\infty }} \frac{\Lambda _{\text{som,max}}}{\Lambda _{\text{som}}^\infty }, \quad c_{\beta _{-,j}}={\frac{v_0}{L_j^\infty }}(1- \rho _j^\infty ) \frac{\Lambda _{\text{som,max}}}{\Lambda _{\text{som,max}}- \Lambda _{\text{som}}^\infty }, \\[5pt] c_{\alpha _{-,j}}&={\frac{v_0}{L_j^\infty }} \frac{\Lambda _{j, \text{max}}}{\Lambda _j^\infty }, \quad c_{\beta _{+,j}}={\frac{v_0}{L_j^\infty }}(1- \rho _j^\infty ) \frac{\Lambda _{j, \text{max}}}{\Lambda _{j, \text{max}}- \Lambda _j^\infty }. \end{align*}

The interesting question of stability of these states will be treated in future work.

5.1. Finite volume scheme

We now present a computational scheme for the numerical solution of model (2.1)–(2.5). The scheme relies on a spatial finite volume discretisation of the conservation law (2.2) and adapted implicit–explicit time-stepping schemes. Starting point for the construction of a discretisation is the transformed equation (3.3). First, we introduce an equidistant grid:

\begin{equation*} 0 = y_{-1/2} \lt y_{1/2} \lt \ldots \lt y_{n_e-1/2} \lt y_{n_e+1/2} = 1 \end{equation*}

of the interval $(0,1)$ and define control volumes $I_k \;:\!=\; (y_{k-1/2},y_{k+1/2})$ , $k=0,\ldots, n_e$ . The mesh parameter is $h=y_{k+1/2}-y_{k-1/2} = (n_e+1)^{-1}$ . The cell averages of the approximate (transformed) solution are denoted by:

\begin{equation*} \bar {f}_{i,j}^k(t) \;:\!=\; \frac{1}{h} \int _{I_k} f_{i,j}(t,y)\,\textrm {d}y,\qquad k=0,\ldots, n_e, \end{equation*}

$i\in \{+,-\}$ , $j\in \{1,2\}$ . To shorten the notation, we omit the vesicle index $j\in \{1,2\}$ in the following. Integrating (3.3) over an arbitrary control volume $I_k$ , $k=0, \ldots, n_e$ , yields

\begin{align*} 0 = \int _{I_k} \Big [ \partial _t f_+ + \frac{1}{L(t)^2} \partial _y\left [- D_T\,\partial _y f_+ + L(t)\,((v_0\,(1-\rho ) - L'(t)\,L(t)\,y)\,f_+)\right ]&\\[5pt] + \frac{L'(t)}{L(t)}\, f_+ - \lambda \,(f_- - f_+) \Big ]\,\textrm{d}y &\\[5pt] = h\,\partial _t \bar{f}_+^k + \left [- \frac{D_T}{L(t)^2}\,\partial _y f_+ + \frac {1}{L(t)}\, \left ((v_0\,(1-\rho ) - L'(t)\,y)\,f_+\right )\right ]_{y_{k-1/2}}^{y_{k+1/2}} &\\[5pt] + h\,\frac{L'(t)}{L(t)}\, \bar{f}_+^k - h\,\lambda \,(\bar{f}_-^k-\bar{f}_+^k)& . \end{align*}

We denote the convective flux by $\textbf{v}_+(t,y) \;:\!=\; v_0\,(1-\rho )-L'(t)\,y$ and use a Lax-Friedrichs approximation at the end points of the control volume:

\begin{equation*} F_+^{k+1/2}(t) \;:\!=\; \{\!\!\{(\textbf {v}_+\,\bar {f}_+)(t,y_{k+1/2})\}\!\!\} - \frac {1}{2}\,\{\!\{\bar {f}_+(t,y_{k+1/2})\}\} \approx (\textbf {v}_+\,f_+)(t,y_{k+1/2}), \end{equation*}

with $\{\{\cdot\}\}$ and $[\![\!\cdot\!]\!]$ denoting the usual average and jump operators. For the diffusive fluxes, we use an approximation by central differences:

\begin{equation*} \partial _y f_+(t,y_{k+1/2}) \approx \frac {\bar {f}_+^{k+1}(t) - \bar {f}_+^{k}(t)}{h}. \end{equation*}

For the inner intervals $I_k$ with $k \in \{1,\ldots, n_e-1\}$ , this gives the equations:

(5.1a) \begin{align} \partial _t \bar{f}_+^k + \frac{D_T}{(h\,L)^2}\left (-\bar{f}_+^{k-1} + 2\bar{f}_+^k - \bar{f}_+^{k+1}\right ) &= \frac 1{h\,L} \left (F_+^{k-1/2}-F_+^{k+1/2}\right ) \nonumber \\[5pt] & \quad + \lambda \,(\bar{f}_-^k - \bar{f}_+^k) - \frac{L'}{L}\,\bar{f}_+^k, \end{align}

while for $k=0$ and $k=n_e$ , we insert the boundary conditions (3.4) to obtain

(5.1b) \begin{align} \partial _t\bar{f}_+^0 + \frac{D_T}{(h\,L)^2}\,(\bar{f}_+^0 - \bar{f}_+^1) &= \frac 1{h\,L}\left (\alpha _+(\Lambda _{\text{som}})\,g_+(\bar{f}_+^0, \bar{f}_-^0) - F_+^{1/2} \right ) \nonumber \\[5pt] & \quad + \lambda \,(\bar{f}_-^0 - \bar{f}_+^0) - \frac{L'}{L}\,\bar{f}_+^0, \end{align}

(5.1c) \begin{align} \partial _t\bar{f}_+^{n_e} + \frac{D_T}{(h\,L)^2}\,(\bar{f}_+^{n_e} - \bar{f}_+^{n_e-1}) &= \frac 1{h\,L} \left (F_+^{n_e-1/2} - \beta _+(\Lambda )\,\bar{f}_+^{n_e} \right ) \nonumber \\[5pt] & \quad + \lambda \,(\bar{f}_-^{n_e} - \bar{f}_+^{n_e}) - \frac{L'}{L}\,\bar{f}_+^{n_e}, \end{align}

almost everywhere in $(0,T)$ . In the same way, we deduce a semi-discrete system for $\bar{f}_-^k$ , $k=0,\ldots, n_e$ , taking into account the corresponding boundary conditions from (3.4).

To treat the time dependency, we use an implicit–explicit time-stepping scheme. We introduce a time grid $t_n = \tau \,n$ , for $n=0,\ldots, n_t$ , and for some time-dependent function $g\colon [0,T]\to X$ we use the notation $g(t_n) =: g^{(n)}$ . To deduce a fully discrete scheme, we replace the time derivatives in (5.1) by a difference quotient $\partial _t\bar{f}_{+}(t_{n+1}) \approx \tau ^{-1}(\bar{f}_+^{(n+1)}-\bar{f}_+^{(n)})$ and evaluate the remaining terms related to diffusion in the successive time point $t_{n+1}$ and all convection and reaction related terms in the current time point $t_n$ . This yields a system of linear equations of the form:

(5.2) \begin{equation} \left (M+\tau \,\frac{D_T}{(L^{(n)})^2}\,A\right )\,\vec f_{\pm }^{(n+1)} = M\,\vec f_{\pm }^{(n)} + \tau \,\vec b_{\pm }^{(n)} + \tau \,\vec c_{\pm }^{(n)}, \end{equation}

with vector of unknowns $\vec f_\pm ^{(n)} = (f_\pm ^{0,(n)}, \ldots, f_\pm ^{n_e,(n)})^\top$ , mass matrix $M$ , diffusion matrix $A$ , and a vector $\vec b_\pm ^{(n)}$ summarising the convection related terms and another vector for the reaction related terms $\vec c_\pm ^{(n)}$ . In the same way, we deduce equations for the discretised ordinary differential equations (2.5) and (2.4) which correspond to a standard backward Euler discretisation:

(5.3) \begin{align} \Lambda _{\text{som}}^{(n+1)} &= \Lambda _{\text{som}}^{(n)} + \tau \,\sum _{j=1,2} \left ( \beta _{-,j}\,\bar{f}_{j,-}^{0, (n+1)} - \alpha _{+,j}(\Lambda _{\text{som}}^{(n+1)})\, g_{+}(\bar{f}_{+,j}^{0,(n+1)}, \bar{f}_{-,j}^{0,(n+1)}) \right ) + \tau \,\gamma _{\text{prod}}^{(n)}, \end{align}

(5.4) \begin{align} \Lambda _j^{(n+1)} &= \Lambda _j^{(n)} + \tau \,\beta _{+,j}(\Lambda _j^{(n+1)})\,\bar{f}_{+,j}^{n_e,(n+1)} \end{align}

(5.5) \begin{align} &\quad -\tau \,\alpha _{-,j}(\Lambda _j^{(n+1)})\,g_{-,j}(\bar{f}_{+,j}^{n_e,(n+1)}, \bar{f}_{-,j}^{n_e,(n+1)}) - \tau \,\chi \, h_j(\Lambda _j^{(n)}, L_j^{(n)}), \nonumber \\[5pt] L_j^{(n+1)} &= L_j^{(n)} + \tau \,h_j(\Lambda _j^{(n+1)}, L_j^{(n+1)}),\qquad j=1,2. \end{align}

Equations (5.2)–(5.5) even decouple. One after the other, we can compute

\begin{equation*} \boldsymbol{f}^0, L_j^0, \Lambda _k^0\;\rightarrow \; \vec f_{\pm, j}^{(1)}\;\rightarrow \; \Lambda _{k}^{(1)}\;\rightarrow \; L_j^{(1)} \; \rightarrow \; \vec f_{\pm, j}^{(2)}\;\rightarrow \; \Lambda _k^{(2)}\;\rightarrow \; L_j^{(2)} \;\rightarrow \;\ldots \end{equation*}

for $k\in \{1,2,\text{som}\}$ and $j\in \{1,2\}$ .

5.2. Non-dimensionalisation of the model

To transform the model to a dimensionless form, we introduce a typical time scale $\tilde{t}$ , a typical length $\tilde{L}$ , etc., and dimensionless quantities $\bar{t},\, \bar{L}$ such that $t = \tilde{t} \bar{t}$ , $L = \tilde{L} \bar{L}$ . This is performed on the original system from Section 2, not on the one transformed to the unit interval, as we want to work with appropriate physical units for all quantities, including the length of the neurites. Realistic typical values are taken from [Reference Humpert, Di Meo, Püschel and Pietschmann19] (see also [Reference Pfenninger27, Reference Tsaneva-Atanasova, Burgo, Galli and Holcman32, Reference Urbina, Gomez and Gupton34]) which yield the following choices: the typical length is $\tilde{L} = 25 \,\mu \text{m}$ , the typical time is $\tilde{t} = 7200 \,\text{s}$ , the diffusion constant is $D_T = 0.5\, \frac{\mu \text{m}^2}{\text{s}}$ , and the velocity is $\tilde{v}_0 = 50 \frac{\mu \text{m}}{\text{min}} = \frac 56\,\frac{\mu \text{m}}{\text{s}}$ . For the reaction rate, we assume $\tilde{\lambda } = \frac{1}{s}$ . The typical influx and outflow velocity is $\tilde{\alpha } = \tilde{\beta } = 0.4\, \frac{\mu \text{m}}{\text{s}}$ . Finally, we choose a typical production of $\tilde{\gamma } = 10\, \text{vesicles}/\text{sec}$ .

The remaining quantities to be determined are the maximal density of vesicles inside the neurites, the factor which translates a given number of vesicles with length change of the neurite and the maximal capacity of soma and growth cones.

Maximal density

We assume the neurite to be tube-shaped, pick a circular cross section at an arbitrary point and calculate the maximal number of circles having the diameter of the vesicles that fit the circle whose diameter is that of the neurite. In this situation, hexagonal packing of the smaller circles is optimal, which allows to cover about $90\,\%$ of the area (neglecting boundary effects). As the typical diameter of one vesicle is $130\,\textrm{nm}$ and the neurite diameter is 1 $\mu$ m, we obtain the condition:

\begin{align*} \underbrace{0.9\cdot n_{\text{max}}\pi \left (\frac{130\,\textrm{nm}}{2}\right )^2}_{\text{area covered by $n_{\text{max}}$ circles of vesicle diameter}}\le \underbrace{\pi \left (\frac{1000\,\textrm{nm}}{2}\right )^2}_{\text{area of neurite cross-section}}, \end{align*}

which implies $n_{\text{max}} \le 65$ . Now for a tube segment of length $1\,\mathrm{\mu m}$ , one can stack $7$ fully packed cross-sectional slices, each of which has the diameter of the vesicles, that is, $130\,\textrm{nm}$ . This results in a maximal density of $455 \frac{\text{vesicles}}{\mu m}$ . As the neurite also contains microtubules and as an optimal packing is biologically unrealistic, we take one-third of this value as maximal density, which yields $ \rho _{\text{max}} = 155 \frac{\text{vesicles}}{\mathrm{\mu m}}.$ The typical density of anterograde and retrograde particles is fixed to $\tilde{f} \;:\!=\; \tilde{f}_+= \tilde{f}_- = 39\, \frac{\text{vesicles}}{\mu \text{m}}$ so that their sum corresponds to a half-filled neurite. Thus, for the scaled variables $\bar{f}_+, \,\bar{f}_-$ , their sum being $\bar{\rho }= \bar{f}_+ + \bar{f}_- = 2$ corresponds to a completely filled neuron. This implies that the term $1- \rho$ has to be replaced by $1-\frac{\bar{\rho }}{2}$ .

Vesicles and growth

We again consider the neurite as a cylinder with a diameter of $1\,\mu \textrm{m}$ . Thus, the surface area of a segment of length $1\,\mu \textrm{m}$ is $ A_{\text{surf}} = 2\pi \frac{1\,\mu \textrm{m}}{2}1\,\mu \textrm{m} \approx 3.14\, (\mu \textrm{m})^2.$ We consider vesicles of $130$ nm diameter, which thus possess a surface area of $0.053$ $(\mu \textrm{m})^2$ . Thus, the number of vesicles needed for an extension of $1\,\mu \textrm{m}$ is

\begin{equation*} \frac {3.14\, \mu \textrm {m}^2}{0.053\,\mu \textrm {m}^2} \approx 59,\quad \text { i.e., we fix } c_1 = c_2 = c_h \;:\!=\; \frac {58.4 \text { vesicles}}{\mu \textrm {m}}. \end{equation*}

Maximal capacities and minimal values

Finally, we fix the maximal amount of vesicles in the pools and soma to $\Lambda _{\text{som,max}} = 6000\,\text{vesicles}$ and $\Lambda _{j,\text{max}}= 400\,\text{vesicles}$ and choose the typical values $\tilde{\Lambda }_{\text{som}}$ and $\tilde{\Lambda }_{\text{cone}}$ as half of the maximum, respectively. It remains to fix the minimal length of each neurite as well as the number of vesicles in the growth cone which defines the switching point between growth and shrinkage. We choose a minimal length of $5\,\mu \textrm{m}$ , while the sign of $h_j$ changes when the number of vesicles in the growth cone reaches a value of $100\,\text{vesicles}$ . This yields the dimensionless quantities $ \bar{\Lambda }_{\text{growth}} = 1, \; \bar{L}_{\text{min}} = 0.1.$ Applying the scaling, model (2.1)–(2.5) transforms to

\begin{align*} \begin{aligned} \partial _t \bar{f}_{+,j}& +\partial _x \left (\kappa _v\,\bar{f}_{+,j}\left (1- \frac{\bar{\rho }_j}{2}\right )- \kappa _D\,\partial _x \bar{f}_{+,j}\right ) = \kappa _\lambda \bar{\lambda } (\bar{f}_{-,j} - \bar{f}_{+,j}), \\[5pt] \partial _t \bar{f}_{-,j}& - \partial _x\left (\kappa _v\,\bar{f}_{-,j}\left (1- \frac{\bar{\rho }_j}{2}\right )- \kappa _D\,\partial _x \bar{f}_{-,j}\right ) = \kappa _\lambda \bar{\lambda }(\bar{f}_{+,j} - \bar{f}_{-,j}), \end{aligned} \end{align*}

$\text{in } (0, T) \times (0, \bar{L}_j(t))$ , with dimensionless parameters $ \kappa _v = \frac{v_0 \tilde{t} }{\tilde{L}}, \; \kappa _D = D_T\frac{\tilde{t} }{\tilde{L}^2}, \; \kappa _\lambda = \tilde{t} \tilde{\lambda },$ and with boundary conditions (keeping the choices (4.5), (4.8), (4.9)):

\begin{equation*} \begin {aligned} \bar {J}_{+,j}(t,0) &= \kappa _{\alpha _{+,j}}\frac {\bar {\Lambda }_{\text {som}}(t)}{2}\,\bar {f}_{+,j}(\bar {t},0)\left (1-\frac {\bar {\rho }_j(t,0)}{2}\right ), \\[5pt] -\bar {J}_{-,j}(t,0) &= \kappa _{\beta _{-,j}}\left (1-\frac {\bar {\Lambda }_{\text {som}}(t)}{2}\right )\,\bar {f}_{-,j}(t,0),\\[5pt] \bar {J}_{+,j}(t, L_j(t)) - \bar {L}'_j(t) \bar {f}_{+,j}(t, \bar {L}_j(t)) &= \kappa _{\beta _{+,j}}\left (1-\frac {\bar {\Lambda }_{j}(t)}{2}\right )\,\bar {f}_{+,j}(\bar {t}, \bar {L}_j(\bar {t})), \\[5pt] -\bar {J}_{-,j}(t, L_j(t)) + \bar {L}'_j(t) \bar {f}_{-,j}(t, \bar {L}_j(t))&= \kappa _{\alpha _{-,j}}\frac {\bar {\Lambda }_j(t)}{2}\,\bar {f}_{-,j}(\bar {t},0)\left (1-\frac {\bar {\rho }_j(t,0)}{2}\right ), \end {aligned} \end{equation*}

with $\kappa _{\alpha _{+,j}} = \frac{\tilde{t}}{\tilde{L}} c_{\alpha _{+,j}}, \kappa _{\alpha _{-,j}} = \frac{\tilde{t}}{\tilde{L}} c_{\alpha _{-,j}}, \kappa _{\beta _{+,j}} = \frac{\tilde{t}}{\tilde{L}} c_{\beta _{+,j}}, \kappa _{\beta _{-,j}} = \frac{\tilde{t}}{\tilde{L}} c_{\beta _{-,j}}$ . It remains to fix the values of the constants appearing in the functions $\alpha _{\pm, j},\,\beta _{\pm, j}$ . As they correspond to velocities, we fix them to the typical in-/outflux velocity:

\begin{equation*} \tilde {c} \;:\!=\; c_{\alpha _{+,j}} = c_{\alpha _{-,j}} = c_{\beta _{+,j}}= c_{\beta _{-,j}} = 0.4 \,\frac {\mu \text {m}}{\text {s}}. \end{equation*}

For the soma and the growth cones, we choose half of the maximal amount of vesicles as typical values, that is, $\tilde{\Lambda }_{\text{som}} = 3000\,\text{vesicles}$ , $\tilde{\Lambda }_{j} = 200\,\text{vesicles}$ , $j=1,2$ . We obtain

\begin{equation*} \begin {aligned} & \bar {\Lambda }'_{\text {som}}(t)= \kappa _{\text {som}}\sum _{j=1,2} \left [\left (1-\frac {\bar {\Lambda }_{\text {som}}(t)}{2}\right )\,\bar {f}_{-,j}(t,0) - \frac {\bar {\Lambda }_{\text {som}}(t)}{2}\,\bar {f}_{+,j}(\bar {t},0)\left (1-\frac {\bar {\rho }_j(t,0)}{2}\right ) \right ] \\[5pt] & \qquad \qquad \qquad + \kappa _\gamma \bar {\gamma }_{\text {prod}}(t), \\[5pt] &\bar {\Lambda }'_{j}(t) = \kappa _{\text {cone}}\left [\left (1-\frac {\bar {\Lambda }_{j}(t)}{2}\right )\,\bar {f}_{+,j}(\bar {t}, \bar {L}_j(\bar {t}))- \frac {\bar {\Lambda }_j(t)}{2}\,\bar {f}_{-,j}(\bar {t}, \bar {L}_j(t))\left (1-\frac {\bar {\rho }_j(\bar {t}, \bar {L}_j(t))}{2}\right )\right ] \\[5pt] & \qquad \qquad \qquad -\kappa _{h} \bar {h}_j(\bar {\Lambda }_j(t), \bar {L}_j(t)), \end {aligned} \end{equation*}

with $ \kappa _{\text{som}} = \frac{\tilde{t}}{\tilde{\Lambda }_{\text{som}}}\tilde{c} \tilde{f}, \kappa _\gamma = \tilde{\gamma } \frac{\tilde{t}}{\tilde{\Lambda }_{\text{som}}}, \kappa _{\text{cone}} = \frac{\tilde{t}}{\tilde{\Lambda }_{\text{cone}}}\tilde{c} \tilde{f}, \kappa _{h} = \frac{\tilde{t}}{\tilde{\Lambda }_{\text{cone}}} \tilde{h} \chi .$ Finally, for the scaled production function $\bar{h}$ in (2.4), we make the choice $\bar{h}_j(\bar{\Lambda }, \bar{L}) = \operatorname{atan}(\bar{\Lambda } - \bar{\Lambda }_{\text{growth}})H(\bar{L} - \bar{L}_{\text{min}})$ , $j=1,2$ , where $H$ is a smoothed Heaviside function. We have

\begin{align*} \bar{L}_j'(t) = \kappa _{L}\, \bar{h}_j(\bar{\Lambda }_j(t), \bar{L}_j(t)), \; \text{ with } \kappa _L = \frac{\tilde{t}}{\tilde{L}}\,\tilde{h}. \end{align*}

Figure 3.

The vesicle densities $f_{\pm, j}$ , $j=1,2$ , and pool capacities $\Lambda _k$ , $k\in \{\text{som},1,2\}$ , for the example from Section 6.1 plotted at different time points.

Figure 4.

The neurite lengths $L_j$ , $j=1,2$ , and pool capacities $\Lambda _k$ , $k\in \{\text{som},1,2\}$ , for the example from Section 6.1 plotted over time.

Figure 5.

The vesicle densities $f_{\pm, j}$ , $j=1,2$ , and pool capacities $\Lambda _k$ , $k\in \{\text{som},1,2\}$ , for the example from Section 6.2 plotted at different time points.

Figure 6.

The neurite lengths $L_j$ , $j=1,2$ , and pool capacities $\Lambda _k$ , $k\in \{\text{soma},1,2\}$ , for the example from Section 6.2 plotted over time.

6.1. Fast growth by local effects

The first example shows that an initial length deficiency of a neurite can be overcome by a local advantage of vesicles on the growth cone. In this set-up, we fixed (all scaled quantities) the following initial data: $L_1^0 = 1.1$ , $L_2^0 = 0.9$ , $\Lambda _{\text{soma}}^0 = 1$ , $\Lambda _1^0= 0.65$ , $\Lambda _2^0= 1.15$ , $\boldsymbol{f}_1^0 = \boldsymbol{f}_2^0 = (0.2,0.2)$ . Furthermore, the functions

\begin{align*} g_\pm (f_{+},f_{-}) &= 1-\frac 12\left (f_{+}+f_{-}\right ),\quad \alpha _+(\Lambda ) = 0.05 \,\frac{\Lambda }2,\quad \alpha _-(\Lambda ) = 0.1\, \frac{\Lambda }2,\\[5pt] \beta _{\pm }(\Lambda ) &= 0.7 \, \left (1-\frac{\Lambda }2\right ),\quad h(\Lambda, L) = \frac{\tan ^{-1}(\Lambda - \Lambda _{\textrm{growth}})}{1+\exp (\!-4\,(L - L_{\textrm{min}}-0.2))},\quad v_0=0.1 \end{align*}

were chosen in this example. The choices of $\alpha _+$ and $\alpha _-$ are motivated as follows: both are proportional to $\Lambda / 2$ as the typical values within the soma and growth cones, respectively, are chosen to be half the maximum which means that $\Lambda / 2 = 1$ if this value is reached. Thus, relative to their capacity, the outflow rates from soma and growth cone behave similarly. Now, since we are interested in the effect of vesicle concentration in the respective growth cones, we chose a small constant in $\alpha _+$ in order to limit the influence of new vesicles entering from the soma, relative to $\alpha _-$ . This is a purely heuristic choice to examine if such a local effect can be observed in our model at all. Figure 3 shows snapshots of the simulation at different times, while Figure 4a shows the evolution of neurite lengths and vesicle concentrations over time. The results demonstrate that the local advantage of a higher vesicle concentration in the growth cone of the shorter neurite is sufficient to outgrow the competing neurite. Yet, this requires a weak coupling in the sense that the outflow rate at the soma is small, see the constant $0.05$ in $\alpha _+$ . Increasing this value, the local effect does not prevail and indeed, the longer neurite will always stay longer while both neurites grow at a similar rate as shown in Figure 4b. Thus, we consider this result as biologically not very realistic, in particular as it cannot reproduce cycles of extension and retraction that are observed in experiments.

6.2. Oscillatory behaviour due to coupling of soma outflow rates to density of retrograde vesicles

In order to overcome the purely local nature of the effect in the previous example, it seems reasonable to include effects that couple the behaviour at the growth cones to that of the soma via the concentrations of vesicles in the neurites. We propose the following two mechanisms: first, we assume that a strongly growing neurite is less likely to emit a large number of retrograde vesicles as it wants to use all vesicles for the growth process. In addition, we assume that the soma aims to reinforce strong growth and is doing so by measuring the density of arriving retrograde vesicles. The lower it becomes, the more anterograde vesicles are released. Such behaviour can easily be incorporated in our model by choosing

\begin{align*} g_+(f_+, f_-) &= \left (\sqrt{\max (0,1-3\,f_-)^2 + 0.1}+0.5\right )\,\left (1-\frac 12\,(f_++f_-)\right ),\\[5pt] \alpha _+(\Lambda ) &= 0.6\frac{\Lambda }2,\quad \alpha _-(\Lambda )=(1-\frac \Lambda 2)\,\frac \Lambda 2,\quad v_0=0.04. \end{align*}

The remaining functions are defined as in Section 6.1. The initial data in this example are $L_1^0 = 1.1$ , $L_2^0 = 1$ , $\Lambda _{\textrm{som}}^0 = 1$ , $\Lambda _{1}^0=\Lambda _2^0 = 0.9$ and $f_{\pm, 1}^0 = f_{\pm, 2}^0 = 0$ .

The results are presented in Figures 5 (snapshots) and 6 and are rather interesting: first, it is again demonstrated that the shorter neurite may outgrow the larger one. Furthermore, as a consequence of the non-local coupling mechanism, the model is able to reproduce the oscillatory cycles of retraction and growth that are sometimes observed, see for example, [Reference Cooper9, Reference Winans, Collins and Meyer35]. Also the typical oscillation period of 2–4 hours observed in [Reference Winans, Collins and Meyer35, Figure 1] can be confirmed in our computation. Finally, the model predicts one neurite being substantially longer than the other which one might interpret as axon and dendrite.