Front propagation in the shadow wave-pinning model

Abstract

In this paper we consider a non-local bistable reaction–diffusion equation which is a simplified version of the wave-pinning model of cell polarization. In the small diffusion limit, a typical solution u(x, t) of this model approaches one of the stable states of the bistable nonlinearity in different parts of the spatial domain $\Omega$, separated by an interface moving at a normal velocity regulated by the integral $\int _\Omega u(x,t)\,dx$. In what is often referred to as wave-pinning, feedback between mass-conservation and bistablity causes the interface to slow and approach a fixed limit. In the limit of a small diffusivity $\varepsilon ^2\ll 1$, we prove that for any $0<\gamma <1/2$ the interface can be estimated within $O(\varepsilon ^\gamma )$ of the location as predicted using formal asymptotics. We also discuss the sharpness of our result by comparing the formal asymptotic results with numerical simulations.

Appendix A. The differential algebraic equation and its solvability

In this appendix we reformulate the system (1.10) as a differential algebraic equation (DAE) that more easily lends itself to numerical calculations and analysis. To this end we first define

$$\begin{aligned} W(s) \equiv |\{x\in \Omega \,| \text {dist}(x,\Gamma _0)<s \}|. \end{aligned}$$

Since s is the distance of the interface from its initial position, it is easy to see that ds/dt indicates its speed which, by the method of matched asymptotic expansions, corresponds to $\alpha$. Specifically, we deduce that (1.10) is equivalent to the system

$$\begin{aligned}&\frac{ds}{dt} = \alpha (V(s)), t>0;\qquad s(0) =0, \end{aligned}$$

(A.1a)

$$\begin{aligned}&V(s) + W(s)h^+(V(s)) + (1-W(s))h^-(V(s)) = M_0. \end{aligned}$$

(A.1b)

It is then straightforward to recover ${\hat{v}}(t)$ and $\hat{\Omega }(t)$ by using

$$\begin{aligned} {\hat{v}}(t) = V(s(t)),\qquad \hat{\Omega }(t) = \bigl \{x\in \Omega \,| \text {dist}(x,\Gamma _0) < s(t)\bigr \}, \end{aligned}$$

from which ${\hat{u}}(x,t)$ is then obtained using (1.10c). While solving the DAE (A.1) is a relatively simple task, the calculation of W(s) may be more difficult depending on properties of the initial interface $\Gamma _0$. However, this reformulation has the benefit that once the initial interface $\Gamma _0$ is known, W(s) can be precomputed for a sufficiently large range of s values.

In addition to simplifying numerical calculation of the leading order solution, it is also easier to deduce the existence of solutions to (A.1). It suffice to show that the right hand side of (A.1a) is Lipschitz in s. To show this we first define $G:(v_{\min },v_{\max })\times (0,1)\rightarrow {\mathbb {R}}$ by

$$\begin{aligned} G(x,y)\equiv x + yh^+(x) + (1-y)h^-(x) = y\bigl ( h^+(x) + x\bigr ) + (1-y)\bigl (h^-(x) + x \bigr ) \end{aligned}$$

Then for $x_2>x_1$ we calculate

$$\begin{aligned} G(x_2,y)-G(x_1,y) =\,&y(x_2-x_1)\textstyle \int _0^1\left( 1 + \tfrac{dh^+}{dv}|_{sx_2+(1-s)x_1}\right) ds \\&+ (1-y)(x_2-x_1)\textstyle \int _0^1\left( 1 + \tfrac{dh^-}{dv}|_{sx_2+(1-s)x_1}\right) ds \\ >\,&C_0 y (x_2-x_1) + C_0(1-y)(x_2-x_1) = C_0(x_2-x_1), \end{aligned}$$

where the first inequality follows from (1.12) and in particular $dh^\pm /dv>-1$. On the other hand

$$\begin{aligned} G(x,y_2)-G(x,y_1) \le |h^+(x)-h^-(x)||y_2-y_1| \le A|y_2-y_1|. \end{aligned}$$

Now let $s_1$ and $s_2$ satisfy (A.1b) and assume that $V(s_2)>V(s_1)$. Then

$$\begin{aligned} G(V(s_2),W(s_1))-G(V(s_1),W(s_1)) = G(V(s_2),W(s_1))- G(V(s_2),W(s_2)), \end{aligned}$$

with which the above inequalities give

$$\begin{aligned} C_0|V(s_2)-V(s_1)| < A|W(s_2)-W(s_1)|. \end{aligned}$$

(A.2)

Now from the definition of W(s) we deduce $|W(s_2)-W(s_1)|<C_1|s_2-s_1|$ for some constant $C_1>0$ depending only on $\Gamma _0$. From (1.17) we then deduce

$$\begin{aligned} |\alpha (V(s_2))-\alpha (V(s_1))| \le A|V(s_2)-V(s_1)| \le \frac{A^2 C_1}{C_0}|s_2-s_1|. \end{aligned}$$

(A.3)