Sharp interface limit in a phase field model of cell motility

Figure 1.  Illustration of the ansatz (34). Function $\rho_\varepsilon$ decays to a non-zero constant of the order $\varepsilon$ for $x\to\pm\infty$ and to a constant slightly different from 1 for $-a\leq x\leq a$ (solid). Dashed line represents the limiting profile, which is the characteristic function of $(-a, a)$

Figure 2.  Left: $\Phi_\beta(V)$ for $\beta=150$ and $W_{\text{as}}(\rho)=\frac{1}{4}\rho^2(1+\rho^2)(\rho-1)^2$, positive slope illustrates $\Phi'_\beta(0)>0$; Right: $\theta_0$, standing wave for the Allen-Cahn equation for $W_{\text{sym}}(\rho)=\frac{1}{4}\rho^2(\rho-1)^2$ (dashed) and $W_{\text{as}}(\rho)=\frac{1}{4}\rho^2(1+\rho^2)(\rho-1)^2$ (solid)

Figure 3.  Left: Plot of function $\Phi_\beta(V)$ for $\beta =150>\beta_{\text{cr}}$; {\it Right}: Plot $c_0V- \Phi_\beta(V)$ for $\beta=150$ vs $F$. For $-F=1.5$ there is one intersection ((61) has one root). For each $-F=1.762$ and $-F=2.264$ there are two intersections ((61) has two roots). For $-F=2$ there are three intersections ((61) has three roots)

Figure 4.  Sketch of the function $F(V)=-c_0V+\Phi_\beta (V)$; $F(V)$ has one local minimum, $F_{\text{min}}=F(V_{\text{min}})$, and one local maximum, $F_{\text{max}}=F(V_{\text{max}})$. Left: Until $F<F_{\max}$ we stay on the left branch. When $F$ exceeds $F_{\max}$ we jump on the right branch; Right: Until $F>F_{\min}$ we stay on the right branch; When $F$ becomes less than $F_{\min}$ we jump on the left branch. Red arrows on both figures illustrate jumps in velocities.

Figure 5.  Hysteresis loop in the problem of cell motility. Simulations of $V=V(F)$ Left: (61) Jumping from the left to the right branches and back; Right: PDE system (57)-(58). On both figures arrows show in what direction the system $(V(t), F(t))$ evolves as time $t$ grows; red curve is for $F_{\uparrow}(t)$, blue curve is for $F_{\downarrow}(t)$