Rigorous computation of non-uniform patterns for the

2-dimensional Gray-Scott reaction-diffusion equation.

Roberto Castelli ∗

May 13, 2014

Abstract

In this paper the Gray-Scott reaction diffusion system defined on a 2-dimensional bounded domain

Ω is considered. Some analytical results about existence and non-existence of non uniform patterns are

recalled, then, by means of a computer-assisted proof, the existence of several non-uniform stationary pat-

terns is proved, providing rigorous enclosure around numerical approximations. A non local bifurcation

diagram of families that bifurcate from the homogenous states is depicted.

The rigorous enclosure is obtained by adapting the radii polynomial technique as presented by

M. Gameiro and J. Lessard, ( SIAM Journal on Numerical Analysis, 51(4):2063–2087, 2013.). How-

ever due to a constant factor that weakens the effect of the linear operator, the technique does not allow

to rigorously compute the aimed solutions in reasonable time. Then we introduce a new computational

parameter that allows to obtain sharper bounds for the nonlinear terms and to perform successful com-

putations. The modification is not problem dependent and can be applied to solve other problems that

may reveal the same complications.

Keywords

Rigorous numerics · Radii polynomials · 2-dimensional Gray-Scott reaction diffusion equation ·

Contraction mapping theorem · Non uniform stationary solutions · Pattern dynamics

Mathematics Subject Classification (2000)

35J60 · 35K57 · 35B32 · 65G20 · 65G40∗Dept. of Mathematics, VU University Amsterdam, de Boelelaan 1081, 1081 HV Amsterdam, the Netherlands. Email:

r.castelli@vu.nl.

1

1 Introduction

Formation and self-organisation of patterns are fascinating phenomena that occur in several natural systems,

as for instance in semiconductors, ferroelectric and magnetic materials, in combustion systems, in biological

processes and chemical reactions. See [1] for a survey.

Since Turing [2], systems of reaction-diffusion equations have been largely adopted to describe the dy-

namics of patterns and the Gray-Scott model is a prototype. In chemistry the Gray-Scott is a model for a

pair of reactions involving cubic autocatalysis [3] and it consists in the following reaction-diffusion system Ut = d1∆U − UV 2 + F (1− U), x ∈ ΩVt = d2∆V + UV 2 − (F + κ)V x ∈ Ω (1)where U and V are the concentrations of chemical reactants U and V, d1, d2 are the diffusion coefficients, F

is the feed rate and (F + κ) the removal rate of V.

The numerical investigation reported by Pearson [4] reveals a rich and complex structure in the solu-

tions for the Gray-Scott equation. These solutions include for instance self-replicating patterns, oscillating

patterns, spots annihilation, stripe patterns, irregular patterns, spatio-temporal chaos. Since then, the be-

haviour and the dynamics of the patterns in the Gray-Scott system has been extensively studied through

experimental observation, numerical technique and analytic approach. A wide literature has been produced

on this topic and we attempt only a brief overview.

Experiments and numerical simulations can be found for instance in [5, 6]. Analytical results mainly

concern the 1-dimensional case: for example single and multi-pulse solutions, stationary waves, periodic

patterns and their stability have been addressed in [7, 8, 9, 10, 11, 12, 13], kinks and solitons solutions in

form of heteroclinic and homoclinic connections are explicitly presented and analysed in [14, 15] and, by

means of a computer-assisted proof, in [16].

In higher dimensional case, existence of single spot solutions is proved in [17, 18] and a formal asymp-

totic analysis on the construction and stability of two and three-dimensional radially-symmetric static spike

autosolitons is given in [19, 20]. Also, ring-like solitons and stripe patterns have been found and analysed

in [21, 22, 23]. More recently an hybrid analytic-numerical approach is adopted in the investigation of

multi-spot quasi equilibrium patterns in two dimensional domains [24].

An interesting question concerns the underlying mechanisms that drive the transition from one dynamics

to another. For instance, how to explain the self-replicating pattern dynamics, that is, the itinerary process

that moves a localised trigger to a stable stationary or oscillating pattern by multiple splitting of the pulses.

Or even, which is the geometrical mechanism behind the spatio-temporal chaos. Attempts to answer these

questions are presented in [25, 26, 27]. Looking at the global bifurcation diagram of stationary solutions, in

2

[26] it has been suggested that a hierarchy structure of saddle-node points is responsible for the occurrence

of self-replicating pattern dynamics. In [25] the relationship between different bifurcation branches of steady

states is considered as a criteria for the onset of spatio-temporal chaos. The spatio-temporal chaotic dynamics

it has been explained to emerge by unfolding an itinerant heteroclinic cycle. Also, self-destruction and

annihilation can be explained from a dynamical system point of view as an orbit starting close to an unstable

pattern whose unstable manifold is connected to the homogenous state. From these considerations, it is

evident that the bifurcation diagram of steady states together with the stability analysis plays a key role in

the comprehension of the mechanism behind the complex pattern dynamics.

In this paper we are concerned with the 2-dimensional Gray-Scott equation in case of equal diffusivities

(d1 = d2 = d), defined on a bounded domain Ω and with no-flux boundary condition. Motivated by the above

discussion, the primary goal is to prove the existence and to provide an enclosure of several non homogenous

stationary patterns. Indeed, we are going to depict a nonlocal bifurcation diagram of families of stationary

solutions emanating from the homogeneous states. As often happens when dealing with nonlinear analysis in

dimension higher than one, non-trivial explicit solutions do not exist and analytical techniques can provide

local or perturbative results only, while they fail in solving non-local problems. On the contrary, numerical

simulations can face a larger spectra of problems, providing accurate numerical solutions at the expenses of

the mathematical rigorousness.

In this paper we combine analytical theory and rigorous numerical methods in a computer-assisted tech-

nique, that allows to successfully treat non-local problems while preserving the mathematical rigorousness.

More technically, we are going to use the so called radii polynomial technique, a rigorous computational

method that provides approximate solutions to the given problem together with precise bounds within

which the genuine solutions are guaranteed to exist. In this context, the second contribution of this work is

the presentation of a variant of the computational method. Indeed, as it will be shown, it is necessary to

introduce some modifications to the technique in order to be able to rigorously enclose the solutions we are

interested in. However this variant is not problem dependent and it may be advantageous to be considered

in the analysis of other systems that reveal similar peculiarity as the Gray-Scott system.

In order to introduce the results, let us present more in details the system we are dealing with. Following

[15], by means of a number of space-time-parameter changes, the Gray-Scott system is equivalent to the

following ut = ∆u− uv2 + λ(1− u), x ∈ Ωγdvt = γ∆v + uv

2 − v x ∈ Ω

∂u/∂ν = ∂v/∂ν = 0, x ∈ ∂Ω

. (2)

For any choice of λ and γ, problem (2) admits a stable uniform stationary solution (u, v) = (1, 0) called

3

p1 and, for λ ≥ 4, two branches of uniform stationary solutions denoted by p2(λ) and p3(λ). Regarding

non-constant stationary patterns, some non-existence results stating that solutions may exists only in some

regions of the λ − γ space have been proved, together with apriori upper and lower bounds for the norm

of the solutions, see e.g [28, 29, 30]. By means of local bifurcation analysis, it can be shown that families

of non-constant stationary solutions bifurcate from the homogeneous states at certain values of γ. In the

following sections we are going to recall some of these results and make the statement more precise. However,

up to the author’s knowledge, no results about the existence of solutions far from the bifurcation neither the

analysis of a global bifurcation diagram have been presented. Hence, by means of numerical continuation

and rigorous computations, we investigate various bifurcation branches of non-constant solutions emanating

from the homogenous states. In this direction the goal is to prove existence of several non constant patterns,

significantly far away from being a perturbation of the homogenous ones, providing explicitly the numerical

data together with the enclosing radius. Also, although it is not analysed further in this paper, the computed

solutions give a first picture of the global bifurcation diagram that may be rigorously validated by means of

a parameter-dependent rigorous computational method [31].

−0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Homogenous soluti on = p 2. Paramete rs: λ = 4.5, Ω = [0, 1.1] × [ 0, 0.8]

γ

(0,1)

(0,2)

(0,3)

(1,0)

(1,1)

(1,2)

(1,3)

(2,0)

(2,1)

(2,2)

(2,3)

(3,0)

(3,1)

(3,2)

(3,3)

‖v − v0‖2

(a)

0 0.005 0.01 0.015 0.02 0.025

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

γ

Homogenous soluti on = p 3. Paramete rs: λ = 4.5, Ω = [0, 1.1] × [ 0, 0.8]

(0,1)

(0,2)

(0,3)

(1,0)

(1,1)

(1,2)

(1,3)

(2,0)

(2,1)

(2,2)

(2,3)

(3,0)

(3,1)

(3,2)

(3,3)

‖v − v0‖2

(b)

Figure 1: Bifurcation diagrams of rigorously computed non-uniform stationary solutions for the Gray-Scott system

(2) on the domain Ω = [0, 1, 1] × [0, 0.8] and λ = 4.5. The branches of solutions bifurcate from the homogenous state

(a) p2(λ) and (b) p3(λ). The horizontal axis represents the parameter γ and the vertical axis represents the distance,

measure with the squared L2-norm, of the v-component of the solution from the v-component of the steady state.

Looking at Figure 1, the results can be formulated as follows:

Theorem 1.1. Let λ = 4.5 and Ω = [0, 1.1]× [0, 0.8]. Then

• any bullet in Figure 1a represents a non-uniform stationary solution for system (2) belonging to

branches that bifurcate from the homogenous state p2(λ).

4

• any bullet in Figure 1b represents a non-uniform stationary solution for system (2) belonging to

branches that bifurcate from the homogenous state p3(λ).

Clearly the technique is not working for the solutions stated in Theorem 1.1 only. By changing the

parameter λ and the size of the rectangular domain Ω, many different branches of non-constant solutions

can be computed and rigorously validated. Also, most of the families depicted in Figure 1 can be easily

continued further.

In the next theorem we focus on one family of solutions and we do continuation until the method provides

successful computation in reasonable time.

0.008 0.01 0.012 0.014 0.016 0.018

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Homogenous solu tion = p 3. Parame te rs: λ = 4 .3, Ω = [0 , 1] × [0 , 0 .7]

γ

(2,0)

‖v − v0‖2

Figure 2: Family of non-uniform stationary solution for system (2) on the domain Ω = [0, 1] × [0, 0.7] and λ = 4.3.

The bifurcation occurs at parameter value γ ≈ 0.018015. Any bullet represents a rigorously validated solution. The

continuation of the branch has been performed until the computational method provided an enclosure in reasonable

time.

Theorem 1.2. Let λ = 4.3 and Ω = [0, 1] × [0, 0.7]. Any bullet in Figure 2 represent a non-uniform

stationary solution for system (2).

As anticipated, the computer assisted proof is based on the radii-polynomial technique, introduced in

[32] and widely used in the last decade to solve nonlinear differential problems. In the years the method

passed through various adaptations and modifications with the target both to enlarge the number of treatable

problems and to improve the performance of the computation. Afterwards a section is dedicated to the radii

polynomial technique, therefore we postpone the detailed discussion of the method and of the literature. For

the moment we only mention that the method is not working for our business, at least in the form it has

5

been developed so far. More precisely, the computational cost and the time required for the enclosure of a

single solution stated in Theorem 1.1 would be so high as to be practically unfeasible.

Let us briefly anticipate the reason for this failure: the method is based on the Banach fixed point theorem

and aims at proving that a certain infinite dimensional nonlinear operator is a contraction in some set. The

proof comes by verifying a finite number of computable conditions (e.g. the radii polynomials), that are

defined combining rigorous numerical computations on a finite dimensional approximation and analytical

estimates on a infinite dimensional Banach space.

As we will see later on, the linear part in the differential system enforces the contractivity, while the

nonlinear part acts somehow in the opposite direction. For this it is necessary to provide an accurate

estimates of the nonlinear terms and it is advantageous to have a high order linear operator. In our case,

the nonlinearity consists in a 2-dimensional cubic polynomial and, due to the curse of dimensionality, a quite

large finite dimensional projection would be necessary to produce a sharp estimate. Also, the linear part is an

operator of the second order, not so high for our purposes, even more multiplied by a constant factor γ whose

value could be so small to dramatically weaken the contribution of the linear term itself. As a counterpart, a

much sharper estimate of the nonlinear part is required. The result is that the finite dimensional parameter

and other computational parameters must be chosen so large to make the computation unfeasible.

Since we can not change the linear operator, the only possibility is to improve the enclosure of the

nonlinear terms. The contribution of this paper consists in the introduction of a sharpening parameter that

allows to obtain more precise estimates of the nonlinear terms without enlarging the dimension of the finite

dimensional approximation.

The paper is organised as follows. In section 2 some preliminary analytical results are recalled. We intro-

duce the stationary solutions for system (2) and we report some results about non-existence of non stationary

patterns. Also the local bifurcation analysis is presented providing the values of γ in correspondence of which

the bifurcation branches emerge. Section 3 concerns the radii polynomial technique. First we present the

computational method in the general form and the notion of radii polynomial is introduced. Then, in section

3.2 some fundamental formulas for the analytic estimates of the convolution sums are presented. In section 4

we adapt the computational method to the Gray-Scott system. First we introduce the proper Banach space

and the fixed point operator and we write the radii polynomial as they would be without the modifications.

In section 4.1 we explain in more details the reason why the radii polynomials technique, so far presented,

fails in the enclosure of the stationary solutions of (2). Then the sharpening parameter is introduced and

the new scheme to bound the nonlinear terms is discussed. According to this, the new version of the radii

polynomials is written. Finally, in section (5) we are concerned with the computational results.

6

2 Some analytical results

In this section we present some results about the stationary solutions of (1), that is solution of the elliptic

problem

∆u− uv2 + λ(1− u) = 0, x ∈ Ω

γ∆v + uv2 − v = 0 x ∈ Ω

∂u/∂ν = ∂v/∂ν = 0, x ∈ ∂Ω .

(3)

As pointed out in the Introduction, different aspects regarding the stationary solutions have been ad-

dressed in the literature, both taking into account the complete parameters set (d1, d2, F, κ) and the restricted

one (λ, γ), where the hypothesis of equal diffusivities is considered. Among them the existence and non exis-

tence of non constant stationary patterns, a priori bounds for the behaviour of the solutions, the stability of

uniform states under the flow induced by (1), the study of the bifurcation from the constant solutions have

been discussed, see e.g. [28, 30, 29, 33].

We now report only few of the results and we refer to the quoted paper for a more exhaustive overview.

Throughout this section, we denote by hn the eigenvalues of −∆u = hu x ∈ Ω∂u/∂ν = 0, x ∈ ∂Ω (4)so that 0 = h0 < h1 < h2 < . . . and by φn the associated eigenfunctions.

2.1 Uniform stationary solutions

For any value of λ and γ, system (3) admits the uniform state (u, v) = (1, 0), denoted by p1. At λ = 4 a

bifurcation occurs and a new solution arises (u, v) = ( 12 , 2), say p′1. For larger value of λ the solution p

′1

disappears and two branches of uniform solutions are created. Hence the set of uniform solutions is the

following

p1 = (1, 0), λ < 4

p1 = (1, 0), p′1 = (

12 , 2), λ = 4

p1 = (1, 0), p2(λ) =(λ+√λ2−4λ2λ ,

λ−√λ2−4λ2

), p3(λ) =

(λ−√λ2−4λ2λ ,

λ+√λ2−4λ2

)λ > 4 .

Remark 2.1. Note that for any solution (u∗, v∗), except for the solution p1, it holds u∗v∗ = 1.

At first we are concerned with the stability of the uniform solutions under the flow generated by the

system (2). Following [28], we see if any modes of the spatial operator are exponentially growing under the

7

linearized flow. Denoting by (u∗, v∗) any of the uniform solutions, the Frechet derivative at (u∗, v∗) is given

by the linear operator ut = ∆u− v2∗u− 2u∗v∗v − λuvt = d∆v + dγ (v2∗u+ 2u∗v∗v − v) .Setting u = ehtα, v = ehtβ, it results ∆α− (v2∗ + λ+ h)α− 2u∗v∗βd∆β + ( dγ 2u∗v∗ − dγ − h)β + dγ v2∗α .We can now project the function α, β on the basis of eigenfunctions φn and write (α, β) = (θ1, θ2)φn, thus

the question of stability reduces to the eigenvalues problem −hn − v2∗ − λ −2u∗v∗dγ v

2∗ −dhn + dγ 2u∗v∗ −

dγ

θ1θ2

= ν θ1

θ2

.In case of the uniform solution p1, (u∗, v∗) = (1, 0), the eigenvalues are

ν1 = −hn − λ, ν2 = −dhn −d

γ

that are negative proving that p1 is linearly stable. For the solutions p′1, p2(λ) and p3(λ), the above matrix

is given by

M =

−hn − v2∗ − λ −2dγ v

2∗ −dhn + dγ

.It can be shown that the solution p2(λ) is a saddle point, while p3(λ) could be stable, unstable or a spiral

node, depending on the particular choice of the parameters, see [28] for details.

2.2 Existence and non existence of non constant stationary solutions

Concerning the a priori bounds and the existence and non-existence of nontrivial solutions we report the

following results.

Lemma 2.2. Let (u, v) 6= (1, 0) be any solution of (3). Then

0 ≤ u ≤ 1, v > 0, ∀x ∈ Ω̄.

Proof. Let x and x̄ be the points where u(x) = minx∈Ω̄ u(x) and u(x̄) = maxx∈Ω̄ u(x). For the maximum

principle,

−u(x)v(x)2 + λ(1− u(x)) ≥ 0

−u(x̄)v(x̄)2 + λ(1− u(x̄)) ≤ 0.

8

Hence

u(x) ≥ λλ+ v(x)2

> 0, u(x̄) ≤ λλ+ v(x̄)2

≤ 1.

For the same reason, if x1 is the point where v(x) achieves the minimum, then u(x1)v(x1)2 − v(x1) ≤ 0.

Being u(x1) ≥ 0 it follows v(x1) ≥ 0. However, if v(x) = 0 in one point, the maximum principle applied

to γ∆v − v ≤ 0 in Ω implies that v(x) is zero everywhere. Therefore v(x) > 0 for any solution except for

(u, v) = (1, 0). �

The previous result is not optimal, indeed it can be proved that there exist positive constants C1, C2

depending on λ, γ,Ω so that any solution, except (1, 0), satisfies C1 ≤ u(x), v(x) ≤ C2. See [29]. In particular

maxx∈Ω̄

v(x) ≤ max{γ−1, λ}. (5)

Theorem 2.3. If λ ≤ 4, system (3) does not admit non constant solution for any γ > 14 .

Proof. Denote w(x) = u(x) + γv(x)− 1. Then, from the two equations of the system (3), it follows

∆w − λw = (1− λγ)v.

If λγ ≤ 1 then ∆w − λw ≥ 0 and the maximum principle implies that w(x) ≤ 0 for any x ∈ Ω̄. Therefore

from the second in (3), γ∆v = v(1− uv) ≥ v(1− (1− γv)v) ≥ 0 for any γ ≥ 14 . Taking the product ∆v · v

and integrating over Ω, we conclude∫

Ω|∇v|2 dx ≤ 0. Hence v(x) is constant.

Suppose now λγ > 1 and consider w(x) = v − λ(1− u). It follows

∆w =

[(1− 1

λγ

)v2 + λ

]w +

(1

λγ− 1)v(v2 − vλ+ λ).

We are in the situation ∆w − cw ≤ 0, c > 0. The maximum principle implies w(x) ≥ 0 for any x ∈ Ω.

Moreover, the sum of the two equations in (3) produces ∆u+ γ∆v = λ(u− 1) + v that, integrated, yields∫Ω

λ(u− 1) + v dx = 0.

The last one, together with w(x) ≥ 0, gives λ(u− 1) + v ≡ 0. By inserting u = u(v) into the second of (3)

and being λ ≤ 4 we infer that λγ∆v ≥ 0. Arguing as before, we conclude that v(x) is constant. �

Theorem 2.4. If λ, γ satisfy one of the following alternatives, then system (3) does not admit non-constant

solutions:

i) λγ > 1, and γ(1 + γh1) < λ(1 + λh−11 )

ii)1

γ> max{λ, 1/2}, and γ > 1

2

(γ + h−11 )

γ(h1 + λ)+

1

h1

(1

2

1

γ2+

2

γ− 1).

where h1 is the first positive eigenvalue of (4).

9

Proof. Denote by ū = 1|Ω|∫

Ωudx, v̄ = 1|Ω|

∫Ωvdx, ϕ = u− ū, ψ = v − v̄.

Adding the two equations of system (3) and integrating, we obtain

λ

∫Ω

u dx+

∫Ω

v dx = λ|Ω|

that is λū+ v̄ = λ. Define w(x) = u(x) + γv(x). From the previous relation it follows

−∆w = −λϕ− ψ, ∂νw = 0, on ∂Ω . (6)

Multiply by w and integrate. Since∫

Ωϕ, dx =

∫Ωψ dx = 0, it results

∫Ω|∇w|2 dx = −

∫Ω

(λϕ+ ψ)(ϕ+ ū+ γψ + γv̄) dx

= −λ∫

Ωϕ2 dx− (λγ + 1)

∫Ωϕψ, dx− γ

∫Ωψ2 dx

(7)

proving that ∫Ω

ϕψ dx ≤ 0. (8)

Multiply (6) by ϕ and integrate: we obtain∫Ω

∇w · ∇ϕdx = −λ∫

Ω

ϕ2 dx−∫

Ω

ϕψ dx.

On the other side, for the definition of w, it holds∫

Ω∇w · ∇ϕdx =

∫Ω|∇ϕ|2 dx + γ

∫Ω∇ϕ · ∇ψ dx, that,

compared with the previous relation, produces

γ

∫Ω

∇ϕ · ∇ψ dx = −λ∫

Ω

ϕ2 dx−∫

Ω

ϕψ dx−∫

Ω

|∇ϕ|2 dx.

Let us now compute∫Ω|∇w|2 dx =

∫Ω|∇ϕ|2 dx+ γ2

∫Ω|∇ψ|2 dx+ 2γ

∫Ω∇ϕ · ∇ψ dx

=∫

Ω|∇ϕ|2 dx+ γ2

∫Ω|∇ψ|2 dx− 2

(λ∫

Ωϕ2 dx+

∫Ωϕψ dx+

∫Ω|∇ϕ|2 dx

).

The last equality, together with (7) yields the following relation

λ

∫Ω

ϕ2 dx+

∫Ω

|∇ϕ|2 dx+ (1− λγ)∫

Ω

ϕψ = γ

∫Ω

ψ2 dx+ γ2∫

Ω

|∇ψ|2 dx. (9)

Recall the Poincaré inequality ∫Ω

ϕ2 dx ≤ 1h1

∫Ω

|∇ϕ|2 dx (10)

and the same for ψ.

Consider now case i), when λγ > 1. Therefore, from (5) maxx∈Ω̄ v(x) < λ. Relations (9) and (8) imply

γ

∫Ω

ψ2 dx+ γ2∫

Ω

|∇ψ|2 dx ≤ λ∫

Ω

φ2 dx+

∫Ω

|∇ϕ|2 dx

10

then , for (10) ∫Ω

ψ2 dx ≤ 1 + λ/h1γ(1 + γh1

∫Ω

|∇ϕ|2 dx. (11)

Multiplying the first equation of system (3) by ϕ and integrating, it descends∫Ω|∇ϕ|2 dx = −

∫Ω

(uv2 + λ(1− ū− ϕ))ϕdx

= −λ∫

Ωϕ2 −

∫Ωuv2ϕdx = −λ

∫Ωϕ2 dx−

∫Ωuv(v̄ + ψ)ϕdx

= −λ∫

Ωϕ2 dx−

∫Ωuvv̄ϕ dx−

∫Ωuvψϕdx = −λ

∫Ωφ2 dx−

∫Ωu(v̄ + ψ)v̄ϕ dx−

∫Ωuvψϕdx

= −λ∫

Ωϕ2 dx−

∫Ω

(ū+ ϕ)v̄2ϕdx−∫

Ωuv̄ψϕ−

∫Ωuvψϕdx

= −(λ+ v̄2)∫

Ωϕ2 dx−

∫Ωu(v + v̄)ϕψ dx.

Since |u| ≤ 1 and |v| ≤ λ∫Ω

|∇ϕ|2 dx ≤ −λ∫

Ω

ϕ2 dx+ λ2∫

Ω

|ϕ||ψ| dx ≤ λ∫

Ω

ψ2dx.

The last together with (11) implies∫Ω

|∇ϕ|2 dx ≤ λ(1 + λh−11 )

γ(1 + γh1)

∫Ω

|∇ϕ|2 dx (12)

then, for the second condition in i) we conclude that u is constant.

In case ii) we have λγ < 1. Relation (9) and the Poincare inequality imply that∫Ω

ϕ2 dx ≤ γ(γ + h−11 )

h1 + λ

∫Ω

|∇ψ|2 dx.

We now multiply the second equation in (3) by ψ and integrate. Similar computations as above lead to

γ

∫Ω

|∇ψ|2 dx ≤ 1γ2

∫Ω

|ϕ||ψ| dx+(

2

γ− 1)∫

Ω

|ψ|2 dx.

If 1γ ≥12 , the Cauchy inequality, equation (12) and the Poincaré inequality give

γ∫

Ω|∇ψ|2 dx ≤ 12

1γ2

∫Ωϕ2 dx+

(12

1γ2 +

2γ − 1

) ∫Ωψ2 dx

≤ 121γ

(γ+h−11 )h1+λ

∫Ω|∇ψ|2 dx+

(12

1γ2 +

2γ − 1

)1h1

∫Ω|∇ψ|2 dx

from where we infer the non existence of non constant solutions. In the last case, when 1γ <12 , Theorem 2.3

provides the non existence result. �

2.3 Bifurcation of stationary solutions

In this section we discuss the existence of branches of non constant stationary patterns for system (1)

bifurcating from the family of uniform states p2(λ), p3(λ). The aim is to detect the values of γ where the

bifurcations occur and the proper perturbation we should apply to the constant solution so to move on the

11

bifurcating branch. Then, the perturbed function is regarded as a first guess for a subsequent numerical

computation of a non-constant solution of (3). From one approximate solutions, by means of a continuation

argument, different solutions on the same branch will be computed.

Different authors discussed the local bifurcation of the branches of nontrivial solutions both for the Gray-

Scott model and for a wider set of systems of reaction-diffusion equations, see for instance [34]. Here we

study the local bifurcations from a more direct approach.

Again, denote by (u∗, v∗) any of the constant stationary solutions p2(λ), p3(λ) and recall that hn is an

eigenvalue of −∆ with Neumann boundary condition and φn the associated eigenvector.

For any n > 0 define

un(ε) = u∗ + εaφn + o(ε), vn(ε) = v∗ + εbφn + o(ε)

a2 + b2 = 1

a perturbation of the steady state in the direction of the eigenfunction φn. Following the standard bifur-

cation theory, a bifurcation occurs at value γn if for any ε small enough it exists a solution of the form

(un(ε), vn(ε), γ(ε)), being γ(ε) = γn + εγ′ + o(ε).

Inserting (un, vn) into the elliptic problem (3) and using the fact that (u∗, v∗) is a solution for any γ, it

remains the following

a∆φn − 2u∗v∗bφn − v2∗aφn − λaφn + o(ε) = 0

γ(ε)b∆φn + 2u∗v∗bφn + v2∗aφn − bφn + o(ε) = 0 .

Neglecting the infinitesimal terms, for Remark 2.1, it remains to solve the system

−ahn − 2u∗v∗b− v2∗a− λa = 0

−γnbhn + 2u∗v∗b+ v2∗a− b = 0

whose solution is given by

a2 + b2 = 1, a = − 2bhn + v2∗ + λ

(13)

and

γn =1

hn

(λ+ hn − v2∗hn + v2∗ + λ

). (14)

Therefore we have the following

Lemma 2.5. For any n > 0, a branch of non-constant stationary solutions for (1) bifurcates from the

uniform solution (u∗, v∗) at value γn given in (14). The proper perturbation to be applied to the steady state

(u∗, v∗) is (aφn, bφn) where a, b solve (13).

12

3 The Computational method

The existence of several non constant stationary solutions for the Gray-Scott equation is proved by means

of a rigorous computational method.

The method we are going to present is one of the so called verification methods [35] and targets to a

computer-assisted proof of the aimed statements. Whatever f and x would be, suppose we are looking for

solutions of f(x) = 0 for x in some Banach space (X, ‖ · ‖X). The goal of a verification methods is to prove

the existence of a genuine solution x∗ of f(x) = 0 close to a numerical approximate solution x̄ and to provide

an upper bound of ‖x∗ − x̄‖X . In this paper we adopt the radii polynomial technique that relies on the

Banach fixed point theorem. Other techniques based on the contraction mapping principle are, for instance,

the Krawczyk approach [36] and the methods discussed in [37, 38]. However, the main difference is that the

radii polynomials technique aims to an optimal enclosure, that is to the smaller r so that ‖x∗ − x̄‖X < r

rather than to guess a value for r and to verify the enclosure.

It is worthy to mention that the radii polynomial technique and the verification methods in general,

represent only one of the possible way to approach the nonlinear problems by means of computer-assisted

proof. Indeed, in the last decades, different rigorous computational methods have been developed to study

differential equations and dynamical systems, based for instance on topological arguments as the Conley

index theory, the Shauder fixed point theorem, covering relation, the cone condition, see for example [39, 40,

41, 42, 43] and the references therein.

Before presenting the radii polynomial technique in the general form, let us fix some notation. Boldface

is used to denote multi-indeces as for instance k = (k(1), . . . , k(d)) ∈ Zd. When applied to multi-indexes,

the absolute value | · | acts component-wise, |k| := (|k(1)|, . . . , |k(d)|). Inequality in between multi-indexes is

intended component-wise, that is for k,n ∈ Zd, k < n means that k(j) < n(j) for all j = 1, . . . , d. The same

for ≥,≤, >. In the following we will introduce the multi-indexes m and M̄ and M all in Nd, respectively

called finite dimensional parameter, FFT dimension parameter and computational parameter, satisfying

m ≤ M̄ ≤M . Associated to them, we define the set of indices Fm, FM̄ , FM as Fm := {k ∈ Zd : |k|

3.1 Overview of the radii polynomial method

In the last ten years the radii polynomial technique has been exploited to study a variety of problems in the

areas ranging from, but not restricted to, ODEs, PDEs, delay differential equations and dynamical systems.

For instance the method has been applied to prove existence of equilibria and periodic orbits for partial

[44, 32, 31, 45], delay [46] and ordinary [47] differential equations, to find connecting orbits between steady

states [16], to study the spectra properties of bounded states for NLS [48], to solve initial value problem [49].

In [50] it is discussed the rigorous computation of the Floquet normal form of the fundamental solutions of

linear non autonomous systems of ODEs. This result allows the rigorous enclosure of the stable and unstable

bundle of hyperbolic periodic orbits and enables the parametrization of the local invariant manifolds [51].

Note that in [49, 16] the radii polynomial technique has been applied to the 1-dimensional Gray-Scott system

to establish the existence of symmetric homoclinic orbits to the steady state p1 = (1, 0).

We describe the different steps of the technique in the context of solving the PDE

L(u) + cpup = 0 (15)

defined on a bounded domain Ω ⊂ Rd, subjected to prescribed boundary conditions. Here L : D(L) ⊂ H →

H is a linear operator densely defined on a Hilbert space H and p is the degree of nonlinearity, cp ∈ R.

The first step of the method consists in writing an equivalent problem

f(x) = 0

defined on a suitable Banach space, so that the zeros of f(x) correspond univocally to solutions of (15). That

could be achieved, for instance, by spectral decomposition. We assume that the Hilbert space H admits an

orthogonal basis formed by the eigenfunctions of L(u), say {ψk}k with eigenvalues {µk}k and assume that

the domain of L is given by

D(L) =

u = ∑k∈Zd

xkψk

∣∣∣ ∑k∈Zd

µkxkψk converge

.By expanding (15) on the basis {ψk}k, it follows that the solutions of (15) correspond to the zeros of f(x)

given by

f = {fk}k∈Zd

fk(x) = µkxk + cp < up, ψk >H .

In this paper we assume that the eigenfunction basis {ψk}k is such that the polynomial nonlinearity up is

translated into convolutions sums, that is

fk(x) = µkxk + cp∑

k1+···+kp=kkj∈Zd

xk1 . . . xkp , k ∈ Zd.

14

For a vector s = (s(1), . . . , s(d)), let the weight ωsk be defined as

ωsk :=

d∏j=1

ws(j)

k(j) , wsk =

1, k = 0|k|s, k 6= 0 .Under certain regularity conditions of the solutions u, the sequence of Fourier coefficients {xk}k decay

exponentially fast to zero, then we can look for solutions of f(x) = 0 on the Banach space Xs of sequences of

coefficients decaying algebraically at least as fast as { 1ωsk }k∈Zd . Defining the s-norm of a sequence x = {xk}kas ‖x‖s := supk∈Zd{ωsk|xk|}, the Banach space is given by

Xs := {x : ‖x‖s

The method aims at determining a ball B(x̄, r) ⊂ Xs within which T is a contraction. That is done

through the notion of the radii polynomials, a finite set of inequalities in the variable r, the verification of

which implies that T is a contraction on some ball B(x̄, r). The construction of the radii polynomials relies

on precise bounds on the action of T as like as on the contractivity of T and it is based on the following

requirements.

Assume that two sequences {Yk}k and {Zk(r)}k are provided such that, for any k ∈ Zd,∣∣∣[T (x̄)− x̄]k

∣∣∣ ≤ Yk, supb,c∈B(r)

∣∣∣[DT (x̄+ b)c]k

∣∣∣ ≤ Zk(r) . (17)Furthermore, for a choice of a computational parameter M = (M (1), . . . ,M (d)), assume that ỸM and Z̃M (r)

have been found so that

Yk ≤r

ωskỸM , Zk(r) ≤

r

ωskZ̃M (r), ∀k /∈ FM . (18)

Define the radii polynomials {pk(r)}k∈FM by

pk(r) := Yk + Zk(r)−r

ωsk

and the tail polynomial

p̃M (r) := ỸM + Z̃M (r)− 1.

The next lemma clarifies the role of the radii polynomials:

Lemma 3.1. If there exists r > 0 so that pk(r) < 0 for all k ∈ FM and p̃M (r) < 0, then there is an unique

x ∈ B(x̄, r) such that f(x) = 0.

Proof. The result follows as an application of the Banach fixed point theorem on the operator T . See for

instance [32, 37, 50] for more details. �

Before discussing some analytical estimates used in the construction of the bounds Y and Z and, af-

terwards, presenting explicitly the radii polynomials for the Gray-Scott system, let us close this section by

remarking some limitations and ongoing developments of the technique.

Remark 3.2.

• The method requires that the eigenvalues and eigenfunctions of the linear operator L(u) are explicitly

known. That is the reason why we will consider only rectangular domain Ω with Neumann boundary

conditions and polynomial nonlinearities. However, only few and trivial modifications are necessary if

periodic, rather than Neumann, boundary conditions are posed.

16

• The radii polynomial technique here presented aims at a rigorous enclosure in the Ck category of

functions. Recent developments [53] extend the technique so to enclose the solution in the space of

analytic functions.

• The fact that the polynomial nonlinearity is transformed into convolution sums is a peculiarity of the

exponential function basis. Indeed, in the application, we are considering the cosine basis. However,

when a different complete function basis is chosen the polynomial up may be translated in something

similar to the convolution sum, as in the case of the Chebyshev function basis ( that are nothing more

that Fourier series in disguise [49]), or in something very far from a convolution sum as in the case of

the Hermite function basis. In the last case, all the machinery developed to estimate the convolution

sums is useless and a different approach must be followed. That is the subject of an ongoing research

project.

3.2 Analytical estimates of the convolution sums

The construction of the bound Y and Z is one of the most important steps in the development of the

method. These bounds are not uniquely defined, indeed they only have to satisfy the inequalities (17) and

(18). However, one has to consider two constraints: first the bounds have to be sharp enough, in particular

for what it concerns the bound Z, so to control the contractility of T . On the other hand, the construction

of Y , Z and the evaluation of the polynomials pk should not be too expensive, from the point of view of the

computational cost.

The most delicate issue is to handle the non linear part of the map fk: in this regard, one has to deal

with precise bounds for the convolution sums. In this direction a deep and systematic analysis have been

performed in [52], where analytical estimates have been proved for any convolutions terms and a general

scheme for the construction of the bounds Y , Z is proposed. Such a scheme requires the computation of

several convolution sums of the form ∑k1+k2+k3=k

kj∈FM

ak1bk2ck3 .

If, from one side the Fast Fourier Transform (FFT) could be invoked to compute in a fast way the convolutions

sums, still the computational time dramatically increases as far as M , and in particular the dimension d,

increase. Moreover, aiming at a rigorous enclosure, these operations must be performed in interval arithmetics

regime, hence requiring an even larger computational time.

These complications and in particular the curse of dimensionality have been addressed in [54]. In that

work the authors suggested a new construction of the polynomials where the nonlinear terms are bounded

by products of one dimensional estimates rather than to perform large FFT computations. Moreover they

17

improved the analytical estimates introduced in [52]. Since we follow this approach in the construction of

the radii polynomials for the Gray-Scott equation and we will use the same estimates when introducing the

new modifications, it is convenient to explicitly report some of such analytical results.

In the following we recall the properties of the bound α(n)k and ε

(n)k firstly introduced in [52], later

improved in [54] and reported (in the improved version) in Appendix.

The first set of estimates concerns the upper bound of the convolution sums when the indices are free to

move in the full set Z.

Lemma 3.3. (Quadratic estimates). Given s ≥ 2 and M > 6, let α(2)k be defined as in (35). Then, for any

k ∈ Z, ∑k1+k2=kkj∈Z

1

ωsk1ωsk2

≤α

(2)k

wsk.

(General estimates). Given s ≥ 2 and M > 6, let α(n)k be recursively defined as in (36). Then, for any

k ∈ Z, ∑k1+···+kn=k

kj∈Z

1

ωsk1 . . . ωskn

≤α

(n)k

wsk.

Proof. See Lemma A2 and Lemma A3 in [54] �

As an extension in the multidimensional case, we have the following.

Lemma 3.4. Let α(n)k be defined as in (38). Then, for any k ∈ Zd,

∑k1+···+kn=k

kj∈Zd

1

ωsk1 . . . ωskn

≤α

(n)k

ωsk

Proof. See Lemma 2.1 in [52]. �

Now, we consider the case when some of the indices are restricted to take value outside a certain set.

Lemma 3.5. Given n ≥ 3, s ≥ 2 and M ≥ M̄ ≥ 6, let the sequence ε(n)k be defined as in (37). Then for

any ` = 1, . . . , n and 0 ≤ k ≤ M̄ − 1, it holds∣∣∣∣∣∣∣∑

k1+···+kn=kmax{|k1|,...,|k`|}≥M̄

ω−sk1 . . . ω−skn

∣∣∣∣∣∣∣ ≤ `ε(n)kProof. See Lemma A.4 in [54]. �

As for the multidimensional case, suppose M̄ = (M̄ (1), . . . , M̄ (d)) has be defined so that M̄ (j) ≥ 6 and

M ≥ M̄ . The notation {k1, . . . ,k`+1} 6⊂ FM̄ means that at least one of the kj is not in FM̄ . Such a set

corresponds to the complement of {k1, . . . ,k`+1 : kj ∈ FM̄}.

18

Lemma 3.6. For any k ∈ FM̄ and ` = 0, . . . , n− 1∑k1+···+kn=k

{k1,...,k`+1}6⊂FM̄

ω−sk1 . . . ω−skn≤ (`+ 1)ε(n)k .

Proof. See Lemma 2.2 in [52]. �

4 The radii polynomials for the enclosure of stationary solutions

of the Gray-Scott system

In this section we write explicitly how the radii polynomials look like when adapted to the problem of

enclosing non constant solutions of the Gray-Scott system (3). As said, we follow the approach presented in

[54] where the FFT parameter M̄ was introduced for the first time. The role of the FFT parameter is to

reduce the computational cost required to build the radii polynomial in comparison with previous approaches.

Then, in section 4.1 we discuss the reason why the presence of M̄ , although important, is not sufficient to

assure that the non constant solutions for the Gray-Scott equation would be computed in reasonable time.

Hence, in section 4.2 the new approach for the definition of the Z bound is presented.

For the reader’s convenience, let us rewrite system (3)∆u− uv2 + λ(1− u) = 0, x ∈ Ω

γ∆v + uv2 − v = 0 x ∈ Ω

∂u/∂ν = ∂v/∂ν = 0, x ∈ ∂Ω .

We consider the domain Ω = [0, L1]× [0, L2], (z, y) ∈ Ω, and look for solutions of (3) in the form

u(z, y) = u(0,0) +∑k∈Z2∗

uk cosk, v(z, y) = v(0,0) +∑k∈Z2∗

vk cosk (19)

where Z2∗ := Z2 \ {(0, 0)},

uk, vk ∈ R, u|k| = uk, v|k| = vk (20)

and, for k = (k(1), k(2)),

cosk := cos

(k(1)

π

L1z

)cos

(k(2)

π

L2y

).

Note that:

i) The sequence {cosk}k∈N2 is a complete basis for {u ∈ L2(Ω)}∩{∂νu = 0, (z, y) ∈ ∂Ω}. Therefore in the

expansion of the function u and v (19) it would be enough to consider k ∈ N2. However, the equivalent

choice k ∈ Z2 together with the symmetry conditions (20) allows to easily expand the product uv2.

19

ii) The boundary conditions in (3) are automatically satisfied by any u, v given in (19).

Denote by

hk = π2

((k(1))2

L21+

(k(2))2

L22

).

Since

∆ cosk = −hk cosk and < uv2, cosk >L2=∑

k1+k2+k3=k

uk1vk2vk3 ,

solving system (3) corresponds to solve for the unknowns x := {uk, vk}k∈N2 the system

f(x) = 0

given by

f(x) = {fk(x)}k∈N2

where

f(0,0)(x) :=

λ(1− u(0,0))−

∑k1+k2+k3=(0,0)

uk1vk2vk3

−v(0,0) +∑

k1+k2+k3=(0,0)

uk1vk2vk3

and, for k 6= (0, 0),

fk(x) :=

−(hk + λ)uk −

∑k1+k2+k3=k

uk1vk2vk3

−(γhk + 1)vk +∑

k1+k2+k3=k

uk1vk2vk3

. (21)

In each fk we replace the second line with the sum of the first and the second and we obtain the equivalent

system

f(0,0)(x) :=

λ(1− u(0,0))−

∑k1+k2+k3=(0,0)

uk1vk2vk3

−v(0,0) + λ(1− u(0,0))

fk(x) :=

−(hk + λ)uk −

∑k1+k2+k3=k

uk1vk2vk3

−(γhk + 1)vk − (hk + λ)uk.

(22)

From the second equation it follows

v(0,0) = λ(1− u(0,0))

vk = −hk + λ

γhk + 1uk

20

then, plugging vk into the first equation, we end up with a system of equations in the variables {uk}k only.

However, to easy the notation, we continue to use the variable vk, having in mind that vks are functions of

uk.

Let us define

qk := −hk + λ

γhk + 1, q := {qk}k∈Z2

and, for any given sequence x = {xk}k, we adopt the notation qx as for the sequence qx = {qkxk}k. In

particular v = qu.

The system we need to solve is

f(u) = {fk(u)}k∈N2 = 0

where

f(0,0)(u) := λ(1− u(0,0))−∑

k1+k2+k3=(0,0)

uk1vk2vk3 = 0

fk(u) := −(hk + λ)uk −∑

k1+k2+k3=k

uk1vk2vk3 = 0, ∀k ∈ N2∗(23)

in the unknown u = {uk}k and we look for solutions in the Banach space

Xs = {u : ‖u‖s

and, for any k = (k(1), k(2)) 6= 0

D(j, uk) =∑

k1+k2=j−k

vk1vk2 +∑

k1+k2=j+k

vk1vk2 +∑

k1+k2=j−k̄

vk1vk2 +∑

k1+k2=j+k̄

vk1vk2︸ ︷︷ ︸if k(1) 6= 0, k(2) 6= 0

D(j, vk) = 2

∑

k1+k2=j−k

uk1vk2 +∑

k1+k2=j+k

uk1vk2 +∑

k1+k2=j−k̄

uk1vk2 +∑

k1+k2=j+k̄

uk1vk2︸ ︷︷ ︸if k(1) 6= 0, k(2) 6= 0

having k̄ = (−k(1), k(2)).

Then, for a choice of the finite dimensional parameter m, suppose that a numerical solution ūFm has been

computed so that f (m)(ūFm) ≈ 0. Denote by ū = (ūFm , 0Im) and v̄ = qū. We define the finite dimensional

matrix J−1m and the infinite dimensional operator J−1 as in (16).

Let the computational parameter M be chosen, so that M ≥ 3(m− 1)− 1.

Note that, since ūk = 0 for k 6∈ Fm, the convolutions terms (u ∗ v ∗ v)k =∑

k1+k2+k3=k

uk1vk2vk3 = 0 for

any k 6∈ FM . Therefore, the construction of Yk readily follows from the first of the requirements (17)

Yk :=

[∣∣J−1m f (m)(ūFm ;λ, γ)∣∣]k , if k ∈ Fm∣∣µ−1k fk(ū;λ, γ)∣∣ , if k ∈ FM \ Fm0, if k 6∈ FM

.

and ỸM = 0.

The polynomial Zk(r) is defined as

Zk(r) =

Z(0)k r +

(|J−1m |Z

(1)Fm

)kr +

(|J−1m |Z

(2)Fm

)kr2 +

(|J−1m |Z

(3)Fm

)kr3, for k ∈ Fm

|µ−1k |(Z

(1)k r + Z

(2)k r

2 + Z(3)k r

3), for k 6∈ Fm

where

Z(0)k := [|I − J

−1m Df

(m)(ūFm)|ω−sFm ]k, ∀k ∈ Fm.

For what it concerns the other bounds Z(i)k , j = 1, . . . , 3, the procedure is as follows: define the operator J̃

as

[J̃(u)]k :=

[Df (m)(uFm)]k, if k ∈ Fmµkuk, if k 6∈ Fm (25)and consider the quantity [(Df(ū+b)− J̃)c]k as b, c vary in B(r). Writing b = rs and c = rt with s, t ∈ B(1),

one defines the sequences C(j)k = C

(j)k (ū, s, t) so that

[(Df(ū+ b)− J̃)c]k = C(1)k r + C(2)k r

2 + C(3)k r

3.

22

The sequences Z(1)k , Z

(2)k , Z

(3)k are then constructed so that |C

(j)k | ≤ Z

(j)k , uniformly as s, t ∈ B(1).

Let us write explicitly the action of the operator J̃ and of Df(ū+ b) on the sequence c = {ck}k. It holds

[J̃c]k =

µkck−

∑k1+k2+k3=k

|ki|∈Fm

v̄k1 v̄k2ck3 − 2∑

k1+k2+k3=k

|ki|∈Fm

ūk1 v̄k2(qc)k3 k ∈ Fm

µkck k 6∈ Fm

and

[Df(ū+ b)c]k = µkck − [(v(ū+ b) ∗ v(ū+ b) ∗ c+ 2(ū+ b) ∗ qc ∗ v(ū+ b)]k

where v(ū+ b) = q(ū+ b) = v(ū) + qb = v̄ + qb. Hence

[Df(x̄+ b)c]k = µkck − [v̄ ∗ v̄ ∗ c+ 2v̄ ∗ c ∗ qb+ qb ∗ qb ∗ c+ 2(ū ∗ v̄ ∗ qc+ ū ∗ qb ∗ qc+ v̄ ∗ b ∗ qc+ b ∗ qb ∗ qc)]k.

Plugging b = rs and c = rt into [(Df(x̄+ b)− J̃)c]k and collecting on the powers of r, we obtain

C(1)k =

−∑

k1+k2+k3=k

|k1|,|k2|∈Fm,|k3|/∈Fm

v̄k1 v̄k2tk3 − 2∑

k1+k2+k3=k

|k1|,|k2|∈Fm,|k3|/∈Fm

ūk1 v̄k2(qt)k3 , k ∈ Fm

−∑

k1+k2+k3=k

|k1|,|k2|∈Fm,k3∈Z2

v̄k1 v̄k2tk3 − 2∑

k1+k2+k3=k

|k1|,|k2|∈Fm,k3∈Z2

ūk1 v̄k2(qt)k3 , k 6∈ Fm

and, for any k

C(2)k = −2

∑k1+k2+k3=k

|k1|∈Fm,k2,k3∈Z2

v̄k1(qs)k2tk3 − 2∑

k1+k2+k3=k

|k1|∈Fm,k2,k3∈Z2

ūk1(qs)k2(qt)k3 − 2∑

k1+k2+k3=k

|k1|∈Fm,k2,k3∈Z2

v̄k1sk2(qt)k3

C(3)k = −

∑k1+k2+k3=k

ki∈Z2

(qs)k1(qs)k2tk3 − 2∑

k1+k2+k3=k

ki∈Z2

(s)k1(qs)k2(qt)k3

(26)

We recall that Z(1)k , Z

(2)k , Z

(3)k are defined so that |C

(j)k | ≤ Z

(j)k as {tk}, {sk} ∈ B(1). The usual procedure

to obtain the upper bounds is to take the absolute value of all the factors in C(j)k and to use the uniform

bound |tk|, |sk| ≤ ω−sk . For what concerns the extra factor q, define

q̂ :=1

γ

so that |qk| ≤ q̂ for any k.

Any of the sum appearing in the definition of C(j)k is indeed a series, with an infinite number of contribu-

tions. Therefore the procedure consists now in separating any series into a finite sum and a tail series. The

23

first is rigorously computed, while the second is analytically bounded by means of the lemmas discussed in

section 3.2. The criteria for such a separation and the sharpness of the analytical estimates determine both

the computational cost necessary for the definition of Z(j)k and the accuracy of Z

(j)k as a bound of C

(j)k .

Let the FFT dimension parameter M̄ be chosen so that

m ≤ M̄

The tail coefficient is defined as

Z̃M =1

µ̃M

(Z̃

(1)M + Z̃

(2)M r + Z̃

(3)M r

2)

with Z̃(j)M ≥ Z

(j)k ω

sk for any k 6∈ FM . It is enough to define

Z̃(1)M := (‖v̄‖

2s + 2q̂‖ū‖s‖v̄‖s)α̃

(3)M , Z̃

(2)M := (4||v̄||sq̂ + 2||ū||sq̂

2)α̃(3)M , Z̃

(3)M := 3q̂

2α̃(3)M (28)

with α̃(3)M given in (39),

4.1 The role of the linear terms

Let us have a deeper analysis on the way the radii polynomials are defined so to explain under which

conditions the radii polynomials would produce a successful computation.

For k ∈ FM the polynomial pk(r) is on the form pk(r) = a0 +a1r+a2r2 +a3r3. The constant term a0 is

positive and it is related to the quality of the numerical solution. The closer the numerical approximation ū

is to the genuine solution, the smaller the coefficient a0 will be. In this regard, if the aimed solution u is an

equilibria of a PDE, then it is known to have an high degree of regularity and its Fourier coefficients should

decrease exponentially fast. Therefore, up to increase the finite dimensional projection m, a numerical

approximation could be produced ( at least theoretically) within any accuracy.

Also a2 and a3 are positive and they are related to the rate of expansion of the operator T due to the

nonlinear terms. Therefore a necessary condition for the polynomial pk(r) to be negative somewhere is that

a1 is negative. Looking at the case k ∈ FM \ Fm, the coefficient a1 is of the form a1 = 1|µk|Z(1)k −

1ωsk

. The

chances for a1 to be negative are placed both on the sharpness of the bound Z(1)k and, especially, on the

growth rate of the |µk|s. Remember that µks are the eigenvalues of the linear operator in the PDE. Therefore,

higher the order of the linear operator is, larger the |µk| will be, and easier the radii polynomial would be

negative. For instance, if the linear part of the PDE is given by ∆u and Ω is the square [0, 2π]2 with periodic

boundary conditions, then the eigenvectors are ψk(x, y) = eik1xeik

2y and the eigenvalues µk = (k1)2 + (k2)2.

If rather than a second order operator the linear part is given by a fourth order operator, then µk would

growth as ‖k‖4, hence increasing the contractivity of the operator T . Let us inspect more in detail how the

growth rate of µk influences the choice of M̄ and M , the context of a cubic nonlinear PDE L(u) + u3 = 0.

For k ∈ FM \ FM̄ , it results that

Z(1)k = 3‖x̄‖

2s

α(3)k

ωsk

giving

a1 =

(3‖x̄‖2s

α(3)k

|µk|− 1

)1

ωsk.

25

Since a1 must be negative, the parameter M̄ could be chosen as small as µk remains reasonable larger than

α(3)k for any k /∈ FM̄ . Therefore a larger growth for µk allows a smaller choice of M̄ .

The role played by the eigenvalues of the linear part is even more evident in the tail polynomial, that for

the cubic nonlinear PDE is of the form

p̃M := ỸM + Z̃M − 1 =

(3α̃

(3)M

µ̃M− 1

)+

1

µ̃M3α̃

(3)M

(2‖x̄‖sr + r2

).

Since p̃M has been factorized by r, in that case it must be negative the 0-th order term. Up to constant

multiplications, it is necessary that µ̃M > α̃(3)M . This inequalities provides an estimates on the magnitude of

M . Again, the larger the growth of µk is, the smaller the parameter M can be fixed. Note that the choice

of M is somehow imposed by the uniform bounds µ̃M and α̃(3)M and it could be significantly large. Previous

versions of the construction of the radii polynomials, as done for instance in [52], required the computation

of convolution sums for any k ∈ FM . Rather, in the latest version, the convolution sums are computed for

any k ∈ FM̄ only. Hence we realise the importance of the parameter M̄ , introduced in [54], in avoiding

unnecessary and costly convolutions computations.

Concerning the computation of the non stationary solutions for the Gray-Scott mode, the arguments

above discussed should convince the reader that the radii polynomial technique, in the form so far exposed,

would yields to complicated, if not unfeasible, computations. Indeed the linear part of the Gray Scott system

consists in the operator

L

uv

= ∆− λ

γ∆− 1

uv

whose eigenvalues, in case of rectangular domain with Neumann boundary conditions, are given by

µk =

−hk − λ−γhk − 1

, hk = π2( (k(1))2L21

+(k(2))2

L22

).

Clearly, |hk| ∼ ‖k‖2, and that growth would be enough to ensure successful computations. The inconvenient

is the factor γ in front of hk. As shown in section 5, for the solutions we are interested in, the factor γ is of

magnitude 10−3 so the choice of the computational parameters is highly influenced.

Remark 4.1. To be more precise, in the construction of the radii polynomial discussed in the previous

section, the eigenvalues µk are given by (24) and the factor γ does not appear anymore. One may think that

the influence of γ has been suppressed, but that is not the case. Indeed the reduction we did by expressing v

as function of u moved γ from the linear to the nonlinear part. As a result the presence of γ is now in the

bounds Zk through q̂ = γ−1.

26

Consider the tail polynomial first, that provides a necessary condition for the choice of M . Looking at

(28) the computational parameter M need to be fixed so that Cα̃

(3)M

µ̃Mγ< 1, with C > 1.

To have a clear idea, fix λ = 4.5, L1 = L2 = 1, s = (2, 2) and M = (M,M). If M = 200, we compute

α̃(3)M ≈ 4 · 103 and µ̃M ≈ 3.9 · 105. It results α̃

(3)M /µ̃M ≈ 10−2, that is clearly less than 1, but not less than γ.

Hence, due to the presence of γ, a larger parameter M must be chosen. For instance, in figure 1a the family

of solutions labeled with (0, 3), bifurcates at the value γ ≈ 0.0087. Taking into account all the constant

factors, due to the s-norm of ū and v̄ it would be necessary to chose M larger than 480, so to ensure that

the 0-th order term of p̃M is negative.

On the other side, we pointed out that a large value of M does not affect so much the computational

cost, provided the FFT parameter M̄ is small. However, the factor γ plays the same role in the selection of

M̄ . For instance, in the same example as before, setting M̄ = (M̄, M̄) one needs to chose M̄ at least larger

than 250. Then, one has to compute cubic convolutions of sequences with around (500)2 terms. That is a

serious computational task.

In conclusion, we are not claiming that the method fails when applied to the Gray-Scott model, but that,

just for one tentative enclosure, one would need to perform a number of computations so large that could

not be addressed by a standard computer in reasonable time. Therefore, to the goal of rigorously prove

existence of families of solutions, the method so far described is not feasible. To handle this hurdles, in the

next section we introduce some modifications.

4.2 New definition of the bound Z(j)k and Z̃M

The procedure that we are introducing aims at sharpening those terms in Z(1)k and Z

(2)k that involve q̂. For

instance, we are concerned with the factors q̂||ū||s||v̄||sε(3)k , q̂||ū||s||v̄||sα

(3)k

ωskthat have been introduced as

upper bounds for the series ∑k1+k2+k3=k

|k1|,|k2|∈Fm,|k3|/∈FM̄

|ūk1 ||v̄k2 ||qk3 |ω−sk3 , k ∈ FM̄ , (29)

and ∑k1+k2+k3=k

|k1|,|k2|∈Fm,k3∈Z2

|ūk1 ||v̄k2 ||qk3 |ω−sk3 , k 6∈ FM̄ ,

respectively. The first bound have been obtained using the fact that |ūk| ≤ ‖ū‖sω−sk , |v̄k| ≤ ‖v̄‖sω−sk and

Lemma 3.6, as follows:∑k1+k2+k3=k

|k1|,|k2|∈Fm,|k3|/∈FM̄

|ūk1 ||v̄k2 ||qk3 |ω−sk3 ≤ q̂‖ū‖s‖v̄‖s∑

k1+k2+k3=k

|k1|,|k2|∈Fm,|k3|/∈FM̄

ω−sk1 ω−sk2ω−sk3 ≤ q̂||ū||s||v̄||sε

(3)k

27

Similarly, the second bound have been defined applying Lemma 3.4.

Rather that using an uniform bound for all the |ūk|, |v̄k| as done above, we can separate in the series (29)

the significative contributions from the smaller one. Indeed, it could be the case that few of the ūks and v̄ks

are large, while the others are extremely small. Such a behaviour is expectable because the solutions are

bifurcating from the constant solution and only one mode is initially perturbed.

For a given �, let us define

Fm(ū, �) := {k : |k| ∈ Fm, |ūk|ωsk > �}

Fm(v̄, �) := {k : |k| ∈ Fm, |v̄k|ωsk > �}.

(30)

We refer to � as the sharpening parameter and we assume that � < min{‖ū‖s, ‖v̄‖s} so that both Fm(ū, �),

Fm(v̄, �) are not empty. Rewrite (29) as ∑k1+k2+k3=k

|k1|,|k2|∈Fm,|k3|/∈FM̄

|ūk1 ||v̄k2 ||qk3 |ω−sk3 =

∑k1∈Fm(ū,�)

|ūk1 |+∑

k1∈Fm\Fm(ū,�)

|ūk1 |

∑k2∈Fm(v̄,�)

|v̄k2 |+∑

k2∈Fm\Fm(v̄,�)

|v̄k2 |

·

(qω−s)k−k1−k2 if |k − k1 − k2| /∈ FM̄0 otherwise .It follows∑

k1+k2+k3=k

|k1|,|k2|∈Fm|k3|/∈FM̄

|ūk1 ||v̄k2 ||qk3 |ω−sk3 =∑

k1+k2+k3=k

|k1|∈Fm(ū,�)|k2|∈Fm(v̄,�)|k3|/∈FM̄

|ūk1 ||v̄k2 ||qk3 |ω−sk3 +∑

k1+k2+k3=k

|k1|∈Fm\Fm(ū,�)|k2|∈Fm(v̄,�)|k3|/∈FM̄

|ūk1 ||v̄k2 ||qk3 |ω−sk3

+∑

k1+k2+k3=k

|k1|∈Fm(ū,�)|k2|∈Fm\Fm(v̄,�)

|k3|/∈FM̄

|ūk1 ||v̄k2 ||qk3 |ω−sk3 +∑

k1+k2+k3=k

|k1|∈Fm\Fm(ū,�)|k2|∈Fm\Fm(v̄,�)

|k3|/∈FM̄

|ūk1 ||v̄k2 ||qk3 |ω−sk3 .

The first sum on the right hand side provides the main contribution and it will be rigorously computed,

while the others are uniformly bounded by means of the bound ε(3)k , see Lemma 3.6. At this stage, anytime

the coefficient |k1| ranges outside Fm(ū, �) ( resp. |k2| ranges outside Fm(v̄, �)) the estimate |ūk1 | ≤ �ω−sk1holds (reap. |v̄k2 | ≤ �ω−sk2 ). Explicitly we obtain∑

k1+k2+k3=k

|k1|,|k2|∈Fm|k3|/∈FM̄

|ūk1 ||v̄k2 ||qk3 |ω−sk3 ≤∑

k1+k2+k3=k

|k1|∈Fm(ū,�)|k2|∈Fm(v̄,�)|k3|/∈FM̄

|ūk1 ||v̄k2 ||qk3 |ω−sk3 + q̂�‖v̄‖sε(3)k + q̂�‖ū‖sε

(3)k + �

2q̂ε(3)k .

28

For the case k /∈ FM̄ the same strategy and Lemma 3.4 lead to

∑k1+k2+k3=k

|k1|,|k2|∈Fm,k3∈Z2

ūk1 v̄k2 |qk3 |ω−sk3 ≤∑

k1+k2+k3=k

|k1|∈Fm(ū,�)|k2|∈Fm(v̄,�)|k3|∈Z2

|ūk1 ||v̄k2 ||qk3 |ω−sk3 + (q̂�‖v̄‖s + q̂�‖ū‖s + �2q̂)

α(3)k

ωsk.

The advantage of this approach is to enfeeble the effect of γ−1 hidden in q̂. On the other side we have to

calculate one more sum at any stage, requiring further computational time. However, if the mass of ū is

concentrated in few Fourier coefficients, such a computation will be easy and fast. Indserting these newly

defined bounds, we redefine Z(1) as

Z(1)k :=

∑k1+k2+k3=k

|k1|,|k2|∈Fm|k3|∈FM̄\Fm

|v̄k1 ||v̄k2 |ω−sk3 + 2∑

k1+k2+k3=k

|k1|,|k2|∈Fm|k3|∈FM̄\Fm

|ūk1 ||v̄k2 ||qk3 |ω−sk3

+2∑

k1+k2+k3=k

|k1|∈Fm(ū,�)|k2|∈Fm(v̄,�)|k3|/∈FM̄

|ūk1 ||v̄k2 ||qk3 |ω−sk3 + ||v̄||2s�

(3)k + 2q̂�(‖v̄‖sε

(3)k + ‖ū‖sε

(3)k + �ε

(3)k )

k ∈ Fm

∑k1+k2+k3=k

|k1|,|k2|∈Fm|k3|∈FM̄

|v̄k1 ||v̄k2 |ω−sk3 + 2∑

k1+k2+k3=k

|k1|,|k2|∈Fm|k3|∈FM̄

|ūk1 ||v̄k2 ||qk3 |ω−sk3

+2∑

k1+k2+k3=k

|k1|∈Fm(ū,�)|k2|∈Fm(v̄,�)|k3|/∈FM̄

|ūk1 ||v̄k2 ||qk3 |ω−sk3 + ||v̄||2s�

(3)k + 2q̂�(‖v̄‖sε

(3)k + ‖ū‖sε

(3)k + �ε

(3)k )

k ∈ FM̄ \ Fm

2∑

k1+k2+k3=k

|k1|∈Fm(ū,�)|k2|∈Fm(v̄,�)|k3|∈Z2

|ūk1 ||v̄k2 ||qk3 |ω−sk3 + ||v̄||2s

α(3)k

ωsk+ 2q̂�(‖v̄‖s + ‖ū‖s + �)

α(3)k

ωsk, k /∈ FM̄ .

(31)

Concerning the bound Z(2)k , we apply the same technique to obtain sharper estimates for the series∑

k1+k2+k3=k

|k1|∈Fm,{|k2|,|k3|}6⊂FM̄

|v̄k1 |ω−sk2 |qk3 |ω−sk3,

∑k1+k2+k3=k

|k1|∈Fm,|k2|,|k3|∈Z2

|ūk1 ||qk2 |ω−sk2 |qk3 |ω−sk3. (32)

For k ∈ FM̄ , the former can be estimated by

29

∑k1+k2+k3=k

|k1|∈Fm,{|k2|,|k3|}6⊂FM̄

|v̄k1 |ω−sk2 |qk3 |ω−sk3

=∑

k1+k2+k3=k

|k1|∈Fm(v̄,�),{|k2|,|k3|}6⊂FM̄

|v̄k1 |ω−sk2 |qk3 |ω−sk3

+∑

k1+k2+k3=k

|k1|∈Fm\Fm(v̄,�),{|k2|,|k3|}6⊂FM̄

|v̄k1 |ω−sk2 |qk3 |ω−sk3

≤ q̂∑

|k1|∈Fm(v̄,�)

|v̄k1 |∑

k2+k3=k−k1{|k2|,|k3|}6⊂FM̄

ω−sk2 ω−sk3

+ �q̂∑

k1+k2+k3=k

{|k2|,|k3|}6⊂FM̄

ω−sk1 ω−sk2ω−sk3

≤ q̂∑

|k1|∈Fm(v̄,�)

|v̄k1 |ε(2)k−k1 + 2εq̂ε

(3)k .

(33)

and, for k /∈ FM̄ , ∑k1+k2+k3=k

|k1|∈Fm,|k2|,|k3|∈Z2

|v̄k1 |ω−sk2 |qk3 |ω−sk3≤ q̂

∑|k1|∈Fm(v̄,�)

|v̄k1 |α

(2)k−k1ωsk−k1

+ 2�q̂α(3)

ωsk.

Similarly, for the second in (32) ∑k1+k2+k3=k

|k1|∈Fm,|k2|,|k3|∈Z2

|ūk1 ||qk2 |ω−sk2 |qk3 |ω−sk3≤

∑k1+k2+k3=k

|k1|∈Fm,|k2|,|k3|∈FM̄

|ūk1 ||qk2 |ω−sk2 |qk3 |ω−sk3

+ q̂2∑

|k1|∈Fm(ū,�)

|ūk1 |ε(2)k−k1 + 2�q̂

2ε(3)k k ∈ FM̄

q̂2∑

|k1|∈Fm(ū,�)

|ūk1 |α

(2)k−k1ωsk−k1

+ 2�q̂2α(3)

ωsk, k /∈ FM̄ .

Collecting all the terms, we define

Z(2)k :=

4∑

k1+k2+k3=k

|k1|∈Fm,|k2|,|k3|∈FM̄

|v̄k1 |ω−sk2 |qk3 |ω−sk3

+ 2∑

k1+k2+k3=k

|k1|∈Fm,|k2|,|k3|∈FM̄

|ūk1 ||qk2 |ω−sk2 |qk3 |ω−sk3

+4q̂

∑|k1|∈Fm(v̄,�)

|v̄k1 |ε(2)k−k1 + 2�ε

(3)k

+ 2q̂2 ∑|k1|∈Fm(ū,�)

|ūk1 |ε(2)k−k1 + 2�ε

(3)k

k ∈ FM̄

4q̂

∑|k1|∈Fm(v̄,�)

|v̄k1 |α

(2)k−k1ωsk−k1

+ 2�α

(3)k

ωsk

+ 2q̂2 ∑|k1|∈Fm(ū,�)

|ūk1 |α

(2)k−k1ωsk−k1

+ 2�α

(3)k

ωsk,

k /∈ FM̄The same sharpening procedure can not be applied to the sequence Z

(3)k because no terms ūk, v̄k appear in

(26) , hence we report the same sequence as it was previously defined

Z(3)k :=

3

∑k1+k2+k3=k

|ki|∈FM̄

|qk1 |ω−sk1 |qk2 |ω−sk2ω−sk3 + 9q̂

2ε(3)k k ∈ FM̄

3q̂2α

(3)k

ωsk, k /∈ FM̄

.

30

Let us explain now how to manage with the uniform bound Z̃M . By definition, we have to find a constant

Z̃(1)M so that

Z̃(1)M ≥ sup

k/∈FMωskZ

(1)k .

Looking at the last line in (31), we see three different terms: according to [52] the two latest can be easily

bounded by means of the uniform bound α̃(3)M given in (39). The first one is less than 2q̂Sk where

Sk :=∑

k1+k2+k3=k

|k1|∈Fm(ū,�)|k2|∈Fm(v̄,�)|k3|∈Z2

|ūk1 ||v̄k2 |ω−sk3 .

Having in mind that k = (k(1), k(2)), k1 = (k(1)1 , k

(2)1 ),k2 = (k

(1)2 , k

(2)2 ), M = (M

(1),M (2)), s = (s(1), s(2)),

let us introduce the following quantities.

Definition 1. For a choice of � and M , define

M̂ := min{M (1),M (2)}

k̂ := max{k(1)1 , k(2)1 , k

(1)2 , k

(2)2 : |k1| ∈ Fm(ū, �), |k2| ∈ Fm(v̄, �)}

χ(1)(M , �) := max

ωs(1)

k(1)

ωs(1)

k(1)−k(1)1 −k(1)2

: |k(1)| < M (1), |k1| ∈ Fm(ū, �), |k2| ∈ Fm(v̄, �)

χ(2)(M , �) := max

ωs(2)

k(2)

ωs(2)

k(2)−k(2)1 −k(2)2

: |k(2)| < M (2), |k1| ∈ Fm(ū, �), |k2| ∈ Fm(v̄, �)

.The next Lemma provides an upper bound for ωskSk as k 6∈ FM .

Lemma 4.2. Let � be fixed and k̂, M̂ , χ(1)(M , �), χ(2)(M , �) be as in Definition 1. Then

supk/∈FM

ωskSk ≤ max

∣∣∣∣∣1− 2k̂M̂

∣∣∣∣∣−s(1)−s(2)

,

∣∣∣∣∣1− 2k̂M̂∣∣∣∣∣−s(2)

χ(1)(M , �),

∣∣∣∣∣1− 2k̂M̂∣∣∣∣∣−s(1)

χ(2)(M , �)

∑|k1|∈Fm(ū,�)|k2|∈Fm(v̄,�)

|ūk1 ||v̄k2 |.

(34)

Proof. It holds

supk/∈FM

ωskSk = sup{

sup|k(1)|≥M(1),|k(2)|≥M(2)

ωskSk, sup|k(1)|≥M(1),|k(2)|

=∑

|k1|∈Fm(ū,�)|k2|∈Fm(v̄,�)

|ūk1 ||v̄k2 |1∣∣∣∣1− k(1)1 +k(1)2k(1) ∣∣∣∣s(1) ∣∣∣∣1− k(2)1 +k(2)2k(2) ∣∣∣∣s(2)

≤ 1∣∣∣1− 2k̂M̂

∣∣∣s(1)+s(2)∑

|k1|∈Fm(ū,�)|k2|∈Fm(v̄,�)

|ūk1 ||v̄k2 |.

Hence

sup|k(1)|≥M(1),|k(2)|≥M(2)

ωskSk ≤1∣∣∣1− 2k̂

M̂

∣∣∣s(1)+s(2)∑

|k1|∈Fm(ū,�)|k2|∈Fm(v̄,�)

|ūk1 ||v̄k2 |.

If |k(1)| < M (1) and |k(2)| ≥M (2) we have

ωskSk = ωs(1)

k(1) |k(2)|s

(2) ∑k1+k2+k3=k

|k1|∈Fm(ū,�)|k2|∈Fm(v̄,�)|k3|∈Z2

|ūk1 ||v̄k2 |ω−sk3 =∑

|k1|∈Fm(ū,�)|k2|∈Fm(v̄,�)

|ūk1 ||v̄k2 |ωs

(1)

k(1)|k(2)|s(2)

ωs(1)

k(1)−k(1)1 −k(1)2

|k(2) − k(2)1 − k(2)2 |s

(2)

=∑

|k1|∈Fm(ū,�)|k2|∈Fm(v̄,�)

|ūk1 ||v̄k2 |ωs

(1)

k(1)

ωs(1)

k(1)−k(1)1 −k(1)2

∣∣∣∣1− k(2)1 +k(2)2k(2) ∣∣∣∣s(2)≤ 1∣∣∣1− 2k̂

M̂

∣∣∣s(2) χ(1)(M , �)∑

|k1|∈Fm(ū,�)|k2|∈Fm(v̄,�)

|ūk1 ||v̄k2 |.

Similarly, in case |k(1)| ≥M (1) and |k(2)| < M (2)

ωskSk ≤1∣∣∣1− 2k̂M̂

∣∣∣s(1) χ(2)(M , �)∑

|k1|∈Fm(ū,�)|k2|∈Fm(v̄,�)

|ūk1 ||v̄k2 |.

�

Denoting by η the right hand side of the (34) we can define

Z̃(1)M := 2q̂η +

(‖v̄‖2s + 2

(q̂�‖v̄‖s + q̂�‖ū‖s + q̂�2

))α̃

(3)M .

For Z̃(2)M and Z̃

(3)M we adopt the same bounds as in (28). Hence the tail bound Z̃M is now given by

Z̃M :=1

µ̃M

(2q̂η + α̃

(3)M

(‖v̄‖2s + 2

(q̂�‖v̄‖s + q̂�‖ū‖s + q̂�2

))+ α̃

(3)M

(4||v̄||sq̂ + 2||ū||sq̂2

)r + 3α̃

(3)M q̂

2r2).

5 Results

In this section we report the results we obtained by applying the radii polynomial technique, in the form

discussed in the previous section, for the enclosure of non constant solutions for system (3). All the rigorous

computations and the visualisation have been performed in Matlab with the interval arithmetic package

Intlab [55].

We remind that the domain Ω is given as the rectangular Ω = [0, L1] × [0, L2] and, for simplicity, the

various computational parameters have been chosen homogeneous, that is

m = (m,m), M̄ = (M̄, M̄), M = (M,M), s = (s, s).

32

In any computation the decay rate s is fixed equal to s = (2, 2). We did not investigate the performance of

the enclosure as s varies. At this regard, an interesting analysis has been performed [45].

For convenience, let us recall the homogenous states

p2(λ) =

(λ+√λ2 − 4λ2λ

,λ−√λ2 − 4λ2

)p3(λ) =

(λ−√λ2 − 4λ2λ

,λ+√λ2 − 4λ2

).

First we discuss how to compute the numerical solutions, then we report some details about the rigorous

enclosure.

5.1 Computing the numerical solution

For a choice of the domain sizes L1, L2 and of the parameter λ we numerically compute several patterns lying

on the same family bifurcating from one of the homogenous state p2(λ), p3(λ). Each solutions is intended in

Fourier space, that is we numerically find a zero of a finite m-dimensional projection of system (22) in the

unknowns X = {γ, {uk}k∈Fm , {vk}k∈Fm}.

The eigenvalues of the eigen-problem (4) are labeled by k = (k(1), k(2)), k(1), k(2) ≥ 0 and are given by

hk = π2

((k(1))2

L21+

(k(2))2

L22

)therefore, in light of lemma 2.5, the bifurcations occur at values

γk =1

hk

(λ+ hk − v2∗hk + v2∗ + λ

)where v∗ is the value of the v-component in p2(λ) and p3(λ). According to the analysis performed in section

2.3, for a choice of k we perturb the homogenous state in the direction of the eigenfunction φk. Since

we are working in Fourier space and the basis we have projected on is exactly the basis of eigenfunctions

{φk}k, such a perturbation practically consists in perturbing the corresponding coefficients uk and vk. This

perturbed state, together with γk, is now considered as initial data for a Newton iteration scheme. Note

that, at this stage, the parameter γ is considered as an unknown then we need to append one more equation

to the system. A good choice is for instance u2k + v2k = α, with 0 < α

of the increment of γ, the size m of the finite dimensional projection and also the accuracy of the aimed

solution. In all the computations we assume that the numerical solution x̄ is achieved when the sup-norm

of f (m)(x̄) is less than 10−10. The parameter m = (m,m) ranged in between 12 ≤ m ≤ 23 and different

∆γ-step have been considered. Note that this procedure can not be continued at folds, but apparently, we

did not encounter any.

5.2 Rigorous enclosure results

Some details about the validation of the numerical solutions, obtained following our method, are now pre-

sented. For validation of a numerical solution we mean to provide the radius r of the ball around the

numerical solution in the Banach space Xs within which the exact solution is guaranteed to exist. We

remind that within such a ball the genuine solution is unique.

More explicitly, suppose the numerical solution is given by x̄ = {ūk, v̄k}k∈Fm and the method results an

enclosing radius r in the space Xs. It means that it exists an exact solution x = {uk, vk}Z2 so that

|uk − ūk| ≤ rω−sk , |vk − v̄k| ≤ rω−sk , ∀k ∈ Fm

|uk| ≤ rω−sk , |vk| ≤ rω−sk , ∀k 6∈ Fm.

Remark 5.1. We point out that the computational method does not return one radius r, rather a interval

R = [R−,R+] such that for any r ∈ R the hypothesis of Lemma 3.1 are satisfied. Hence, the information

provided by R is twofold: from one side R− is an upper bound for the distance (in ‖ · ‖s norm) between

the exact solution and the numerical approximation, while R+ is the radius of the smallest ball around the

numerical approximation within which the genuine solution is guaranteed to be isolated. In the tables below

the radius r is in the between of R− and R+.

Example 1. Global bifurcation diagrams for λ = 4.5, Ω = [0, 1.1]× [0, 0.8]

Here we discuss the results stated in Theorem 1.1 and depicted in Figure 1. A magnification of Figure 1a is

given in Figure 3. Any bullet in the figures represents a validated non-constant pattern. The graphs depict

γ against the squared L2 norm of the v component of the solutions, where v0 denotes the v component

of the homogenous state. The different families are labelled according to the bifurcation point, that is the

(k(1), k(2))-branch bifurcates from the homogenous state at value γ(k(1),k(2)). In Figure 4 some solutions are

plotted. They belong to different families that bifurcate from p3.

Table 1 reports the parameters m, M̄,M and the sharpening coefficient � used in the computation of the

solutions marked in Figure 3 and the resulting enclosure radius.

34

2 4 6 8 10 12 14 16

x 10−3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

γ

(0,2)

(0,3)

(1,2)

(1,3)

(2,2)

(2,3)

(3,0)

(3,1)

(3,2)

(3,3)

‖v − v0‖2

Homogenous soluti on = p 2. Paramete rs: λ = 4.5, Ω = [0, 1.1] × [ 0, 0.8]Magnificati on .

3

2

4

5

6

1

7

Figure 3: Magnification of part of the bifurcation diagram in Figure 1a. The numbers 1, . . . , 7 mark the solutions

analysed in Table 1.

# k γ m M̄ M � r

1 (0,2) 0.0148 6 29 60 10−3 4.011 ·10−5

2 (1,2) 0.0113 14 40 100 10−3 1.128 ·10−5

3 (0,3) 0.0066 12 48 130 10−3 6.731 ·10−7

4 (3,1) 0.0089 21 55 170 10−3 1.468 ·10−5

5 (2,3) 0.0050 21 88 247 10−3 4.785 ·10−6

6 (3,3) 0.0046 21 106 260 10−4 2.429 ·10−6

7 (3,1) 0.0106 12 40 90 10−3 6.216 ·10−6

Table 1: The table reports the values of the parameters used in the computation of the solutions labeled in

Figure 3.

35

0 0.20.4 0.6

0.8 1

0

0.5

1

2.2

2.4

2.6

2.8

3

3.2

3.4

(a) (b) (c)

(d) (e) (f)

(g) (h)

Figure 4: Plot of the v-component of solutions for the Gray-Scott equation in 2D. Each solution belongs to a different

branch of families depicted in Figure 1b, where Ω = [0, 1.1] × [0, 0.8], λ = 4.5. For each figure we report the value of

γ and the label k of the branch. (a) γ = 0.0226,k = (1, 0), (b) γ = 0.0122, k = (0, 2), (c) γ = 0.0215, k = (1, 1),

(d) γ = 0.0061, k = (0, 3), (e) γ = 0.0052, k = (2, 3), (f) γ = 0.0080, k = (3, 1), (g) γ = 0.0096, k = (1, 2), (h)

γ = 0.0043, k = (3, 3).

36

From the data in the Table 3, we realise that a larger finite dimensional parameter and larger compu-

tational parameters are requested as long as γ decreases and the size of k increases. In particular M is

necessarily related to γ: as we already discussed in previous sections, M must be large enough to control the

tail polynomial. Although the introduction of the sharpening coefficient � enables to weaken the aftereffect

of γ, a dependence between the two parameters is still present and a smaller γ requires a larger M . On the

other side, k denotes the Fourier coefficient of u and v that has been initially perturbed at the bifurcation.

Due to the nonlinear effects, such a perturbation is spread on the neighbourhood coefficients. Hence a larger

k requires larger m and M̄ that allow to take into account a large number of contributing coefficients. Also,

when moving on a bifurcation branch, further the solution is from the bifurcation point, more important is

the effect of the nonlinearity. For example, the solutions number 4 and 7 listed above both belong to the

same bifurcation branch. The latter one is close to the bifurcation and only six coefficients of the numerical

solution ū are larger than 10−6, namely ūk with

k = (0, 0), (0, 2), (3, 1), (6, 0), (6, 2), (3, 3).

The former one, that is the solution number 4, is much further along the branch and it holds that |ūk| ≥ 10−6

for

k = (0, 0), (0, 2), (3, 1), (6, 0), (9, 1), (6, 2), (3, 3), (0, 4), (6, 4), (3, 5), (0, 6).

We remark that most of the families depicted in Figure 1 can be easily continued further. In other cases, as

for the branches labeled (2, 2), (2, 3), (3, 2), (3, 3) in Figure 3, no significative continuation could be produced

in reasonable computational time. Two are the reasons for that: or because the numerical continuation failed

in providing numerical solutions, or because the validation of the numerical solution required the choice of

extremely large computational parameters.

The last phenomena is analysed in more detail in the next experiment, where we monitor the performances

of the computational method along one single branch of solutions.

Example 2. Monitoring the performance on a single branch of solutions.

With reference to Theorem 1.2, we compute different patterns on the branch (2,0) and we push forward the

computation up to the point where the radii polynomials does not produce an enclosure in reasonable time.

We see that this branch of solutions is much more longer than the branches depicted in previous figures

where the continuation was stopped after a certain number of solutions have been computed.

In Table 2 we report the computational data for the validation of those solutions highlighted in Figure

5, in particular the parameters used for the rigorous enclosure and the computational time. ( A MacBook

37

Pro, Intel Core 7, 2.8 GHz laptop has been used). Moreover, in Table 3, we provide the v̄(k(1),0) coefficients,

so to have an insight on how the mass of the solution moves to higher modes as long as the solution is far

from the bifurcation. In Figure 6 the solutions #1, 3, 6 are plotted.

0.008 0.01 0.012 0.014 0.016 0.018

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Homogenous soluti on = p 3. Paramete rs: λ = 4.3, Ω = [0, 1] × [ 0, 0.7]

γ

(2,0)6

5

4

3

2

1

‖v − v0‖2

Figure 5: Any bullet represents a rigorously computed non-constant patterns. The family of solutions bifurcates

from p3(λ), λ = 4.3, at parameter value γk, with k = (2, 0).

# γ m M̄ M � time (s)

1 0.0179 6 47 55 10−3 50

2 0.0177 7 48 57 10−3 55

3 0.0169 8 48 63 10−3 56

4 0.0159 8 49 79 10−3 73

5 0.0139 10 56 102 10−3 112

6 0.0098 12 90 172 10−3 385

Table 2: Computational parameters used and computational time spent in the rigorous enclosure of the solutions

labelled in Figure 5.

38

Solution #

k 1 2 3 4 5 6

(0, 0) 2.72404222944 2.73736587082 2.77176648590 2.81456634315 2.89206192722 3.05351814491

(1, 0) -0.00000000000 -0.00000000000 0.00000000000 0.00000000000 0.00000000000 0.00000000000

(2, 0) 0.12017276153 0.21770220312 0.37926915762 0.53816302906 0.80166562525 1.39018208998

(3, 0) -0.00000000000 0.00000000000 0.00000000000 0.00000000000 0.00000000000 -0.00000000000

(4, 0) 0.00110963050 0.00363102273 0.01096673870 0.02208013753 0.05005550913 0.17961820795

(5, 0) -0.00000000000 -0.00000000000 0.00000000000 -0.00000000000 0.00000000000 -0.00000000000

(6, 0) -0.00000278580 -0.00001729057 -0.00010061903 -0.00031538259 -0.00115275977 -0.00449284116

(7, 0) 0.00000000000 -0.00000000000 0.00000000000 0.00000000000 0.00000000000 -0.00000000000

(8, 0) -0.00000014076 -0.00000153128 -0.00001451275 -0.00006132871 -0.00033308297 -0.00423713101

(9, 0) 0.00000000000 -0.00000000000 0.00000000000 0.00000000000 0.00000000000 0.00000000000

(10, 0) -0.00000000103 -0.00000002009 -0.00000032353 -0.00000190854 -0.00001581400 -0.0005012980

(11, 0) 0.00000000000 0.00000000000 0.00000000000 -0.00000000000 -0.00000000000 -0.00000000000

(12, 0) 0.00000000000 0.00000000015 0.00000000478 0.00000004425 0.00000056614 0.00000814732

(13, 0) -0.00000000000 0.00000000000 0.00000000000 -0.00000000000 0.00000000000 0.00000000000

(14, 0) 0.00000000000 0.00000000001 0.00000000044 0.00000000552 0.00000010767 0.00000977801

(15, 0) 0.00000000000 0.00000000000 0.00000000000 -0.00000000000 -0.00000000000 0.00000000000

(16, 0) 0.00000000000 0.00000000000 0.00000000001 0.00000000013 0.00000000399 0.00000112735

Table 3: The (k(1), 0) coefficients of the v component of the six solutions highlighted in Figure 5 are reported. We

can see how the contribution of higher modes coefficients increases as long as the solutions is far from the bifurcations,

due to the effect of the nonlinearity.

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

1.5

2

2.5

3

3.5

4

4.5

5

(a) (b) (c)

Figure 6: Plot of the v-component of the solution number 1 (a), 3 (b) and 6 (c) marked in Figure 5.

Example 3. Effect of the sharpening parameter.

In the last experiment we investigate the role of the newly introduced sharpening parameter �. Remember

that the small is � the higher is its effect, in the sense that the sets Fm(ū, �), Fm(v̄, �) given in (30) are larger.

Consider the pattern in correspondence to γ = 0.0108 on the (1, 2) bifurcating branch in Figure 1b. For

different values �, in Table 4 we list the computational parameters M̄ and M needed to obtain a successful

validation of the numerical solution. The finite dimensional projection parameter has been fixed m = 13.

We see that for large values of � an extremely large value for the computational parameter is requested. The

last row in the table concerns a value close to the largest possible choice of �, according to the condition

� ≤ min{‖ū‖s, ‖v̄‖s}. Therefore, if the sharpening parameter is not considered at all, that is, if we follow the

39

approach of [54], a even larger computational parameters must be selected and a rigorous enclosure would

be practically impossible.

In Figure 7 both the components u and v of the pattern are depicted.

Figure 7: Plot of the u (left) and v ( right) component of the pattern discussed in example 3. The solution belongs

to the family (1,2) reported in Figure 1b at γ = 0.0108.

� M̄ M

0.001 59 79

0.01 74 89

0.1 164 182 *

0.3 280 400 *

Table 4: The values of the computational parameters M and M̄ necessary for the validation of a numerical solution

are reported, for different values of the sharpening parameter �. In the cases marked with ∗ the computation has not

been done rigorously.

6 Conclusion

In this paper a new development of the radii polynomial technique has been proposed with the aim to rig-

orously compute non-uniform stationary patterns for the two dimensional Gray-Scott system. It is demon-

strated that the new approach succeeds in enclosing several complex patterns, solutions that could not be

proved following the previous approach present in the literature. We plan to develop further the technique

and to research in different directions. For instance, the three dimensional case may be considered. We

40

propose to combine the method with the theory introduced in [44] to compute parameter dependent smooth

branches of solutions. Then a systematic analysis of the bifurcation branches of non-uniform patterns may

be performed, to see, for instance, whether some branches undergo secondary bifurcations or reconnect

to the uniform solutions. Also, a more challenging project is to rigorously prove existence of heteroclinic

connections and homoclinic cycles.

Acknowledgements

This work was partially funded by the the MICINN grant MTM2011-24766.

7 Appendix

Suppose s ≥ 2, M ≥ 6. For k ≥ 3 let be defined

γk := 2

[k

k − 1

]s+

[4 ln(k − 2)

k+π2 − 6

3

] [2

k+

1

2