Course 6 Book 6A: The Heat Equation and Its RelativesChapter 4

Maximum Principles

Weak and strong maximum principles, comparison, and invariant regions for systems.

30 min read · Updated Oct 2, 2026

Read with Evans, Partial Differential Equations, section 2.3.3 (the maximum principle for the heat equation), section 6.4 (maximum principles for elliptic operators, including Hopf's lemma) and section 7.1.4 (maximum principles for parabolic operators). Protter and Weinberger, Maximum Principles in Differential Equations, is the classic reference.

In this chapter · 9 sections

4.1No cold spot forms by itself
4.2The core argument
4.3The weak maximum principle
4.4The strong maximum principle and Hopf's lemma
4.5Comparison principles
4.6Systems and invariant regions
4.7Invasion fronts
4.8History
4.9Exercises

The maximum principle is the most important single tool in the analysis of the Ricci flow. Hamilton's 1982 theorem, the Hamilton–Ivey pinching estimate, the preservation of positive curvature, the lower bound on scalar curvature that Perelman uses at every step: all of them are maximum principles. And every one of them rests on an argument that fits in one line, which this chapter isolates, proves in its various forms, and then extends from single equations to systems.

The line is this. At a point where a function first reaches a new maximum, its gradient vanishes, its Hessian is negative semidefinite, and its time derivative is non-negative. So $u_t \geq 0 \geq \Delta u$ there. If an equation says $u_t - \Delta u$ is negative at such a point, the point cannot exist. You have met each half of this argument before: the extreme value theorem (2A.9 Continuous Functions), the derivative at a maximum (2A.10 Derivatives), and the second-derivative test (2B.8 Calculus in Several Variables). This chapter combines them.

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

state and prove the weak maximum principle for parabolic and elliptic operators with lower-order terms;
state Hopf's lemma and the strong maximum principle, and prove Hopf's lemma with a barrier;
prove comparison principles for semilinear equations, and compare a PDE with an ODE;
explain why maximum principles fail for systems in general, and prove that invariant convex sets of the reaction ODE are invariant for the reaction–diffusion system;
recognise the invariant-region argument as the ancestor of Hamilton's tensor maximum principle.

No cold spot forms by itself

In the world Model Thermal design by the maximum principle

Heat flows from hot to cold. In a body with no internal heat sources, conduction can only smooth temperatures out, so no new hot spot or cold spot can form inside: the maximum and minimum temperatures over the whole history are reached at the start or on the surface. This is the weak maximum principle for the heat equation (6A.3 The Heat Equation on ℝⁿ, Theorem 4.1). Engineers use it in both directions. To bound the temperature of a component that conducts heat, it suffices to bound the initial temperature and the temperatures imposed on its surface; the interior can be no worse. And when there are sources, the principle becomes a comparison: if the heating rate is at most $q$ , the temperature is at most that of a body heated uniformly at rate $q$ with the same surface conditions, which can often be computed by hand (Theorem 4.4).

The core argument

The idea The one-line maximum principle

Let $u$ be smooth on a region of space-time, and suppose $(x_0, t_0)$ is a point where $u(\cdot, t_0)$ has a spatial maximum at an interior point $x_0$ , and $u(x_0, \cdot)$ was smaller or equal just before $t_0$ . Then at $(x_0, t_0)$ :

\nabla u = 0, \qquad \nabla^2u \leq 0\ \text{(so } \Delta u \leq 0\text{)}, \qquad u_t \geq 0.

Every maximum principle in this book, and every maximum principle for the Ricci flow in 11A.4 Maximum Principles under Ricci Flow, is this observation plus a way to rule out the equality cases.

The first two are the first- and second-order conditions for an interior maximum (2B.8 Calculus in Several Variables); a negative semidefinite matrix has non-positive trace. The third is the one-sided derivative at a point the function has reached from below (2A.10 Derivatives). More generally, if $(a^{ij})$ is positive semidefinite and $\nabla^2u \leq 0$ , then $\sum a^{ij}\partial_i\partial_ju = \operatorname{tr}(A\nabla^2u) \leq 0$ (the trace of a product of a positive and a negative semidefinite matrix is non-positive, Exercise 4.6). So the argument works for any operator whose second-order part has a positive semidefinite coefficient matrix: any elliptic operator (6A.1 What a PDE Is).

Figure 4.1. The first time

t_0

at which

\max_xu(\cdot, t)

reaches a new value, at an interior point

x_0

. The spatial profile has a peak there (

\nabla u = 0

\nabla^2u \leq 0

), and the value at

x_0

has been rising (

u_t \geq 0

). An equation that forces

u_t - \Delta u < 0

makes this impossible.

The weak maximum principle

Let $U \subset \mathbb{R}^n$ be bounded and open, $U_T = U\times(0, T]$ and $\Gamma_T$ its parabolic boundary (6A.3 The Heat Equation on ℝⁿ). Consider the operator

Lu = u_t - \sum_{i,j}a^{ij}(x, t)\partial_i\partial_ju + \sum_ib^i(x, t)\partial_iu + c(x, t)u,

with continuous bounded coefficients and $(a^{ij})$ positive semidefinite (for the strong principle we will need uniform parabolicity, $\sum a^{ij}\xi_i\xi_j \geq \theta|\xi|^2$ with $\theta > 0$ ).

Theorem 4.1 The weak maximum principle

Let $u \in C^{2,1}(U_T)\cap C(\overline{U_T})$ .

If $c = 0$ and $Lu \leq 0$ in $U_T$ , then $\max_{\overline{U_T}}u = \max_{\Gamma_T}u$ .
If $c \geq 0$ and $Lu \leq 0$ in $U_T$ , then $\max_{\overline{U_T}}u \leq \max_{\Gamma_T}u^+$ , where $u^+ = \max(u, 0)$ .

Proof. (1) Suppose first $Lu < 0$ . If the maximum over $\overline{U_T}$ were attained at $(x_0, t_0)$ with $x_0 \in U$ and $0 < t_0 \leq T$ , then by the core argument $u_t \geq 0$ , $\nabla u = 0$ and $\sum a^{ij}\partial_i\partial_ju \leq 0$ there, so $Lu \geq 0$ : a contradiction. In general let $u^\varepsilon = u - \varepsilon t$ ; then $Lu^\varepsilon = Lu - \varepsilon < 0$ , so $\max u^\varepsilon$ is attained on $\Gamma_T$ , and letting $\varepsilon \to 0$ gives the claim.

(2) If $\max u \leq 0$ there is nothing to prove. Otherwise, on the open set $V = \{u > 0\}\cap U_T$ , the operator $L'u = Lu - cu$ satisfies $L'u = Lu - cu \leq 0$ , since $c \geq 0$ and $u > 0$ there. Apply (1) to $L'$ on (the components of) $V$ : the positive maximum is attained on the parabolic boundary of $V$ , where either $u = 0$ or the point lies on $\Gamma_T$ .

The sign condition on $c$ matters: for $u_t = u_{xx} + u$ , the function $e^t\sin x$ on $(0, \pi)$ is zero on the sides, positive inside, and grows. With $c < 0$ the equation has a source proportional to $u$ , and the maximum can grow. That is exactly what happens in the reaction terms below, and it is handled by comparison with an ODE rather than forbidden.

The elliptic case is the time-independent version: if $-\sum a^{ij}\partial_i\partial_ju + \sum b^i\partial_iu \leq 0$ in a bounded $U$ with $(a^{ij})$ uniformly positive definite, then $\max_{\overline U}u = \max_{\partial U}u$ (Evans, section 6.4.1). The proof perturbs by $\varepsilon e^{\lambda x_1}$ instead of $\varepsilon t$ , using the uniform ellipticity to make the perturbation strictly subsolution (Exercise 4.7).

The strong maximum principle and Hopf's lemma

The weak principle says the maximum is attained on the boundary. The strong principle says it is attained only there, unless the function is constant. For harmonic functions this came from the mean value property (6A.2 Harmonic Functions). For general operators, which have no mean value property, it comes from a lemma about the boundary.

Lemma 4.2 Hopf's lemma

Let $u \in C^2(U)\cap C^1(\overline U)$ satisfy $-\sum a^{ij}\partial_i\partial_ju + \sum b^i\partial_iu \leq 0$ in $U$ , with $(a^{ij})$ uniformly elliptic. Suppose $x_0 \in \partial U$ , $u(x_0) > u(x)$ for all $x \in U$ , and $U$ satisfies an interior ball condition at $x_0$ : there is a ball $B \subset U$ with $x_0 \in \partial B$ . Then the outward normal derivative satisfies

\frac{\partial u}{\partial\nu}(x_0) > 0.

Proof. Translate so that $B = B_r(0)$ . Consider the barrier $v(x) = e^{-\lambda|x|^2} - e^{-\lambda r^2}$ , which is positive in $B$ , zero on $\partial B$ , and has $\frac{\partial v}{\partial\nu} = -2\lambda re^{-\lambda r^2} < 0$ on $\partial B$ . A computation (Exercise 4.8) shows that for $\lambda$ large, $v$ satisfies the strict inequality $-\sum a^{ij}\partial_i\partial_jv + \sum b^i\partial_iv < 0$ in the annulus $R = B_r(0)\setminus\overline{B_{r/2}(0)}$ . On $\partial B_{r/2}$ , $u < u(x_0)$ strictly, so for small $\varepsilon > 0$ , $w = u + \varepsilon v - u(x_0) \leq 0$ there; on $\partial B_r$ , $v = 0$ and $u \leq u(x_0)$ , so $w \leq 0$ . By the weak maximum principle on $R$ , $w \leq 0$ in $R$ , with $w(x_0) = 0$ . So $w$ has a maximum at $x_0$ over $\overline R$ , and $\frac{\partial w}{\partial\nu}(x_0) \geq 0$ , i.e. $\frac{\partial u}{\partial\nu}(x_0) \geq -\varepsilon\frac{\partial v}{\partial\nu}(x_0) > 0$ .

The barrier is the important idea: an explicit function, built to satisfy a strict inequality, that is compared with the unknown solution by the weak maximum principle. Barriers are how boundary behaviour is controlled throughout elliptic and parabolic theory.

Theorem 4.3 The strong maximum principle

Let $U$ be connected and $u \in C^2(U)$ satisfy $-\sum a^{ij}\partial_i\partial_ju + \sum b^i\partial_iu \leq 0$ , with $(a^{ij})$ uniformly elliptic. If $u$ attains its maximum over $U$ at an interior point, then $u$ is constant. The parabolic version holds too: if $u_t - \sum a^{ij}\partial_i\partial_ju + \sum b^i\partial_iu \leq 0$ in $U_T$ , with $U$ connected, and $u$ attains its maximum over $\overline{U_T}$ at $(x_0, t_0)$ with $x_0 \in U$ , $0 < t_0 \leq T$ , then $u$ is constant on $U_{t_0}$ .

Proof. (Elliptic case.) Let $M = \max u$ and $C = \{u = M\}$ , closed in $U$ and non-empty. If $V = \{u < M\}$ were non-empty, choose a point of $V$ closer to $C$ than to $\partial U$ , and the largest ball in $V$ around it; this ball touches $C$ at some $x_0$ in the interior of $U$ , and $V$ satisfies the interior ball condition there. Hopf's lemma applied on that ball gives $\partial_\nu u(x_0) > 0$ . But $x_0$ is an interior maximum of $u$ , so $\nabla u(x_0) = 0$ . So $V$ is empty. (The parabolic case is due to Louis Nirenberg, 1953; Evans, section 7.1.4.)

Where this goes The strong maximum principle for curvature

Hamilton's strong maximum principle for tensors says, for instance, that under the Ricci flow a manifold whose curvature operator is non-negative either has positive curvature operator immediately, or its curvature operator has a kernel that is invariant under parallel transport, and then the manifold locally splits as a product (11A.4 Maximum Principles under Ricci Flow). That is how κ-solutions with a zero curvature direction are shown to split off a line (12B.2 The Structure of κ-Solutions). The scalar strong maximum principle here is its first case.

Comparison principles

Most equations of interest are nonlinear. The maximum principle extends to them through comparison: a solution lying below another at the start and on the boundary stays below.

Theorem 4.4 Comparison for semilinear equations

Let $f : \mathbb{R}\to\mathbb{R}$ be Lipschitz with constant $K$ . Suppose $u$ , $v \in C^{2,1}(U_T)\cap C(\overline{U_T})$ satisfy

u_t - \Delta u - f(u) \leq 0 \leq v_t - \Delta v - f(v) \quad\text{in } U_T,

( $u$ is a subsolution, $v$ a supersolution) and $u \leq v$ on $\Gamma_T$ . Then $u \leq v$ in $U_T$ .

Proof. Let $w = u - v$ . Where $w > 0$ , $f(u) - f(v) = c(x, t)w$ with $c = \frac{f(u) - f(v)}{u - v}$ , $|c| \leq K$ . So $w_t - \Delta w - cw \leq 0$ where $w > 0$ . Let $\tilde w = e^{-2Kt}w$ : then $\tilde w_t - \Delta\tilde w + (2K - c)\tilde w \leq 0$ there, with $2K - c \geq K > 0$ . By part 2 of Theorem 4.1 (applied on the set where $\tilde w > 0$ ), $\max\tilde w \leq \max_{\Gamma_T}\tilde w^+ = 0$ . So $w \leq 0$ .

Two uses recur.

Comparison with an ODE. Spatially constant functions $y(t)$ solve $u_t = \Delta u + f(u)$ exactly when $y' = f(y)$ . So on a domain where boundary terms are controlled (all of $\mathbb{R}^n$ with bounded solutions, a closed manifold, or a domain with suitable boundary values), a solution that starts above $y(0)$ stays above $y(t)$ . The PDE is at least as good as its ODE. For example, a solution of $u_t = \Delta u + u^2$ on a closed manifold with $\min u(\cdot, 0) = m > 0$ satisfies $u \geq \frac{m}{1 - mt}$ and must blow up by time $1/m$ (6A.7 Nonlinear Parabolic Equations).

Sandwiching. A solution can be trapped between an explicit subsolution and an explicit supersolution. This is how front speeds are bounded (Figure 4.3), how blow-up times are estimated, and how, in geometric flows, a flowing curve or surface is compared with explicit shrinking spheres: a closed curve inside a circle under curve shortening stays inside the shrinking circle and so must become extinct before it does (6A.8 Curve Shortening and the First Geometric Flows).

Systems and invariant regions

For a system $u = (u^1, \dots, u^m)$ , maximum principles componentwise generally fail. If the components are coupled in their second-order terms, or in their reaction terms, one component can be pushed above its initial maximum by another. What survives is a geometric statement: a closed convex set in the space of values that the reaction terms never push solutions out of is never left by the solution of the full system.

Theorem 4.5 Invariant regions (Weinberger; Chueh, Conley and Smoller)

Let $F : \mathbb{R}^m \to \mathbb{R}^m$ be locally Lipschitz, and let $K \subset \mathbb{R}^m$ be closed and convex. Suppose $K$ is invariant for the ODE $y' = F(y)$ : solutions that start in $K$ stay in $K$ . Let $u : M\times[0, T] \to \mathbb{R}^m$ be a smooth solution of the reaction–diffusion system

u_t = \Delta u + F(u),

with the same diffusion operator $\Delta$ acting on every component, on a closed manifold $M$ (or on $\mathbb{R}^n$ with bounded solutions, or on a bounded domain with values in $K$ on the boundary). If $u(x, 0) \in K$ for all $x$ , then $u(x, t) \in K$ for all $x$ and $t$ .

Proof. (Sketch.) A closed convex set is the intersection of the closed half-spaces containing it, $K = \bigcap\{y : \ell(y) \leq c_\ell\}$ , over linear functionals $\ell$ supporting $K$ (4A.3 Hahn–Banach and Duality). For each such $\ell$ , the scalar function $\phi = \ell(u)$ satisfies

\phi_t = \Delta\phi + \ell(F(u)),

because $\ell$ is linear and the same $\Delta$ acts on every component. At a point where $\phi$ first reaches $c_\ell$ , the value $u$ is on the boundary of $K$ with $\ell$ supporting $K$ there, and ODE-invariance of $K$ means $F(u)$ does not point out of $K$ , so $\ell(F(u)) \leq 0$ . Then the core argument rules out $\phi$ crossing $c_\ell$ , after an $\varepsilon$ -perturbation as in Theorem 4.1; the full proof replaces this by an argument with the distance from $u$ to $K$ , which handles all the half-spaces at once (Chueh, Conley and Smoller, 1977).

The hypotheses matter. Convexity is used to write $K$ by linear inequalities, which commute with $\Delta$ . Equal diffusion is used so that $\ell(u)$ solves a scalar heat equation; with different diffusion rates in different components only rectangles (products of intervals) remain invariant in general (Exercise 4.11).

The idea From invariant regions to Hamilton's maximum principle

Under the Ricci flow the curvature operator $\operatorname{Rm}$ evolves by a reaction–diffusion equation

\partial_t\operatorname{Rm} = \Delta\operatorname{Rm} + \operatorname{Rm}^2 + \operatorname{Rm}^\#,

where the reaction term is quadratic. Hamilton's tensor maximum principle (1986) is Theorem 4.5 in this setting: a closed convex set of curvature operators, invariant under parallel transport, that is preserved by the ODE $\frac{d}{dt}\operatorname{Rm} = \operatorname{Rm}^2 + \operatorname{Rm}^\#$ , is preserved by the Ricci flow. Positive curvature operator, positive Ricci curvature in dimension 3, and the Hamilton–Ivey pinching set are all proved invariant this way, by checking an ODE (11A.4 Maximum Principles under Ricci Flow, 11A.5 Hamilton–Ivey Pinching). The figure Figure 4.2 is redrawn for curvature in 11A.4 Maximum Principles under Ricci Flow.

Figure 4.2. The reaction field

F(u, v) = \big(u(1 - u - 0.5v),\ v(1 - v - 0.6u)\big)

of a two-species competition model, and the square

K = [0, 1]^2

(shaded). On every edge

F

points into

K

or along the edge, so

K

is invariant for the ODE, and by Theorem 4.5 the reaction–diffusion system

u_t = \Delta u + F(u)

keeps both population densities between

0

and

1

Invasion fronts

In the world Model Fisher–KPP and the spread of the muskrat

In 1937 Ronald Fisher, and independently Andrey Kolmogorov, Ivan Petrovsky and Nikolai Piskunov, studied

u_t = Du_{xx} + ru(1 - u),

a model of a population (or an advantageous gene) that diffuses with coefficient $D$ and grows logistically at rate $r$ , with $u$ the density as a fraction of the carrying capacity. The interval $[0, 1]$ is invariant for the ODE $y' = ry(1 - y)$ , so by comparison with the constant solutions $0$ and $1$ every solution starting between $0$ and $1$ stays there (Exercise 4.9). A population introduced in a small region spreads as a travelling front, and the front moves at the speed

c = 2\sqrt{rD},

the slowest speed for which a front connecting $0$ to $1$ with $u \geq 0$ exists (Exercise 4.10, Figure 4.3).

The linear rate is a striking prediction: the radius of the occupied region grows proportionally to time, not to $\sqrt t$ as for pure diffusion, because growth at the edge keeps refilling it. John Skellam tested it in 1951 on the spread of the muskrat in central Europe, after a few animals were released on an estate southwest of Prague in 1905. Plotting the square root of the area of the occupied range, proportional to its radius, against the year, he found the points close to a straight line, as the model predicts. Similar spreading rates have since been fitted for many invasive species, with the caveats that real landscapes are not uniform and that long-distance jumps can make fronts accelerate.

Figure 4.3. The Fisher–KPP equation

u_t = u_{xx} + u(1 - u)

from a step initial state, at

t = 0, 4, 8, \dots, 24

(computed by an explicit finite-difference scheme). The profiles stay between

0

and

1

, as the invariant region requires, and the front (where

u = \frac12

) advances at a speed approaching the predicted

2\sqrt{rD} = 2

from below.

Chemistry. Concentrations in reaction–diffusion models of chemical systems should stay non-negative. If each reaction rate $F^i(u)$ is non-negative whenever $u^i = 0$ and the other concentrations are non-negative (a substance can't be consumed when there is none of it), then the non-negative orthant is invariant for the ODE, and by the theorem (in its version for products of intervals, which allows different diffusion rates) for the reaction–diffusion system too. Models without this property are physically wrong, and the maximum principle is how one checks.

History

The maximum principle for harmonic functions was known in the nineteenth century; for general second-order elliptic operators the strong maximum principle is due to Eberhard Hopf (1927), and the boundary lemma to Hopf (1952) and, independently, Olga Oleinik (1952), with an earlier version by Stanisław Zaremba (1910). Louis Nirenberg proved the strong maximum principle for parabolic equations in 1953. Invariant regions for reaction–diffusion systems were introduced by Hans Weinberger (1975) and developed by Kuan-Nan Chueh, Charles Conley and Joel Smoller (1977). Richard Hamilton's tensor maximum principle appeared in his 1986 paper on four-manifolds with positive curvature operator. Fisher's and Kolmogorov–Petrovsky–Piskunov's papers are from 1937, and Skellam's from 1951.

Recall Where we stand

At a first interior maximum, $\nabla u = 0$ , $\nabla^2u \leq 0$ and $u_t \geq 0$ ; perturbing by $\varepsilon t$ turns this into the weak maximum principle for parabolic operators, and by $\varepsilon e^{\lambda x_1}$ for elliptic ones, with $c \geq 0$ allowed for non-negative maxima. Hopf's lemma, proved with a barrier, gives the strong maximum principle. Comparison principles extend all this to semilinear equations, and comparison with the ODE $y' = f(y)$ bounds solutions of $u_t = \Delta u + f(u)$ . For systems with equal diffusion, closed convex sets invariant under the reaction ODE are invariant under the PDE: the ancestor of Hamilton's tensor maximum principle. 6A.5 Weak Solutions and Elliptic Regularity turns from pointwise methods to energy methods and weak solutions.

Exercises

Exercise 4.6 A trace inequality

Let $A$ be symmetric positive semidefinite and $H$ symmetric negative semidefinite. Show that $\operatorname{tr}(AH) \leq 0$ . (Diagonalise $A = \sum\lambda_ke_ke_k^T$ with $\lambda_k \geq 0$ , 1A.6 Symmetric Matrices and the Spectral Theorem, and write $\operatorname{tr}(AH) = \sum\lambda_ke_k^THe_k$ .)

Exercise 4.7 The elliptic weak maximum principle

Let $Lu = -\sum a^{ij}\partial_i\partial_ju + \sum b^i\partial_iu$ with $a^{11} \geq \theta > 0$ and $|b| \leq \beta$ . Show that $v = e^{\lambda x_1}$ satisfies $Lv < 0$ if $\lambda > \beta/\theta$ , and use $u + \varepsilon v$ to prove that $Lu \leq 0$ in a bounded $U$ implies $\max_{\overline U}u = \max_{\partial U}u$ .

Solution

$Lv = (-a^{11}\lambda^2 + b^1\lambda)e^{\lambda x_1} \leq \lambda(-\theta\lambda + \beta)e^{\lambda x_1} < 0$ . Then $L(u + \varepsilon v) < 0$ , so $u + \varepsilon v$ cannot have an interior maximum (at one, $Lu + \varepsilon Lv \geq 0$ by the core argument). Hence $\max_{\overline U}u \leq \max_{\overline U}(u + \varepsilon v) = \max_{\partial U}(u + \varepsilon v) \leq \max_{\partial U}u + \varepsilon\max_{\overline U}v$ ; let $\varepsilon \to 0$ .

Exercise 4.8 The Hopf barrier

For $v = e^{-\lambda|x|^2} - e^{-\lambda r^2}$ , compute $\partial_iv = -2\lambda x_ie^{-\lambda|x|^2}$ and $\partial_i\partial_jv = (4\lambda^2x_ix_j - 2\lambda\delta_{ij})e^{-\lambda|x|^2}$ . Show that on $r/2 \leq |x| \leq r$ ,

-\sum a^{ij}\partial_i\partial_jv + \sum b^i\partial_iv \leq e^{-\lambda|x|^2}\big(-4\theta\lambda^2|x|^2 + 2\lambda\operatorname{tr}A + 2\lambda|b||x|\big),

which is negative once $\lambda$ is large.

Exercise 4.9 Fisher–KPP stays in

[0, 1]

Use Theorem 4.4 with $f(u) = u(1 - u)$ (Lipschitz on $[-1, 2]$ , which suffices after a truncation argument) and the constant solutions $0$ and $1$ to show that a solution on $\mathbb{R}$ with $0 \leq u(x, 0) \leq 1$ and bounded derivatives satisfies $0 \leq u \leq 1$ for all time. Then show $u > 0$ for $t > 0$ unless $u \equiv 0$ .

Exercise 4.10 The minimal front speed

Look for travelling waves $u = U(x - ct)$ of $u_t = u_{xx} + u(1 - u)$ with $U(-\infty) = 1$ , $U(+\infty) = 0$ . (a) Derive $U'' + cU' + U(1 - U) = 0$ . (b) Linearise at $U = 0$ : $U'' + cU' + U = 0$ . Show that if $c < 2$ the solutions oscillate around $0$ , so $U$ would become negative, and conclude $c \geq 2$ . (With $D$ and $r$ restored, $c \geq 2\sqrt{rD}$ .) That $c = 2$ is actually attained, and is the speed selected from compactly supported data, is the theorem of Kolmogorov, Petrovsky and Piskunov.

Solution

(a) $u_t = -cU'$ , $u_{xx} = U''$ . (b) The characteristic equation $\mu^2 + c\mu + 1 = 0$ has roots $\mu = \frac{-c \pm\sqrt{c^2 - 4}}{2}$ , complex when $c < 2$ , giving $e^{-cz/2}\cos(\omega z + \phi)$ , which changes sign infinitely often as $z \to \infty$ . With $u_t = Du_{xx} + ru(1 - u)$ , rescale $x \mapsto x\sqrt{r/D}$ and $t \mapsto rt$ to reduce to $D = r = 1$ ; speeds scale by $\sqrt{rD}$ .

Exercise 4.11 Rectangles with different diffusions

For the system $u_t = d_1\Delta u + F^1(u, v)$ , $v_t = d_2\Delta v + F^2(u, v)$ with $d_1 \neq d_2$ , show that the rectangle $[a_1, b_1]\times[a_2, b_2]$ is invariant if $F^1 \geq 0$ on the edge $u = a_1$ , $F^1 \leq 0$ on $u = b_1$ , and similarly for $F^2$ . (Apply the scalar comparison principle to each component separately.) Check that the competition model of Figure 4.2 satisfies these conditions on $[0, 1]^2$ .

Exercise 4.12 Rehearsal: the scalar curvature under Ricci flow

Under the Ricci flow on a closed $n$ -manifold the scalar curvature satisfies $\partial_tR = \Delta R + 2|\operatorname{Ric}|^2$ (11A.2 How Curvature Evolves), and $|\operatorname{Ric}|^2 \geq \frac1nR^2$ (Cauchy–Schwarz on the eigenvalues of $\operatorname{Ric}$ ). (a) Show that $R_{\min}(t) = \min_xR(x, t)$ is non-decreasing. (b) If $R_{\min}(0) = \rho > 0$ , compare with the ODE $y' = \frac2ny^2$ to show $R_{\min}(t) \geq \frac{\rho}{1 - \frac{2\rho}{n}t}$ , so the flow must become singular by time $\frac{n}{2\rho}$ . (c) Check that the round sphere of 6A.1 What a PDE Is's rehearsal becomes singular exactly at this time: for the unit $n$ -sphere $R = n(n - 1)$ , and $\frac{n}{2n(n-1)} = \frac{1}{2(n - 1)}$ . This is the first maximum-principle argument of 11A.4 Maximum Principles under Ricci Flow, and Perelman uses its consequence, a lower bound on $R$ , everywhere.

Solution

(a) $\partial_tR \geq \Delta R$ ; apply the weak maximum principle (part 1) to $-R$ on the closed manifold, which has no boundary. (b) $\partial_tR \geq \Delta R + \frac2nR^2$ ; the ODE solution $y = \frac{\rho}{1 - 2\rho t/n}$ is a subsolution, and comparison (on a closed manifold, with the Lipschitz constant of $\frac2nR^2$ on the bounded range of $R$ up to any time before blow-up) gives $R \geq y$ . Since $y \to \infty$ as $t \to \frac{n}{2\rho}$ , the flow cannot be smooth past that time. (c) $\rho = n(n - 1)$ gives $\frac{n}{2n(n - 1)} = \frac{1}{2(n - 1)}$ , the extinction time computed directly in 6A.1 What a PDE Is.

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4.1No cold spot forms by itself

4.2The core argument

4.3The weak maximum principle

4.4The strong maximum principle and Hopf's lemma

4.5Comparison principles

4.6Systems and invariant regions

4.7Invasion fronts

4.8History

4.9Exercises

No cold spot forms by itself

The core argument

The weak maximum principle

The strong maximum principle and Hopf's lemma

Comparison principles

Systems and invariant regions

Invasion fronts

History

Exercises