Course 6 Book 6A: The Heat Equation and Its RelativesChapter 7

Nonlinear Parabolic Equations

Short-time existence by linearisation, continuation criteria and finite-time blow-up.

28 min read · Updated Oct 3, 2026

Read with Evans, Partial Differential Equations, section 9.2 (fixed point methods: Banach's fixed point theorem applied to reaction–diffusion systems) and section 9.4 (nonexistence of global solutions, including blow-up). For the quasilinear existence theory in Hölder spaces, Lieberman's Second Order Parabolic Differential Equations, chapter 8, is the reference.

In this chapter · 7 sections

7.1Thermal runaway
7.2Semilinear equations: existence by contraction
7.3Quasilinear equations: linearise and solve
7.4Blow-up
7.5When nonlinear diffusion behaves differently
7.6History
7.7Exercises

The Ricci flow is a nonlinear parabolic equation. Before it can be studied, two questions have to be answered for nonlinear parabolic equations in general. Does a solution exist, at least for a short time? And if it stops existing, what happens? This chapter answers both.

The answer to the first is a theorem with a recipe: linearise the equation, check that the linearisation is strictly parabolic, solve the linear problem with the estimates of 6A.6 Parabolic Regularity, and close the argument with a contraction (2B.2 Completeness and Contraction) or the inverse function theorem (2B.9 The Inverse and Implicit Function Theorems). This is how Hamilton proved short-time existence for the Ricci flow in 1982, after overcoming the one obstacle the recipe meets there. The answer to the second is a continuation criterion: a solution can only stop existing if some quantity blows up, exactly as for ODEs (2B.10 Ordinary Differential Equations). For the Ricci flow the quantity is the curvature, and the study of what happens as it blows up is the study of singularities, which occupies the rest of the guide.

And solutions do blow up. The chapter's last part shows how a reaction term can drive a solution to infinity in finite time despite diffusion, and when diffusion wins instead.

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

write a semilinear equation in Duhamel form and prove short-time existence by contraction;
state the blow-up alternative, and explain why the existence time depends on the size of the data;
describe the linearise-and-solve strategy for quasilinear equations, and the role of strict parabolicity;
prove finite-time blow-up by comparison with an ODE and by Kaplan's eigenfunction method, and state Fujita's theorem;
recognise self-similar solutions, blow-up rates and the porous medium equation's finite speed of propagation.

Thermal runaway

In the world Model Frank-Kamenetskii's thermal explosion

A reactive material, such as a pile of damp coal, a store of grain, or a stack of chemicals, generates heat by slow reactions whose rate increases steeply with temperature, and loses heat by conduction to its surroundings. If losses keep up, the material reaches a steady warm state; if not, the temperature runs away and the material ignites. David Frank-Kamenetskii's theory (1939) models the temperature excess $\theta$ by

\theta_t = \Delta\theta + \delta e^\theta,

in dimensionless units, where the exponential approximates the Arrhenius rate of the reaction and the parameter $\delta$ grows with the size of the body and the reactivity of the material. For a slab with its faces held at ambient temperature, a steady state ( $\theta_t = 0$ ) exists only when $\delta \leq \delta_c \approx 0.878$ (Exercise 7.6). Above the critical value no steady state exists, and solutions of the time-dependent equation grow without bound in finite time: the model's way of saying the material ignites.

The prediction that matters in practice is the scaling of $\delta$ : it is proportional to the square of the size of the body, so there is a critical size above which a stored material self-heats to ignition at a given ambient temperature. Safety assessments of bulk storage use this kind of reasoning, with the constants measured in laboratory tests on small samples and scaled up. The model is idealised (it ignores the consumption of the reactant, among other things), and finite-time blow-up in the equation stands for a transition to a different physical regime, not for infinite temperature.

Semilinear equations: existence by contraction

Consider

u_t = \Delta u + f(u) \ \text{ on } \mathbb{R}^n\times(0, T], \qquad u(\cdot, 0) = g,

with $g$ bounded and continuous. By Duhamel's principle (6A.3 The Heat Equation on ℝⁿ), a solution satisfies the integral equation

u(t) = S(t)g + \int_0^tS(t - s)f(u(s))\,ds,

where $S(t)h = \Phi(\cdot, t) * h$ is the heat semigroup. A bounded continuous function satisfying this integral equation is called a mild solution; by the smoothing properties of the heat kernel it is a classical solution for $t > 0$ when $f$ is smooth.

Theorem 7.1 Short-time existence

Let $f$ be Lipschitz with constant $K$ . Then for $T < 1/K$ the integral equation has a unique bounded continuous solution on $\mathbb{R}^n\times[0, T]$ .

Proof. Let $X$ be the space of bounded continuous functions on $\mathbb{R}^n\times[0, T]$ with the sup norm, a complete metric space (2B.5 Uniform Convergence and Arzelà–Ascoli), and define $\Psi(u)(t) = S(t)g + \int_0^tS(t - s)f(u(s))\,ds$ . Since $S(t)$ is convolution with a positive kernel of mass $1$ , $\|S(t)h\|_\infty \leq \|h\|_\infty$ . Hence

\|\Psi(u)(t) - \Psi(v)(t)\|_\infty \leq \int_0^t\|f(u(s)) - f(v(s))\|_\infty\,ds \leq KT\|u - v\|_X.

For $KT < 1$ , $\Psi$ is a contraction of $X$ , and Banach's fixed point theorem (2B.2 Completeness and Contraction) gives a unique fixed point.

The proof is the Picard iteration of 2B.10 Ordinary Differential Equations, with the heat semigroup in place of the identity. As there, the restriction $T < 1/K$ can be removed by restarting, so a globally Lipschitz $f$ gives a solution for all time (Exercise 7.5).

Most interesting reaction terms are not globally Lipschitz: $u^2$ , $e^u$ , $u(1 - u)$ . For locally Lipschitz $f$ , run the same argument in a ball $\{\|u\|_X \leq 2\|g\|_\infty\}$ , where $f$ has a Lipschitz constant $K(\|g\|_\infty)$ ; the time of existence depends on the size of the data. Patching together the solutions obtained by restarting gives a maximal solution on $[0, T_{\max})$ , and:

Theorem 7.2 The blow-up alternative

For locally Lipschitz $f$ , either $T_{\max} = \infty$ , or

\limsup_{t\to T_{\max}}\|u(\cdot, t)\|_{L^\infty} = \infty.

Proof. If $\|u(t)\|_\infty \leq M$ on $[0, T_{\max})$ , the existence time from any starting point is at least a fixed $\tau(M) > 0$ . Restarting at $T_{\max} - \frac12\tau(M)$ extends the solution beyond $T_{\max}$ , a contradiction.

Solutions can only end by blowing up. The quantity that blows up is the one controlling the existence time, here the sup norm. This is the PDE version of the ODE blow-up alternative (2B.10 Ordinary Differential Equations), and for the Ricci flow its analogue is Hamilton's theorem: a solution on a closed manifold exists as long as the curvature stays bounded, $T_{\max} < \infty \Rightarrow \limsup|\operatorname{Rm}| = \infty$ (11A.3 Short-Time Existence and Uniqueness). The proof there uses Shi's estimates (6A.6 Parabolic Regularity) to control all derivatives from the curvature, which is the role the sup norm plays here.

Quasilinear equations: linearise and solve

For a quasilinear equation such as

u_t = a(u, \nabla u)\Delta u + b(u, \nabla u),

the second derivatives appear with coefficients that depend on the solution, and the Duhamel trick fails: the heat semigroup no longer matches the leading part. The method is instead:

Linearise. At a reference function (for instance the initial data extended constantly in time), the equation is a linear parabolic equation for the correction, with coefficients determined by the reference.
Check strict parabolicity of the linearisation. Its principal symbol must be positive definite (6A.1 What a PDE Is). For the equation above this means $a > 0$ .
Solve the linear problem with estimates. The Schauder estimates (6A.6 Parabolic Regularity) say the linear operator is an isomorphism from $C^{2+\alpha,1+\alpha/2}$ (with zero initial values) onto $C^{\alpha,\alpha/2}$ , with bounds.
Close by a fixed point. On a short time interval the nonlinear remainder is small in these norms, so either a contraction (2B.2 Completeness and Contraction) or the inverse function theorem in Banach spaces (2B.9 The Inverse and Implicit Function Theorems) produces a solution.

Theorem 7.3 Short-time existence for quasilinear parabolic equations

Let $u_t = \sum a^{ij}(x, u, \nabla u)\partial_i\partial_ju + b(x, u, \nabla u)$ on a closed manifold (or on $\mathbb{R}^n$ with bounded data, or on a bounded domain with compatible boundary data), with smooth coefficients, and initial data $g \in C^{2,\alpha}$ . If the matrix $a^{ij}(x, g, \nabla g)$ is positive definite everywhere, there is $T > 0$ and a unique solution in $C^{2+\alpha,1+\alpha/2}$ on $[0, T]$ ; it is smooth for $t > 0$ if the coefficients are smooth. The same holds for systems whose linearisation is strongly parabolic.

The proof is the four steps above, carried out carefully (Lieberman, chapter 8). Two things are worth noticing. The time of existence depends on norms of the data, and for most nonlinear equations it can be short when the data are large. And strict parabolicity of the linearisation is the one hypothesis that cannot be weakened: without it, the linear problem in step 3 may have no solution with estimates, and the scheme fails (Figure 7.1).

Figure 7.1. The short-time existence scheme for quasilinear parabolic equations, and where the Ricci flow leaves it. Its symbol is degenerate in the directions of diffeomorphisms; DeTurck's trick adds a term that makes the equation strictly parabolic, solves that, and pulls the solution back by diffeomorphisms (11A.3 Short-Time Existence and Uniqueness).

Where this goes Short-time existence for the Ricci flow

Hamilton's 1982 paper proved short-time existence for $\partial_tg = -2\operatorname{Ric}(g)$ on closed manifolds with the Nash–Moser inverse function theorem, a heavy tool, because the linearisation is not strictly parabolic: its symbol vanishes in the directions coming from diffeomorphisms (6A.1 What a PDE Is). Dennis DeTurck found in 1983 that adding a Lie derivative term $\mathcal L_Wg$ , with $W$ built from the metric, gives a strictly parabolic system, the Ricci–DeTurck flow, to which Theorem 7.3 applies directly; its solution is then pulled back by a family of diffeomorphisms to a solution of the Ricci flow. Everything in steps 1–4 above is used, unchanged, in 11A.3 Short-Time Existence and Uniqueness.

Blow-up

Now take a reaction term that grows superlinearly, the simplest being $f(u) = u^2$ :

u_t = \Delta u + u^2.

Without diffusion this is the ODE $y' = y^2$ , whose solution $y = \frac{y_0}{1 - y_0t}$ blows up at $t = 1/y_0$ (2A.10 Derivatives). Diffusion spreads heat out and lowers peaks. Which wins?

On a closed manifold, or for spatially periodic data, the reaction always wins for positive data. By comparison with the spatially constant solution (6A.4 Maximum Principles), if $\min u(\cdot, 0) = m > 0$ then $u \geq \frac{m}{1 - mt}$ , which blows up by time $1/m$ . So $T_{\max} \leq 1/m$ .

On a bounded domain with the boundary held at $0$ , diffusion carries heat out through the boundary, and small data decay. But large data still blow up, as Stanley Kaplan showed in 1963 with a neat argument.

Theorem 7.4 Kaplan's blow-up criterion

Let $U$ be bounded, $\lambda_1$ and $\varphi_1 > 0$ the first Dirichlet eigenvalue and eigenfunction of $-\Delta$ on $U$ , normalised by $\int_U\varphi_1 = 1$ (6A.5 Weak Solutions and Elliptic Regularity). Let $u \geq 0$ solve $u_t = \Delta u + u^2$ with $u = 0$ on $\partial U$ . If $a(0) = \int_Uu(\cdot, 0)\varphi_1 > \lambda_1$ , then $u$ blows up in finite time.

Proof. Let $a(t) = \int_Uu\varphi_1$ . Then, integrating by parts twice (the boundary terms vanish because $u = \varphi_1 = 0$ on $\partial U$ ),

a'(t) = \int_U(\Delta u + u^2)\varphi_1 = \int_Uu\Delta\varphi_1 + \int_Uu^2\varphi_1 = -\lambda_1a + \int_Uu^2\varphi_1 \geq -\lambda_1a + a^2,

using Jensen's inequality (3A.7 Lᵖ Spaces and Jensen’s Inequality) for the probability measure $\varphi_1\,dx$ : $\int u^2\varphi_1 \geq (\int u\varphi_1)^2$ . The ODE $y' = -\lambda_1y + y^2$ with $y(0) > \lambda_1$ blows up in finite time, and $a \geq y$ by ODE comparison. Since $a(t) \leq \|u(t)\|_\infty$ , the solution blows up.

Figure 7.2. Blow-up of

u_t = u_{xx} + u^2

(0, \pi)

with

u = 0

at the ends, from

u(x, 0) = 6\sin x

(computed by an explicit finite-difference scheme with time steps shrinking as

u

grows). Kaplan's criterion applies:

a(0) = 6\cdot\frac\pi4 \approx 4.7 > \lambda_1 = 1

. The profiles at maximum height

6, 10, 20, 40, 80

are shown with their times; the peak sharpens as it grows, and the times crowd together at the blow-up time.

On all of $\mathbb{R}^n$ there is no boundary, but heat can escape to infinity. Hiroshi Fujita proved in 1966 that for $u_t = \Delta u + u^p$ with $p > 1$ :

if $p < 1 + \frac2n$ , every positive solution blows up in finite time, however small;
if $p > 1 + \frac2n$ , sufficiently small positive data (for instance below a small multiple of a Gaussian) give global solutions.

The critical case $p = 1 + \frac2n$ also blows up (Hayakawa for $n = 1, 2$ ; Sugitani and others in general). For $u^2$ this means: blow-up of all positive solutions in dimensions $1$ and $2$ , but not in dimension $3$ or more. The exponent $1 + \frac2n$ comes from comparing the decay of the heat kernel, $\|\Phi(\cdot, t)\|_\infty \sim t^{-n/2}$ , with the growth of the ODE: small data spread out and decay like $t^{-n/2}$ , and the reaction can only catch up if $\int^\infty t^{-n(p-1)/2}dt$ diverges (Exercise 7.8). Scaling decides the outcome, again.

Blow-up rates and self-similarity. Near the blow-up time $T$ , many solutions blow up at the ODE rate, $\|u(t)\|_\infty \sim \frac{1}{T - t}$ for $u^2$ , and look, after rescaling by that rate, like a self-similar profile. Blow-up at the rate of the ODE is called type I, and faster blow-up type II. Exactly this classification is used for Ricci flow singularities, with $|\operatorname{Rm}|$ in place of $u$ : a type I singularity has $|\operatorname{Rm}| \leq \frac{C}{T - t}$ , the rate of the shrinking sphere (6A.1 What a PDE Is, 11B.4 Singularities). Rescaling a solution near a point of very large $u$ , so that the maximum becomes $1$ , produces a new solution of the same equation; that blow-up rescaling (Exercise 7.10) is the basic move of singularity analysis.

When nonlinear diffusion behaves differently

In the world Model Chemotactic collapse

Some bacteria and slime moulds attract each other chemically: each cell secretes a chemical and moves up its gradient. Evelyn Keller and Lee Segel's 1970 model couples the cell density $\rho$ , which diffuses and drifts up the gradient of the chemical concentration $c$ , to an equation for $c$ produced by the cells. Aggregation competes with diffusion, and the outcome depends on dimension and on the total number of cells. In two dimensions, in the standard normalised form of the model in which $c$ is determined instantly by $\rho$ , there is a sharp critical mass: if the total mass is less than $8\pi$ , solutions exist for all time, and if it exceeds $8\pi$ , they blow up in finite time, concentrating mass at a point (Dolbeault and Perthame 2004; Blanchet, Dolbeault and Perthame 2006). The blow-up is a mathematical idealisation of the dense aggregates that real cells form, which the model, lacking any limit on how closely cells pack, cannot stop.

In the world Model The porous medium equation

Gas flowing through porous rock, or groundwater seeping through soil, obeys Darcy's law combined with conservation of mass, and for a gas the result is the porous medium equation

u_t = \Delta(u^m), \qquad m > 1,

with $u \geq 0$ the density. It is a heat equation whose diffusivity $mu^{m-1}$ vanishes where $u = 0$ , so it is degenerate there. Its behaviour is strikingly different from the heat equation's: from compactly supported data, solutions remain compactly supported, and the edge of the support moves at finite speed. The fundamental solution, found by Yakov Zel'dovich and Alexander Kompaneets (1950) and by Grigory Barenblatt (1952), is the self-similar Barenblatt profile

U(x, t) = t^{-\alpha}\Big(C - k\frac{|x|^2}{t^{2\alpha/n}}\Big)_+^{1/(m-1)}, \qquad \alpha = \frac{n}{n(m - 1) + 2},\quad k = \frac{\alpha(m - 1)}{2mn},

a dome that spreads and flattens with a sharp edge at $|x| \sim t^{\alpha/n}$ (Figure 7.3, Exercise 7.9). The heat equation's infinite speed of propagation is a property of linear diffusion, not of diffusion in general.

Figure 7.3. Barenblatt profiles of

u_t = (u^2)_{xx}

in one dimension (

m = 2

\alpha = \frac13

) at

t = 0.2, 0.5, 1, 2, 4

(computed from the formula, with

C = \frac14

and

k = \frac1{12}

). Each is a parabola cut off at zero, with the same area; the edge of the support (dots) is at

|x| = (C/k)^{1/2}t^{1/3} = \sqrt3\,t^{1/3}

and moves at finite speed. Compare the heat kernel in 6A.3 The Heat Equation on ℝⁿ, which is positive everywhere at every positive time.

History

Frank-Kamenetskii's thermal explosion theory dates from 1939. Kaplan's eigenfunction argument appeared in 1963, and Fujita's theorem on critical exponents in 1966; Kantaro Hayakawa (1973) and Sugitani (1975) settled the critical case. Keller and Segel's model is from 1970, and the sharp $8\pi$ threshold from the work of Dolbeault, Perthame and Blanchet in 2004–06. The Barenblatt solution was found in 1950–52. Short-time existence for quasilinear parabolic equations in Hölder spaces was developed in the 1950s and 60s and summarised in the 1967 book of Ladyzhenskaya, Solonnikov and Ural'tseva. Hamilton's 1982 existence proof for the Ricci flow used the Nash–Moser theorem; DeTurck's simplification appeared in 1983.

Recall Where we stand

Semilinear equations $u_t = \Delta u + f(u)$ are solved by writing them in Duhamel form and applying the contraction mapping principle; for locally Lipschitz $f$ the existence time depends on $\|u_0\|_\infty$ , and a solution can only end by $\|u\|_\infty$ blowing up. Quasilinear equations are solved by linearising, checking that the linearisation is strictly parabolic, solving with Schauder estimates and closing with a fixed point. Superlinear reactions can cause finite-time blow-up: always for positive data on closed manifolds (ODE comparison), for large data on bounded domains (Kaplan), and for all positive data on $\mathbb{R}^n$ when $p \leq 1 + \frac2n$ (Fujita). Degenerate nonlinear diffusion, as in the porous medium equation, can have finite speed of propagation. 6A.8 Curve Shortening and the First Geometric Flows meets the first geometric flows, where the same questions are asked about curves and surfaces.

Exercises

Exercise 7.5 Global existence for globally Lipschitz reactions

Show that if $f$ is Lipschitz with constant $K$ , the solution of Theorem 7.1 can be continued to all $t \geq 0$ , by restarting at times $T, 2T, \dots$ with $T = \frac{1}{2K}$ . Why does this fail for $f(u) = u^2$ ?

Exercise 7.6 The critical Frank-Kamenetskii parameter

Steady states of $\theta_t = \theta_{xx} + \delta e^\theta$ on $(-1, 1)$ with $\theta(\pm1) = 0$ solve $\theta'' + \delta e^\theta = 0$ . (a) Check that $\theta = \log\big(\frac{2b^2}{\delta}\operatorname{sech}^2(bx)\big)$ solves the equation for every $b > 0$ (these are the symmetric solutions). (b) The boundary condition $\theta(1) = 0$ becomes $\delta = 2b^2\operatorname{sech}^2b$ . Show that the right side, as a function of $b > 0$ , has maximum value about $0.878$ , at $b \approx 1.20$ . Conclude that steady states exist only for $\delta \leq 0.878$ (two of them for smaller $\delta$ , one at the critical value).

Solution

(a) With $\theta = \log(A\operatorname{sech}^2bx)$ , $\theta' = -2b\tanh bx$ and $\theta'' = -2b^2\operatorname{sech}^2bx$ , while $\delta e^\theta = \delta A\operatorname{sech}^2bx$ ; so $A = \frac{2b^2}{\delta}$ . (b) Maximise $h(b) = 2b^2\operatorname{sech}^2b$ : $h' = 0$ when $\frac1b = \tanh b$ , i.e. $b\tanh b = 1$ , $b \approx 1.1997$ , where $h \approx 2(1.4393)(0.3050) \approx 0.878$ . For $\delta < 0.878$ the equation $h(b) = \delta$ has two roots, for $\delta = 0.878$ one, and for $\delta > 0.878$ none. (The equation $b\tanh b = 1$ is the same as the one that gives the critical ratio for the catenoid in 6A.9 Calculus of Variations and Gradient Flows.)

Exercise 7.7 Comparison with the ODE

On the circle $\mathbb{R}/2\pi\mathbb{Z}$ , let $u_t = u_{xx} + u^2$ with $u(x, 0) = 2 + \cos x$ . Show that $\frac{1}{1 - t} \leq u$ and use the maximum principle on the other side to show $\max u \leq \frac{3}{1 - 3t}$ . What does this say about the blow-up time?

Solution

$\min u(0) = 1$ and $\max u(0) = 3$ ; the spatially constant solutions $\frac{1}{1 - t}$ and $\frac{3}{1 - 3t}$ are a subsolution and a supersolution below and above $u(0)$ , so by comparison (6A.4 Maximum Principles) $\frac{1}{1 - t} \leq u \leq \frac{3}{1 - 3t}$ as long as all exist. So $\frac13 \leq T_{\max} \leq 1$ .

Exercise 7.8 Where Fujita's exponent comes from

Suppose a small positive solution of $u_t = \Delta u + u^p$ on $\mathbb{R}^n$ behaves for a long time like a heat-kernel solution, of size $u \approx \varepsilon t^{-n/2}$ . Writing the reaction as $u_t = \Delta u + u^{p-1}\cdot u$ , the reaction multiplies $u$ by the growth factor $\exp\int u^{p-1}\,dt$ , with $u^{p-1} \approx \varepsilon^{p-1}t^{-n(p-1)/2}$ . Show that the total growth $\int_1^\infty u^{p-1}dt$ is infinite exactly when $p \leq 1 + \frac2n$ , and explain why this suggests Fujita's dichotomy.

Exercise 7.9 The Barenblatt profile

In one dimension with $m = 2$ , check that $U(x, t) = t^{-1/3}\big(C - \frac{x^2}{12t^{2/3}}\big)_+$ solves $u_t = (u^2)_{xx}$ where $U > 0$ , and that $\int U\,dx$ is independent of $t$ . Where is the edge of the support at time $t$ , and how fast does it move?

Solution

With $\alpha = \frac13$ and $k = \frac{\alpha(m - 1)}{2mn} = \frac{1}{12}$ . Write $U = t^{-1/3}C - \frac{x^2}{12t}$ inside the support. Then $U_t = -\frac13t^{-4/3}C + \frac{x^2}{12t^2}$ , and $(U^2)_{xx} = 2(U_x^2 + UU_{xx}) = 2\big(\frac{x^2}{36t^2} - \frac{U}{6t}\big) = \frac{x^2}{18t^2} - \frac{t^{-4/3}C}{3} + \frac{x^2}{36t^2} = -\frac13t^{-4/3}C + \frac{x^2}{12t^2}$ . The mass is $t^{-1/3}\int_{|x|<x_0}(C - \frac{x^2}{12t^{2/3}})dx$ with $x_0 = \sqrt{12C}\,t^{1/3}$ , which equals $\frac43C\sqrt{12C}$ , independent of $t$ . The edge is at $x_0 = \sqrt{12C}\,t^{1/3}$ and moves at speed $\frac13\sqrt{12C}\,t^{-2/3}$ , finite for $t > 0$ .

Exercise 7.10 Rehearsal: blow-up rescaling

Let $u$ solve $u_t = \Delta u + u^2$ on $\mathbb{R}^n\times[0, T)$ with $\sup u(\cdot, t) \to \infty$ as $t \to T$ . Choose points $(x_k, t_k)$ with $Q_k = u(x_k, t_k) \to \infty$ , and define

v_k(y, s) = \frac{1}{Q_k}u\Big(x_k + \frac{y}{\sqrt{Q_k}},\ t_k + \frac{s}{Q_k}\Big).

(a) Show that each $v_k$ solves the same equation $v_s = \Delta v + v^2$ , with $v_k(0, 0) = 1$ , on a region that grows to all of $\mathbb{R}^n\times(-Q_kt_k, 0]$ . (b) Explain why, if one also knew $v_k \leq C$ on these regions (which is what "type I" and a good choice of points give), a compactness theorem would produce a limit solution defined for all negative times: an ancient solution. This is exactly the procedure for Ricci flow, with $Q_k = |\operatorname{Rm}|(x_k, t_k)$ and the metric rescaled by $Q_k$ ; the limits are the ancient solutions and κ-solutions of 11B.2 Ancient Solutions and the Harnack Inequality and 12B.1 κ-Solutions.

Solution

(a) $\partial_sv_k = \frac{1}{Q_k^2}u_t$ , $\Delta_yv_k = \frac{1}{Q_k^2}\Delta u$ , and $v_k^2 = \frac{1}{Q_k^2}u^2$ . The time $t \in [0, t_k]$ corresponds to $s \in [-Q_kt_k, 0]$ , and $Q_kt_k \to \infty$ if $t_k \to T > 0$ . (b) Bounded solutions of the equation on larger and larger regions have bounded derivatives on compact sets, by the parabolic estimates of 6A.6 Parabolic Regularity; Arzelà–Ascoli (2B.5 Uniform Convergence and Arzelà–Ascoli) and a diagonal argument give a subsequence converging on compact subsets of $\mathbb{R}^n\times(-\infty, 0]$ to a solution with $v(0, 0) = 1$ .

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7.1Thermal runaway

7.2Semilinear equations: existence by contraction

7.3Quasilinear equations: linearise and solve

7.4Blow-up

7.5When nonlinear diffusion behaves differently

7.6History

7.7Exercises

Thermal runaway

Semilinear equations: existence by contraction

Quasilinear equations: linearise and solve

Blow-up

When nonlinear diffusion behaves differently

History

Exercises