Jost J. Partial Differential Equations

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5.2 Reaction-Diﬀusion Systems 127

(x, t, u) are assumed to be continuous w.r.t. x, t and Lipschitz continuous

w.r.t. u, as in the preceding section. Again, the u-dependence here is the

important one.

We note that in (5.2.1), the diﬀerent components u

are only coupled through

the non-linear terms F (x, t, u) while the left hand side of (5.2.1) for each α

only involves u

, but no other component u

for β = α. Here, we allow some of

the diﬀusion constants d

to vanish. The corresponding equation for u

(x, t)

then becomes an ordinary diﬀerential equation with the spatial coordinate x

assuming the role of a parameter. If we ignore the coupling with other com-

ponents u

with positive diﬀusion constants d

, then such a u

(x, t)evolves

independently for each position x. In particular, in the absence of diﬀusion,

it is no longer meaningful to impose a Dirichlet boundary condition. When

is positive, however, diﬀusion between the diﬀerent spatial positions takes

place. – We have already explained in §4.1 why the diﬀusion constants should

not be negative.

We ﬁrst observe that, when we assume that the d

are positive, the proofs of

Theorem 5.1.1 and Corollary 5.1.1 extend to the present case when we make

corresponding assumptions on the initial and boundary values. The reason

is that the proof of Theorem 5.1.1 only needs norm estimates coming from

Lipschitz bounds, but no further detailed knowledge on the structure of the

right hand side. Thus

Corollary 5.2.1: Let the diﬀusion constants d

all be positive. Under the

assumptions of Theorem 5.1.1 for the right hand side components F

,and

with the same type of boundary conditions for the components u

, suppose

that the solution u(x, t)=(u

(x, t),...,u

(x, t) of (5.2.1) satisﬁes the a-

priori bound

sup

x∈

Ω,0≤τ ≤t

|u(x, τ )|≤K (5.2.2)

for all times t for which it exists, with some ﬁxed constant K. Then the

solution u(x, t) exists for all times 0 ≤ t<∞.



For the following considerations, it will be simplest to assume homoge-

neous Neumann boundary conditions

∂u

(x, t)

∂ν

=0forx ∈ ∂Ω, t > 0,α=1,...,n. (5.2.3)

We also assume that F is independent of x and t,thatis,F = F(u).

Again, we assume that the solution u(x, t) stays bounded and consequently

exists for all time. We want to compare u(x, t) with its spatial average ¯u

deﬁned by

¯u

(t):=

Ω



(x, t)dx (5.2.4)

128 5. Reaction-Diﬀusion Equations and Systems

where Ω is the Lebesgue measure of Ω.

We also assume that the right hand side F is diﬀerentiable w.r.t. u,and

sup

x,t



dF (x, t, u(x, t))

≤L. (5.2.5)

Finally, let

d := min

α=1,...,n

> 0 (5.2.6)

and λ

> 0 be the smallest Neumann eigenvalue of Δ on Ω, according to

Theorem 9.5.2 below. We then have

Theorem 5.2.1: Assume that u(x, t) is a bounded solution of (5.2.1) with

homogeneous Neumann boundary conditions (5.2.3). Assume that

δ := dλ

− L>0. (5.2.7)

Then





i=1

(x, t)

dx ≤ c

−δt

(5.2.8)

for a constant c

,and



|u(x, t) − ¯u(t)|

dx ≤ c

−δt

(5.2.9)

for a constant c

Thus, under the conditions of the theorem, spatial oscillations decay expo-

nentially, and the solution asymptotically behaves like its spatial average. In

the next §5.3, we shall investigate situations where this does not happen.

Proof: We shall leave out the summation over the index α in our notation,

that is, write u

or u

in place of



α=1

and so on.

We put, as in §4.2,

E(u(·,t)) =





i=1

and compute

∂

∂t

E(u(·,t)) =





i=1





i=1

∂(Δu + F(u))

∂x

= −



(Δu)

dx +





i=1

∂F

∂u

, since

∂u(x, t)

∂ν

=0 forx ∈ ∂Ω

≤ (−λ

+ L)





i=1

dx ≤ 2δE(u(·,t)), (5.2.10)

5.2 Reaction-Diﬀusion Systems 129

using Corollary 9.5.1 below and (5.2.7). This diﬀerential inequality by inte-

gration readily implies (5.2.8).

By Corollary 9.5.1 again, we have



|u(x, t) − ¯u(t)|

dx ≤





i=1

(x, t)

dx, (5.2.11)

and so (5.2.8) implies (5.2.9). 

We now consider the case where all the diﬀusion constants d

are equal.

After rescaling, we may then assume that all d

= 1 so that we are looking

at the system

(x, t) − Δu

(x, t)=F

(x, t, u)forx ∈ Ω,t > 0. (5.2.12)

We then have

Theorem 5.2.2: Assume that u(x, t) is a bounded solution of (5.2.12) with

homogeneous Neumann boundary conditions (5.2.3). Assume that

δ = λ

− L>0. (5.2.13)

Then

sup

x∈Ω

|u(x, t) − ¯u(t)|≤c

−δt

(5.2.14)

for a constant c

Proof: Again, we shall leave out the summation over the index α in our

notation, that is, write u

or u

in place of



α=1

and so on.

As in §4.2, we compute



∂

∂t

− Δ



= u

− u

Δu

−



i=1

= u

∂

∂t

− Δu) −



i=1

≤ Lu

−



i=1

)

. (5.2.15)

Therefore, by Corollary 9.5.1,

∂

∂t



(

∂

∂t

− Δu

) ≤ (L − λ

)



≤ 0 (5.2.16)

by (5.2.7). According to Theorem 4.1.1, therefore

v(t):=sup

x∈Ω



∂u(x, t)

∂t



130 5. Reaction-Diﬀusion Equations and Systems

is a nonincreasing function of t. In particular,

∂u(x,t)

∂t

remains uniformly

bounded in t. Writing our equation for u

Δu

(x, t)=

(x, t) − F

(x, t, u)), (5.2.17)

we may then apply Theorem 11.1.2a below to obtain C

1,σ

bounds on u(x, t)

as a function of x that are independent of t, for some 0 <σ<1. Then, ﬁrst

using the Sobolev embedding theorem 9.1.1 for some p>d(d here is the

dimension of the domain Ω, not to be confused with the minimum of the

diﬀusion constants), and then these pointwise, time-independent bounds on

u(x, t)and

∂u(x,t)

∂x

sup

x∈Ω

|u(x, t) − ¯u(t)|≤



|u(x, t) − ¯u(t)|

dx +





∂

∂x

(u(x, t) − ¯u(t))|

≤



|u(x, t) − ¯u(t)|

dx +





∂u(x,t)

∂x

dx.

From (5.2.8) and (5.2.9), we then obtain (5.2.14). 

A reference for reaction-diﬀusion equations and systems that we have used

in this chapter is [21].

5.3 The Turing Mechanism

The Turing mechanism is a reaction-diﬀusion system that has been proposed

as a model for biological and chemical pattern formation. We discuss it here in

order to show how the interaction between reaction and diﬀusion processes

can give rise to structures that neither of the two processes is capable of

creating by itself. The Turing mechanism creates instabilities w. r. t. spatial

variables for temporally stable states in a system of two coupled reaction-

diﬀusion equations with diﬀerent diﬀusion constants. This is in contrast to the

situation considered in the previous §, where we have derived conditions under

which a solution asymptotically becomes spatially constant (see Theorems

5.2.1, 5.2.2). – In this section, we shall need to draw upon some results about

eigenvalues of the Laplace operator that will only be established in §9.5 below

(see in particular Theorem 9.5.2).

Thesystemisoftheform

= Δu + γf(u, v),

= dΔv + γg(u, v).

(5.3.1)

where the important parameter is the diﬀusion constant d that will subse-

quently be taken > 1. Its relation with the properties of the reaction functions

For this step, we no longer need the assumption that the d

are all equal, and

so, we keep them in the next formula.

5.3 The Turing Mechanism 131

f,g will drive the whole process. The parameter γ>0 is only introduced for

the subsequent analysis, instead of absorbing it into the functions f and g.

Here u, v : Ω × R

→ R for some bounded domain Ω ⊂ R

of class C

∞

,and

we ﬁx the initial values

u(x, 0),v(x, 0) for x ∈ Ω,

and impose Neumann boundary conditions

∂u

∂n

(x, t)=0=

∂v

∂n

(x, t) for all x ∈ ∂Ω,t ≥ 0.

One can also study Dirichlet type boundary condition, for example

u = u

,v = v

on ∂Ω where u

are a ﬁxed point of the reaction sys-

tem as introduced below. In fact, the easiest analysis results when we assume

periodic boundary conditions.

In order to facilitate the mathematical analysis, we have rescaled the in-

dependent as well as the dependent variables compared to the biological

or chemical models treated in the literature on pattern formation. We now

present some such examples, again in our rescaled version. All parameters

a, b, ρ, K, k in those examples are assumed to be positive.

(1) Schnakenberg reaction

= Δu + γ(a − u + u

v),

= dΔv + γ(b −u

v).

(2) Gierer-Meinhardt system

= Δu + γ(a − bu +

= dΔv + γ(u

− v).

(3) Thomas system

= Δu + γ(a − u −

ρuv

1+u + Ku

= dΔv + γ(α(b −v) −

ρuv

1+u + Ku

A slightly more general version of (2) is

(2’)

= Δu + γ



a − u +

v(1 + ku

)



= dΔv + γ(u

− v).

132 5. Reaction-Diﬀusion Equations and Systems

We turn to the general discussion of the Turing mechanism. We assume

that we have a ﬁxed point (u

) of the reaction system:

f(u

)=0=g(u

We furthermore assume that this ﬁxed point is linearly stable. This means

that for a solution w of the linearized problem

= γAw, with A =



) f

)

) g

)



, (5.3.2)

we have w → 0fort → 0. Thus, all eigenvalues λ of A must have

Re(λ) < 0,

as solutions are linear combinations of terms behaving like e

λt

The eigenvalues of A are the solutions of

− γ(f

+ g

)λ + γ

− f

) = 0 (5.3.3)

(all derivatives of f and g are evaluated at (u

)), hence

1,2



+ g

) ±

+ g

)

− 4(f

− f

)



. (5.3.4)

We have Re(λ

) < 0 and Re(λ

) < 0if

+ g

< 0,f

− f

> 0. (5.3.5)

The linearization of the full reaction-diﬀusion system about (u

)is



0 d



Δw + γAw. (5.3.6)

We let 0 = λ

<λ

≤ λ

≤ ... be the eigenvalues of Δ on Ω with

Neumann boundary conditions, and y

be a corresponding orthornormal basis

of eigenfunctions, as established in Theorem 9.5.2 below,

Δy

+ λ

=0 inΩ,

∂y

∂n

=0 on∂Ω.

When we impose the Dirichlet boundary conditions u = u

,v = v

on ∂Ω in

place of Neumann conditions, we should then use the Dirichlet eigenfunctions

established in Theorem 9.5.1. We then look for solutions of (5.3.6) of the form

5.3 The Turing Mechanism 133

λt



αy

βy



λt

with real α, β.

Inserting this into (5.3.6) yields

λw

= −



0 d



+ γAw

. (5.3.7)

For a nontrivial solution of (5.3.7), λ thushastobeaneigenvalueof



γA −



0 d





The eigenvalue equation is

+ λ(λ

(1 + d) − γ(f

+ g

))

+ dλ

− γ(df

+ g

)λ

+ γ

− f

)=0.

(5.3.8)

We denote the solutions by λ(k)

1,2

(5.3.5) then means that

Re λ(0)

1,2

< 0 (recall λ

=0).

We now wish to investigate whether we can have

Re λ(k) > 0 (5.3.9)

for some higher mode λ

Since by (5.3.5), λ

> 0,d> 0, clearly

(1 + d) − γ(f

+ g

) > 0,

we need for (5.3.9) that

dλ

− γ(df

+ g

)λ

+ γ

− f

) < 0. (5.3.10)

Because of (5.3.5), this can only happen if

+ g

> 0.

Computing this with the ﬁrst equation of (5.3.5), we thus need

d =1,

< 0.

134 5. Reaction-Diﬀusion Equations and Systems

If we assume

> 0,g

< 0, (5.3.11)

then we need

d>1. (5.3.12)

This is not enough to get (5.3.10) negative. In order to achieve this for

some value of λ

, we ﬁrst determine that value μ of λ

for which the lhs of

(5.3.10) is minimized, i. e.

μ =

(df

+ g

), (5.3.13)

and we then need that the lhs of (5.3.10) becomes negative for λ

= μ.This

is equivalent to

(df

+ g

)

− f

. (5.3.14)

If (5.3.14) holds, then the lhs of (5.3.10) has two values of λ

where it

vanishes, namely



(df

+ g

) ±

(df

+ g

)

− 4d(f

− f

)





(df

+ g

) ±

(df

− g

)

+4df



(5.3.15)

and it becomes negative for

−

<λ

<μ

. (5.3.16)

We conclude

Lemma 5.3.1: Suppose (5.3.14) holds. Then (u

) is spatially unstable

w. r. t. the mode λ

, i. e. there exists a solution of (5.3.7) with

Re λ>0

if λ

satisﬁes (5.3.16), where μ

are given by (5.3.15).

(5.3.14) is satisﬁed for

d>d

= −

− f



− f

) . (5.3.17)

Whether there exists an eigenvalue λ

of Δ satisfying (5.3.16) depends on

the geometry of Ω. In particular, if Ω is small, all nonzero eigenvalues are

5.3 The Turing Mechanism 135

large (see Corollaries 9.5.2, 9.5.3 for some results in this direction), and so it

may happen that for a given Ω, all nonzero eigenvalues are larger than μ

In that case, no Turing instability can occur.

We may also view this somewhat diﬀerently. Namely, given Ω,wehave

the smallest nonzero eigenvalue λ

. Recalling that μ

in (5.3.15) depends on

the parameter γ,wemaychooseγ>0sosmallthat

<λ

Then, again, (5.3.16) cannot be solved, and no Turing instability can occur.

In other words, for a Turing instability, we need a certain minimal domain

size for a given reaction strength, or a certain minimal reaction strength for

a given domain size.

If the condition (5.3.16) is satisﬁed for some eigenvalue λ

,itisalsoof

geometric signiﬁcance for which value of k this happens. Namely, by Courant’s

nodal domain theorem (see the remark at the end of §9.5), the nodal set

=0} of the eigenfunction y

divides Ω into at most (k +1)regions.On

any of these regions, y

then has a ﬁxed sign, i. e. is either positive or negative

on that entire region. Since y

is the unstable mode, this controls the number

of oscillations of the developing instability.

We summarize

Theorem 5.3.1: Suppose that at a solution (u

) of

f(u

)=0=g(u

we have

+ g

< 0,f

− f

> 0.

Then (u

) is linearly stable for the reaction system

= γf(u, v),

= γg(u, v).

Suppose that d>1 satisﬁes

+ g

> 0,

(df

+ g

)

− 4d(f

− f

) > 0.

Then (u

) as a solution of the reaction-diﬀusion system

= Δu + γf(u, v),

= dΔv + γg(u, v)

is linearly unstable against spatial oscillations with eigenvalue λ

whenever

satisﬁes (5.3.16).

136 5. Reaction-Diﬀusion Equations and Systems

Since we assume that Ω is bounded, the eigenvalues λ

of Δ on Ω are

discrete, and so it also depends on the geometry of Ω whether such an eigen-

value in the range determined by (5.3.16) exists. The number k controls the

frequency of oscillations of the instability about (u

), and thus determines

the shape of the resulting spatial pattern.

Thus, in the situation described in Theorem 5.3.1, the equilibrium state

) is unstable, and in the vicinity of it, perturbations grow at a rate

Reλ

,whereλ solves (5.3.8).

Typically, one assumes, however, that the dynamics is conﬁned within a

bounded region in (R

)

. This means that appropriate assumptions on f and

g for u =0orv =0,orforu and v large ensure that solutions starting in the

positive quadrant can neither become zero nor unbounded. It is essentially

a consequence of the maximum principle that if this holds for the reaction

system, then it also holds for the reaction-diﬀusion system, see the discussion

in §5.1 and §sec4a2.

Thus, even though (u

) is locally unstable, small perturbations grow

exponentially, this growth has to terminate eventually, and one expects that

the corresponding solution of the reaction-diﬀusion system settles at a spa-

tially inhomogeneous steady state. This is the idea of the Turing mechanism.

This has not yet been demonstrated in full rigour and generality. So far, the

existence of spatially heterogeneous solutions has only been shown by sin-

gular perturbation analysis near the critical parameter d

in (5.3.17). Thus,

from the global and non-linear perspective adopted in this book, the topic

has not yet received a complete and satisfactory mathematical treatment.

We want to apply Theorem 5.3.1 to the example (1) above. In that case

we have

= a + b,

(a + b)

(of course, a, b > 0)

and at (u

)then

b − a

a + b

=(a + b)

= −

a + b

= −(a + b)

− f

=(a + b)

> 0.

Since we need that f

and g

have opposite signs (in order to get df

0 later on), we require