Reaction–Diffusion: Fronts, Patterns and Pinching

6A.8 Nonlinear Parabolic Equations ended with the two fates of a nonlinear heat equation: the solution exists for all time, or a reaction term drives it to infinity in finite time. Between those extremes lies most of what reaction–diffusion equations do in the world. A population spreads into empty territory at a constant speed. A nerve carries an impulse of fixed shape and speed. A uniform chemical soup breaks up into spots and stripes. Two phases of an alloy separate, and the boundary between them moves by its curvature. Each of these is an equation of the form

or a system of such equations, in which a reaction $f$, which by itself would be an ordinary differential equation at each point, competes with diffusion, which by itself would smooth everything out. This chapter studies that competition: how the length and time scales of the two set the critical size of a habitat, how a front is a heteroclinic orbit of an ODE and how its speed is selected, why a bistable reaction produces interfaces that move by mean curvature, and how two diffusions at different rates can create pattern where one would destroy it.

The reason the chapter belongs in a book about the Ricci flow is the last section. Under the Ricci flow the curvature tensor evolves by $\partial_t\operatorname{Rm} = \Delta\operatorname{Rm} + Q(\operatorname{Rm})$ (11A.3 Curvature in a Moving Frame), a reaction–diffusion system whose reaction term $Q$ is quadratic. Hamilton's whole approach to the flow, from his 1982 theorem to the pinching estimates that Perelman takes as given, is the study of this one system: find sets that the reaction ODE preserves, prove that diffusion cannot spoil them (the maximum principle of 6A.5 Maximum Principles), and understand singularities as places where reaction overwhelms diffusion. The scalar examples of this chapter are the place to learn that way of thinking where every step can be checked by hand.

By the end of this chapter you will be able to:

• nondimensionalise a reaction–diffusion equation, identify its intrinsic length $\sqrt{D/r}$ and time $1/r$, and compute the critical patch size of a habitat;
• reduce travelling waves to a phase-plane problem, prove that Fisher–KPP fronts exist for every speed $c \geq 2\sqrt{rD}$, and explain why the minimal speed is the one selected;
• solve the bistable (Nagumo) equation explicitly, prove that the sign of the front speed is the sign of $\int_0^1f$, and describe the stability of bistable fronts;
• derive formally that interfaces of the Allen–Cahn equation move by mean curvature, and state what has been proved;
• derive Turing's conditions for diffusion-driven instability by Fourier modes, and explain the roles of the wavelength and of the domain size;
• describe excitable media and nerve pulses;
• read the curvature equations of the Ricci flow as a reaction–diffusion system, prove the ODE comparison for the scalar curvature, and prove the pinching of the three-dimensional curvature ODE.

9.1Reaction against diffusion

Scales. In $u_t = Du_{xx} + rf(u)$ with $f$ dimensionless (for example $f(u) = u(1 - u)$), the coefficient $D$ has units of length squared per time and $r$ of inverse time. There is exactly one length and one time that can be made from them:

Substituting $x = \ell\tilde x$ and $t = \tau\tilde t$ removes both constants: $u_{\tilde t} = u_{\tilde x\tilde x} + f(u)$. So every statement about the dimensionless equation is a statement about the original one in units of $\ell$ and $\tau$, and every speed is a multiple of $\ell/\tau = \sqrt{rD}$. This is why the Fisher–KPP speed of 6A.5 Maximum Principles is $2\sqrt{rD}$: the number $2$ is the speed of the dimensionless equation, and $\sqrt{rD}$ is the only speed available.

The length $\ell$ is the distance over which diffusion acts during one reaction time. On scales much smaller than $\ell$, diffusion wins and the solution is nearly uniform; on scales much larger, each point runs its own ODE and diffusion only connects neighbouring regions through thin transition layers. Almost everything in this chapter happens at scale $\ell$, where the two are in balance.

Critical patch size. Suppose a population grows at rate $r$ when sparse and diffuses with coefficient $D$, in a habitat $(0, L)$ outside which it cannot survive ($u = 0$ at the ends). Near $u = 0$ the equation is $u_t = Du_{xx} + ru$, and its modes are $\sin\frac{k\pi x}{L}$ (6A.4 Fourier Methods and Scale Space) with growth rates $r - D\frac{k^2\pi^2}{L^2}$. The first mode grows exactly when $L > \pi\sqrt{D/r} = \pi\ell$. A habitat smaller than $\pi\ell$ loses individuals across its boundary faster than they are born, however large the population inside.

The same principle appears in the Ricci flow in a different guise. Perelman's functional $\lambda(g)$ (12A.2 Ricci Flow as a Gradient Flow) is the first eigenvalue of the operator $-4\Delta + R$, and its monotonicity is the statement that the flow pushes this "growth rate" in one direction. 6B.16 Eigenvalue Problems and Perelman’s λ develops the theory of first eigenvalues and positive ground states that this needs.

9.2Travelling fronts: a PDE reduced to an ODE

A travelling wave is a solution $u(x, t) = U(x - ct)$ that keeps its shape and moves at constant speed $c$. Substituting into $u_t = u_{xx} + f(u)$, with $z = x - ct$, gives the ordinary differential equation

A front joins two zeros of $f$: $U(-\infty) = 1$ and $U(+\infty) = 0$, say. Writing $V = U'$ turns this into the planar system

and a front is an orbit from the equilibrium $(1, 0)$ to the equilibrium $(0, 0)$: a heteroclinic orbit. The PDE question "does a front of speed $c$ exist?" becomes the ODE question "does the unstable manifold of $(1, 0)$ land on $(0, 0)$?", and the answer depends on the type of the equilibria, which depends on $f'(0)$, $f'(1)$ and $c$.

Note the mechanical reading. $U$ is the position of a particle moving in time $z$ in the potential $F(U) = \int_0^Uf$, with friction coefficient $c$: $U'' = -F'(U) - cU'$. The particle starts at rest at the top of a hill at $U = 1$ and must come to rest at $U = 0$. Friction is what makes the speed matter.

9.2.1Monostable fronts: Fisher–KPP

For $f(u) = u(1 - u)$, the state $u = 0$ is unstable for the ODE and $u = 1$ is stable: the reaction is monostable. Linearising the planar system: at $(1, 0)$ the Jacobian is $\begin{pmatrix}0 & 1\\ 1 & -c\end{pmatrix}$, with eigenvalues of opposite signs, a saddle; at $(0, 0)$ it is $\begin{pmatrix}0 & 1\\ -1 & -c\end{pmatrix}$, with eigenvalues $\frac{-c \pm \sqrt{c^2 - 4}}{2}$. For $c \geq 2$ both are real and negative, a stable node; for $0 < c < 2$ they are complex, a stable spiral, and an orbit approaching it would make $U$ negative, which a density cannot be (6A.5 Maximum Principles, exercise on the minimal speed). So $c \geq 2$ is necessary. It is also sufficient.

The proof uses the inequality $\mu + \frac1\mu \leq c$, and that inequality has a meaning. Look for solutions of the linearised equation $u_t = u_{xx} + u$ of the form $e^{-\mu(x - ct)}$: they exist exactly when $c\mu = \mu^2 + 1$, that is $c = \mu + \frac1\mu$. Each decay rate $\mu$ of the front's leading edge determines a speed, and the smallest speed, $c = 2$, comes from $\mu = 1$ (Exercise 9.5). The front is pulled by its leading edge, where $u$ is tiny and the equation is effectively linear: its speed is determined by $f'(0)$ alone, and would be the same for $f(u) = u(1 - u^2)$ or $f(u) = u(1 - u)(1 + u)$, so long as $f(u) \leq f'(0)u$.

Which speed is chosen. Fronts exist for every $c \geq 2$, but the one that appears from localised initial data is the slowest. This is the theorem of Kolmogorov, Petrovsky and Piskunov (1937), for a step initial state: the solution converges in shape to the front of speed $2$, with the position of the level set $u = \frac12$ advancing like $2t + o(t)$. Faster fronts arise only from initial data with slowly decaying tails, $u_0 \sim e^{-\mu x}$ with $\mu < 1$, where the tail itself, growing like $e^{t}$ by the linear equation, prescribes the speed $\mu + \frac1\mu$. Donald Aronson and Hans Weinberger (1975, 1978) extended the result to $\mathbb{R}^n$: from any compactly supported nonnegative nonzero data, the population eventually occupies a ball expanding at speed $2\sqrt{rD}$, however small the initial population (the hair-trigger effect). Maury Bramson (1983) found the next term: for a step initial state the front is at

a logarithmic delay that is invisible in the leading-order theory and reflects the fact that the front is pulled by a region where fluctuations are large; Bramson's proof goes through branching Brownian motion, whose maximum particle the KPP equation describes.

9.2.2Bistable fronts: a unique speed

Now let $f(u) = u(1 - u)(u - a)$ with $0 < a < 1$. Both $0$ and $1$ are stable states of the ODE, separated by the unstable state $a$: the reaction is bistable, as for a population with an Allee effect (below the density $a$, individuals cannot find mates and the population declines), or for an excitable membrane, or for two phases of a material. The planar system now has saddles at both $(1, 0)$ and $(0, 0)$, and a heteroclinic orbit joining two saddles is non-generic: it exists only for one value of $c$. There is a unique front speed, determined by the whole nonlinearity rather than by its derivative at one point.

For the cubic, $\int_0^1u(1 - u)(u - a)\,du = \frac{1 - 2a}{12}$, so the state $1$ invades when $a < \frac12$, the state $0$ invades when $a > \frac12$, and at $a = \frac12$ the front stands still. In mechanical terms, the particle must lose exactly the height difference $F(1) - F(0)$ to friction, and the friction coefficient that does this is unique. The cubic is special in having an explicit front,

(Exercise 9.6), which is classical.

Bistable fronts are stable in a strong sense: Paul Fife and Bryce McLeod (1977) proved that any initial data that is near $1$ far to the left and near $0$ far to the right, with values above $a$ and below $a$ respectively on the two sides, converges exponentially fast to a translate of the front. Monostable fronts are more delicate: perturbations ahead of the front, in the unstable state, can grow and change the speed. The distinction between pulled (monostable, speed from linearisation) and pushed (bistable, speed from the whole nonlinearity) fronts organises much of the physics literature on front propagation, reviewed by Wim van Saarloos (2003).

9.3Bistable interfaces move by mean curvature

In higher dimensions, a bistable equation produces regions where $u \approx 1$ and regions where $u \approx 0$, separated by thin layers in which $u$ follows the one-dimensional front profile across the layer. How do the layers move? Consider the symmetric case, where the two states are $\pm1$ and the front is stationary in one dimension: the Allen–Cahn equation

introduced by Samuel Allen and John Cahn (1979) for the motion of antiphase boundaries in alloys. It is the gradient flow (6A.11 Calculus of Variations and Gradient Flows) of the energy

with time rescaled by $\varepsilon$. The potential $W$ has two wells, at $\pm1$, and the gradient term penalises sharp transitions between them. A transition across a layer costs a definite energy per unit area: the one-dimensional stationary profile is $q(s) = \tanh\frac{s}{\sqrt2}$, with $s$ the signed distance divided by $\varepsilon$, and its energy is

(Exercise 9.7). As $\varepsilon \to 0$, $E_\varepsilon$ concentrates on the interfaces and approaches $\sigma$ times their area: this is the theorem of Luciano Modica and Stefano Mortola (1977), a $\Gamma$-convergence of $E_\varepsilon$ to the perimeter functional. The gradient flow of area is the mean curvature flow, and so one expects the interfaces of Allen–Cahn to move by mean curvature.

The formal computation. Suppose the interface at time $t$ is a hypersurface $\Gamma_t$, and near it $u(x, t) \approx q\big(d(x, t)/\varepsilon\big)$, with $d$ the signed distance to $\Gamma_t$. Then, with $q$ and its derivatives evaluated at $d/\varepsilon$,

because $|\nabla d| = 1$. The profile satisfies $q'' = W'(q)$, so the terms of order $\varepsilon^{-2}$ cancel, and what is left at order $\varepsilon^{-1}$ is

Take $d$ positive outside the region enclosed by $\Gamma_t$. On $\Gamma_t$, $d_t$ is the speed at which the interface moves inwards (towards $d < 0$), and $\Delta d = H$ is its mean curvature, the sum of the principal curvatures, positive for a sphere: for $d = |x| - R$, $\Delta d = \frac{n - 1}{|x|}$. So the interface moves inwards with normal speed equal to its mean curvature. A sphere of radius $R$ in $\mathbb{R}^n$ shrinks by $R' = -\frac{n - 1}{R}$ and disappears at time $\frac{R_0^2}{2(n - 1)}$ (Exercise 9.8): a droplet of one phase inside the other shrinks and vanishes, which is the coarsening that Allen and Cahn set out to explain.

The formal computation has been made rigorous in several ways. Xinfu Chen (1992) and Piero de Mottoni and Michelle Schatzman (1995) proved that the interfaces converge to the classical mean curvature flow as long as that flow is smooth; Lawrence Evans, Halil Mete Soner and Panagiotis Souganidis (1992) proved convergence for all time to the level-set (viscosity) solution of mean curvature flow (6C.9 Hamilton–Jacobi Equations and Viscosity Solutions), which continues through singularities; and Tom Ilmanen (1993) proved convergence to a weak flow in the sense of Brakke. The Allen–Cahn equation is now also a numerical method: to compute a mean curvature flow, or a flow with singularities and topological changes, one solves a semilinear heat equation with small $\varepsilon$, and the interfaces take care of themselves.

So a bistable reaction–diffusion equation, a single semilinear heat equation for a scalar function, contains the mean curvature flow in its singular limit. 6A.10 Curve Shortening and the First Geometric Flows studies that flow directly for curves, where it is curve shortening.

9.4Turing patterns: instability from diffusion

For a single equation, diffusion only stabilises: the linearisation of $u_t = \Delta u + f(u)$ at a stable equilibrium $u_*$ ($f'(u_*) < 0$) has modes $e^{ik\cdot x}$ with growth rates $f'(u_*) - |k|^2 < 0$. Alan Turing (1952) observed that for two species diffusing at different rates this can fail, and that the instability that results has a preferred wavelength. 6C.12 Quasilinear Parabolic Systems uses his example to show that parabolic systems have no maximum principle in general; here we do the analysis that biologists and chemists use.

Consider

with a uniform equilibrium $(u_*, v_*)$, $f = g = 0$ there, and Jacobian $J = \begin{pmatrix}f_u & f_v\\ g_u & g_v\end{pmatrix}$ at it. A small perturbation $\eta e^{ik\cdot x + \sigma t}$, with $\eta \in \mathbb{R}^2$, solves the linearised system when $\sigma$ is an eigenvalue of

This is the Fourier analysis of 6A.4 Fourier Methods and Scale Space applied to a system: each wave vector $k$ evolves independently, by its own $2\times2$ matrix. The equilibrium is stable without diffusion ($k = 0$) when $\operatorname{tr}J < 0$ and $\det J > 0$. Since $\operatorname{tr}J_k = \operatorname{tr}J - |k|^2(d_u + d_v) < 0$ for all $k$, the only way a mode can grow is that $\det J_k < 0$, which gives a real positive eigenvalue. Now

a quadratic in $q$ that is positive at $q = 0$. It becomes negative for some $q > 0$ exactly when its minimum is negative and occurs at positive $q$:

These are Turing's conditions. Together with $f_u + g_v < 0$ they force $f_u$ and $g_v$ to have opposite signs: one species, say $u$, is an activator ($f_u > 0$, it promotes its own production) and the other an inhibitor ($g_v < 0$). And they force $d_v > d_u$: from $f_u + g_v < 0$ we get $|g_v| > f_u$, and from $d_vf_u > d_u|g_v|$ then $d_v > d_u$. In words: local activation and long-range inhibition. The activator, diffusing slowly, amplifies any local excess of itself; the inhibitor it produces diffuses quickly and suppresses activation in a surrounding ring. At the threshold of instability, the first unstable wave number is $q_c = |k_c|^2 = \sqrt{\det J/(d_ud_v)}$, so the pattern has wavelength $\frac{2\pi}{|k_c|}$, set by the chemistry and the diffusion coefficients, not by the size of the domain.

The domain decides which modes exist. On an interval $(0, L)$ with no-flux boundary conditions, the allowed modes are $\cos\frac{m\pi x}{L}$, $q = \frac{m^2\pi^2}{L^2}$, and a pattern forms only if one of these lies in the unstable band. A domain much shorter than the wavelength is uniform; as the domain grows, more modes enter the band, and the pattern adds stripes. Shigeru Kondo and Rihito Asai (1995) observed exactly this in the angelfish Pomacanthus: as the fish grows, its stripes do not stretch but split and insert new stripes, keeping their spacing, as a Turing system on a growing domain predicts.

Beyond linearisation. Linear analysis predicts which wavelength grows, not what the final pattern looks like. Near threshold, weakly nonlinear analysis (amplitude equations) decides between stripes, hexagonal spots and other planforms; far from threshold, the patterns can be complex. John Pearson (1993) computed, for the two-species Gray–Scott model of Peter Gray and Stephen Scott, a zoo of spots that divide, labyrinths and travelling fronts, all from one pair of semilinear heat equations.

9.5The curvature of a Ricci flow is a reaction–diffusion system

Under the Ricci flow $\partial_tg = -2\operatorname{Ric}$, the curvature evolves by equations of exactly the kind studied here (11A.2 How Curvature Evolves, 11A.3 Curvature in a Moving Frame):

the second in Uhlenbeck's moving frame, with $Q$ a quadratic expression in the curvature operator. Diffusion is the Laplacian of the evolving metric; the reaction is quadratic, like the $u^2$ of 6A.8 Nonlinear Parabolic Equations. Everything in this chapter has a counterpart.

ODE comparison for the scalar curvature. Since $|\operatorname{Ric}|^2 \geq \frac1nR^2$ (Cauchy–Schwarz on the eigenvalues of $\operatorname{Ric}$, whose sum is $R$), the scalar curvature is a supersolution of $u_t = \Delta u + \frac2nu^2$. On a closed manifold the maximum principle compares $R_{\min}(t)$ with the solution of the ODE $y' = \frac2ny^2$:

On a surface, $\operatorname{Ric} = \frac R2g$ and the equation is exactly $\partial_tR = \Delta R + R^2$: the semilinear equation of 6A.8 Nonlinear Parabolic Equations, with the Laplacian of the evolving metric. The round sphere is its spatially constant solution, $R = \frac{2}{1 - 2t}$ for the unit sphere, which blows up at the ODE rate, and Hamilton's study of the flow on surfaces (11A.12 Ricci Flow on Surfaces) is the study of how far a general solution stays from this ODE behaviour.

Invariant sets of the reaction. In dimension three, the eigenvalues $\lambda \geq \mu \geq \nu$ of the curvature operator evolve, under the reaction ODE alone, by

in Hamilton's normalisation (11A.3 Curvature in a Moving Frame). Hamilton's maximum principle for systems (11A.7 Maximum Principles under Ricci Flow), the tensor version of the invariant-region theorem of 6A.5 Maximum Principles, says that a closed convex set of curvature operators that is invariant under parallel translation and preserved by this ODE is preserved by the PDE. This is the same logic as the invariant rectangles of a two-species model: diffusion averages, averaging keeps you inside a convex set, so only the reaction can push you out. The ODE then does the geometry. If $\nu > 0$, the difference $\lambda - \nu$ grows more slowly than $\nu$ does (Exercise 9.10), so the curvature operator becomes relatively closer to a multiple of the identity, which is the pinching toward constant curvature behind Hamilton's 1982 theorem (11A.11 Hamilton’s 1982 Theorem). The Hamilton–Ivey estimate (11A.10 Hamilton–Ivey Pinching), which says that where curvature is large it is nearly nonnegative, comes from an invariant set of the same ODE, and it is one of the facts that Perelman and Morgan–Tian take as the starting point for the analysis of singularities.

Where reaction wins. The ODE $y' = y^2$ blows up; diffusion resists by spreading curvature out. Near a singularity of the Ricci flow, curvature is so large that, on the scale of the curvature, reaction dominates: this is why rescaled limits of singularities are ancient solutions on which the curvature is pinched to be nonnegative, and why the round shrinking cylinder, whose curvature evolves by the pure ODE, is the model of a neckpinch (11B.5 Singularities, 11B.6 Rotationally Symmetric Flows and the Standard Solution). In a rotationally symmetric neck, the cross-sectional spheres shrink by the ODE while diffusion along the neck tries to even them out; Sigurd Angenent and Dan Knopf (2004) showed that the neck pinches at the ODE rate $|\operatorname{Rm}| \sim \frac{1}{T - t}$, a type I singularity in the language of 6A.8 Nonlinear Parabolic Equations. The balance of reaction and diffusion that sets the width of a Fisher front sets the shape of a neck.

9.6History

Fisher's and Kolmogorov, Petrovsky and Piskunov's papers on the spread of advantageous genes appeared in 1937, and Zel'dovich and Frank-Kamenetskii's theory of flame fronts in 1938. Skellam's study of dispersal and the critical patch was published in 1951 and Kierstead and Slobodkin's in 1953. Hodgkin and Huxley's model of the nerve impulse appeared in 1952, the same year as Turing's "The chemical basis of morphogenesis"; FitzHugh's reduction followed in 1961 and Nagumo, Arimoto and Yoshizawa's circuit in 1962. Gierer and Meinhardt formulated activator–inhibitor models in 1972. Aronson and Weinberger's spreading theorems date from 1975 and 1978, Fife and McLeod's stability theorem for bistable fronts from 1977, Allen and Cahn's equation from 1979 and Bramson's logarithmic correction from 1983. Turing patterns were seen in a chemical reaction in 1990. The convergence of Allen–Cahn interfaces to mean curvature flow was proved between 1992 and 1995. Hamilton introduced the maximum principle for systems in 1986.

9.7Exercises

Solution

(a) $u_t = \mu cu$, $u_{xx} = \mu^2u$, so the equation is $\mu c = \mu^2 + 1$. By AM–GM, $\mu + \frac1\mu \geq 2$ with equality at $\mu = 1$. (b) $\mu c = D\mu^2 + r$, so $c = D\mu + \frac r\mu \geq 2\sqrt{rD}$, with equality at $\mu = \sqrt{r/D}$. (c) Since $f(u) \leq ru$, the exponential $w = Ae^{-\mu(x - ct)}$ with $c = c(\mu)$ satisfies $w_t - Dw_{xx} = rw \geq f(w)$, and the constant $1$ satisfies $0 \geq f(1) = 0$; the minimum of two supersolutions is a supersolution (in the sense of the comparison principle). So a solution starting below $\min\{1, w\}$ stays below it, and its level sets move no faster than $c(\mu)$. Taking $\mu = \sqrt{r/D}$ (possible for data decaying at least that fast, in particular for compactly supported data) gives speed at most $2\sqrt{rD}$. The lower bound requires the KPP theorem.

Solution

(a) With $U' = -\frac1{\sqrt2}U(1 - U)$, differentiate: $U'' = -\frac1{\sqrt2}(1 - 2U)U' = \frac12(1 - 2U)U(1 - U)$. Then $U'' + cU' + f(U) = U(1 - U)\big[\frac12(1 - 2U) - \frac{c}{\sqrt2} + (U - a)\big] = U(1 - U)\big[\frac12 - a - \frac{c}{\sqrt2}\big]$, which vanishes exactly when $c = \sqrt2(\frac12 - a)$. (b) $\int U'^2\,dz = \int_0^1|U'|\,du = \frac1{\sqrt2}\int_0^1u(1 - u)\,du = \frac{1}{6\sqrt2}$, substituting $u = U(z)$, $du = U'dz$. And $\int_0^1u(1 - u)(u - a)\,du = \frac{1 - 2a}{12}$. Their ratio is $\frac{1 - 2a}{12}\cdot6\sqrt2 = \frac{\sqrt2(1 - 2a)}{2} = \sqrt2(\frac12 - a) = c$.

Solution

(a) With $s = \sqrt2y$, $q = \tanh y$, $q' = \frac{1}{\sqrt2}\operatorname{sech}^2y = \frac{1}{\sqrt2}(1 - q^2)$ and $q'' = \frac{1}{\sqrt2}(-2q)q' = -q(1 - q^2) = q^3 - q$. (b) Multiply $q'' = W'(q)$ by $q'$: $\frac{d}{ds}\big(\frac12q'^2 - W(q)\big) = 0$, and at $s \to -\infty$ both terms vanish. (c) By (b) the integrand is $q'^2 = q'\sqrt{2W(q)}$, and substituting $u = q(s)$ gives $\int_{-1}^1\sqrt{2W(u)}\,du = \int_{-1}^1\frac{1 - u^2}{\sqrt2}\,du = \frac{1}{\sqrt2}\cdot\frac43 = \frac{2\sqrt2}{3}$.

Solution

With $r = |x|$ and $s = (r - R)/\varepsilon$: $u_t = -\frac{R'}{\varepsilon}q'$, and the radial Laplacian gives $\Delta u = \frac{1}{\varepsilon^2}q'' + \frac{n - 1}{r\varepsilon}q'$. Since $q'' = W'(q)$, $u_t - \Delta u + \varepsilon^{-2}W'(u) = -\frac{q'}{\varepsilon}\big(R' + \frac{n - 1}{r}\big)$. Near $r = R$, $\frac{n - 1}{r} = \frac{n - 1}{R} + O(\varepsilon)$, so the residual is $O(1)$ exactly when $R' = -\frac{n - 1}{R}$. Then $(R^2)' = -2(n - 1)$, $R(t)^2 = R_0^2 - 2(n - 1)t$, and the droplet vanishes at $t = \frac{R_0^2}{2(n - 1)}$. For $n = 2$ this is $R^2 = R_0^2 - 2t$, the shrinking circle of curve shortening, which vanishes at $\frac{R_0^2}{2}$.

Solution

(a) Adding the equations at equilibrium gives $a + b - u = 0$, so $u_* = a + b$ and $v_* = \frac{b}{(a + b)^2}$. The Jacobian is $f_u = -1 + 2u_*v_* = \frac{b - a}{a + b}$, $f_v = u_*^2 = (a + b)^2$, $g_u = -2u_*v_* = -\frac{2b}{a + b}$, $g_v = -(a + b)^2$. (b) With $a = 0.1$, $b = 0.9$: $u_* = 1$, $v_* = 0.9$, $J = \begin{pmatrix}0.8 & 1\\ -1.8 & -1\end{pmatrix}$, $\operatorname{tr}J = -0.2 < 0$ and $\det J = -0.8 + 1.8 = 1 > 0$: stable. Turing's conditions with $d_u = 1$, $d_v = d$: $0.8d - 1 > 0$ and $(0.8d - 1)^2 > 4d$, that is $16d^2 - 140d + 25 > 0$, whose larger root is $d_c = \frac{35 + 15\sqrt5}{8} \approx 8.57$. (The smaller root, about $0.18$, violates $0.8d > 1$.) So the equilibrium is Turing unstable exactly for $d > d_c$. (c) At threshold $q_c = \sqrt{\det J/(d_ud_v)} = d_c^{-1/2} \approx 0.342$, so $|k_c| \approx 0.585$ and the wavelength is $\frac{2\pi}{|k_c|} \approx 10.7$ in the units of the equation.

Solution

(a) $\lambda' - \nu' = \lambda^2 - \nu^2 + \mu\nu - \lambda\mu = (\lambda - \nu)(\lambda + \nu) - \mu(\lambda - \nu)$. (b) $\frac{d}{dt}\log(\lambda - \nu) = \lambda + \nu - \mu$ and $\frac{d}{dt}\log\nu = \nu + \frac{\lambda\mu}{\nu}$; subtracting gives $\lambda - \mu - \frac{\lambda\mu}{\nu} = \lambda(1 - \frac\mu\nu) - \mu$. Since $\mu \geq \nu$, $1 - \frac\mu\nu \leq 0$, and $\mu > 0$, so the derivative is negative. (c) The ratio $\frac{\lambda - \nu}{\nu}$ measures how far the curvature operator is from a multiple of the identity, relative to its size. The ODE decreases it while all eigenvalues tend to infinity in finite time, so at the blow-up time the curvature is relatively round. Hamilton's maximum principle for systems (11A.7 Maximum Principles under Ricci Flow) transfers invariant sets of the form $\{\lambda - \nu \leq C(\lambda + \mu + \nu)^{1 - \delta}\}$ (a sharper form of this computation, valid under the weaker hypothesis of positive Ricci curvature) from the ODE to the PDE, and rescaling the flow near the blow-up time then gives a metric of constant positive curvature (11A.11 Hamilton’s 1982 Theorem). The diffusion term cannot undo pinching, only the reaction could, and the reaction improves it.

Solution

(a) As in Proposition 9.4 with $n = 2$: $R_{\min}(t) \geq \frac{R_{\min}(0)}{1 - R_{\min}(0)t}$, which tends to infinity at $t = \frac{1}{R_{\min}(0)}$. (b) The unit sphere has $R = 2$ and $g(t) = (1 - 2t)g_0$, so $R(t) = \frac{2}{1 - 2t}$, which solves $R' = R^2$ and becomes infinite at $t = \frac12 = \frac{1}{R_{\min}(0)}$: the bound is sharp for the round sphere. (c) By Gauss–Bonnet, $\int R\,dA = 2\int K\,dA = 4\pi\chi(S^2) = 8\pi$ at every time, so $\operatorname{Area}(t) = A_0 - 8\pi t$, which reaches zero at $t = \frac{A_0}{8\pi}$. The flow exists until then (by Hamilton and Chow's theorem, 11A.12 Ricci Flow on Surfaces), so the extinction time is exactly $\frac{A_0}{8\pi}$; for the unit sphere, $\frac{4\pi}{8\pi} = \frac12$.