Ann. Acad. Rom. Sci.

Ser. Math. Appl.

ISSN 2066-6594

Vol. 18, No. 2/2026

AN IMPROVED MODEL FOR ATOMIC

SHRINKING^∗

Alain Haraux^†

Rafael Ortega^‡

Communicated by G. Moro¸sanu

DOI

10.56082/annalsarscimath.2026.2.65

Abstract

The usual equation for the motion of electrons in the deterministic

Rutherford-Bohr atomic model is conservative with a singular potential

at the origin. When a dissipation is added, new phenomena appear,

mainly a contraction of orbits for large time. A special model is studied

in which the dissipation coeﬃcient varies as the inverse of the square

of the distance to the nucleus, and it is shown that this equation has

contraction properties similar to the case of a linear damping, but

the backward equation shows aﬃne growth of the orbits for large time,

which is more realistic than the exponential growth in the case of linear

damping.

Keywords: gravitation, singular potential, second order diﬀerential

equation, global solutions, asymptotic bounds.

MSC: 34A34, 34C11, 34D05.

1 Introduction

The usual equation for the motion of electrons in the deterministic Rutherford-

Bohr atomic model is conservative with a singular potential at the origin.

^∗Accepted for publication on October 30, 2025

^†alain.haraux@sorbonne-universite.fr, Sorbonne Universit´e, Universit´e Paris-

Diderot SPC, CNRS, INRIA, Laboratoire Jacques-Louis Lions, LJLL, F-75005, Paris,

France

^‡rortega@ugr.es, Departamento de Matema´tica Aplicada, Facultad de Ciencias, Uni-

versidad de Granada, 18071 Granada, Spain

65

An improved model for atomic shrinking

66

When a dissipation is added to the basic equation

q²

u

mu⁰⁰= −

3

4πε₀|u|

modeling Coulomb’s central force (with q the elementary charge, m the mass

of the electron and ε₀the vacuum permittivity) written as

u

u⁰⁰+ c₀

= 0,

(1)

3

|u|

with

q²

4πε₀m

c₀:=

,

we obtain an equation of the general form

u

u⁰⁰+ f(t, u, u⁰)u⁰+ c₀

= 0,

(2)

3

|u|

where f is a suitable non-negative function. In the paper [3] the ﬁrst author

considered the special case where f is a positive constant δ, which leads to

the simple equation

u

u⁰⁰+ δu⁰+ c₀

= 0.

(3)

3

|u|

Such an equation appears when trying to understand the long term dissi-

pation induced by random shocks of wandering cosmic corpuscles with the

atoms, and in [3] it was observed that the behavior of the solutions opens the

door to new explanations of totally independent phenomena: cosmological

redshift (cf. [7, 12 ]), aging of electrical appliances, gigantism of arthropods

and plants in the carboniferous era. Although that equation does not have

any solution for which |u(t)| is an exact decreasing exponential Ce^−ηt, the

type of contraction produced by this kind of strong dissipation is basically

uniform and when we try to guess the size of atoms in the distant past, the

presumably exponential growth for the backward equation is totally unreal-

istic.

This observation leads us to study the eﬀect of a somewhat diﬀerent

dissipation mechanism which has already been studied in the literature, in

particular by the second author, with a diﬀerent purpose, cf. eg. [9 ], in

connection with a conjecture of Euler (cf. [2]) on long term collapse of the

solar system. Here, we assume that the eﬀect of wandering corpuscles is

negligible far from the nucleus and maximal when we approach it. In this

case, a reasonable model improving (3) may be

u⁰

u

u⁰⁰+ δ

+ c₀

= 0,

(4)

a²+ |u|

|u|

2

3

A. Haraux, R. Ortega

67

with a and δ some positive constants.

In the present paper, we report on the asymptotic behavior of the solu-

tions to (2) and more precisely (4). In Section 2 we recall, to make subse-

quent comparisons easier, the Rutherford-Bohr atomic model with its cir-

cular orbits. In Section 3, we collect some general properties of (2). Section

4 is devoted to the special case of (4) and the associate backward problem,

with part of the technicalities in the appendix (Section 6). Special proper-

ties of real solutions are discussed in Section 5. The physical interpretation

of the results is outlined in the concluding Section 7.

2 Recalling the classical Rutherford-Bohr model

The Rutherford model (cf. [10 ]), which served as a basis for Bohr’s model [1]

of the hydrogen atom, is based on a corpuscular conception of protons and

electrons and the application of Coulomb’s law for electrostatic forces. De-

noting by q the common absolute value of the charges of proton and electron,

the equation of motion, considering the unique proton of the nucleus as the

center of coordinates, can be written in the plane of the orbit in complex

form. Denoting by u the position of the electron in the complex plane, we

have

q²

u

mu⁰⁰= −

.

(5)

(6)

3

4πε₀|u|

The electron travels in a circular orbit

|u(t)| = R

so that the solutions take the form

u(t) = R exp(iω(t + t₀)

(7)

(8)

with

q²

ω²=

.

4πε₀mR³

In particular, the constant velocity of the electron is

|u⁰(t)| = v = KR^−1/2

for some positive constant K. This is consistent with the analogous prop-

erty given by the third Kepler’s law for the planets motion driven by the

gravitational ﬁeld of the sun. In the framework of Bohr’s modelization, it

An improved model for atomic shrinking

68

is assumed that the radius R can take only the values of a sequence of the

form r_n= r₀n²with n a positive integer. In particular, the electron cannot

“fall” on the nucleus.

Bohr’s theory explains neither why all electrons so luckily always choose a

circular orbit rather than other elliptic possibilities, nor even how the ﬁrst

atoms of matter could appear, even in the simplest case of the hydrogen

atom. If an electron is left without initial velocity in the close surrounding

of a proton, an atom will never appear since the electron will collide with

the proton in a very short time. The problem of the origin of the initial

kinetic energy of the electron is left open. Besides, there is no explanation

of why some circular orbits would be stable and not the others. An attempt

to circumvent that problem is quantum mechanics leading to Schr¨odinger’s

equation (cf. [11]). For the time being, in the same track as [3, 6], we shall

avoid quantum considerations, even though such a framework is presently

considered necessary to understand the electromagnetic waves produced by

hot matter (and therefore to consider spectral properties and the redshift

problematics).

3 General results for equation (2)

In this section, we investigate the standard relevant questions for the solu-

tions of the general equation (2 ).

3.1

Existence and uniqueness

We introduce the total energy

1

c₀

E(t) := |u⁰|²(t) −

,

2

|u(t)|

where u(t) is a solution of (2 ) on some interval [0, T). Assuming for simplicity

that f ∈ C¹(R⁺× C²; R⁺), it is clear that a unique local solution exists for

any initial data u(0) ∈ C^∗, u⁰(0) ∈ C. Moreover the identity

E⁰(t) = −f(t, u, u⁰)|u⁰|²≤ 0

implies

∀t ∈ [0, T), E(t) ≤ E(0)

providing

1

2

c₀

|u⁰|²(t) ≤

+ E(0).

|u(t)|

A. Haraux, R. Ortega

69

This shows that the only way for the solution to be non-global is vanishing

of u(T), in which case the equation does not make sense beyond T.

3.2

Bounded trajectories

The next result generalizes a boundedness result of [3] for any function f.

Theorem 1. Let u₀= 0 and assume the initial smallness condition

|u₀||u⁰|²< 2c₀.

(9)

0

Then the local solution u of (2) on [0, T) with initial conditions u(0) =

u₀; u⁰(0) = u⁰₀satisﬁes the inequality

2c₀|u₀|

∀t ∈ [0, T),

|u(t)| ≤

.

(10)

0

2

2c₀− |u₀||u₀|

In particular, if u does not vanish in ﬁnite time, u is a global bounded non-

vanishing solution.

Proof. The inequality (9) is equivalent to E(0) < 0. Then the inequality

1

c₀

E(t) := |u⁰|²(t) −

≤ E(0) =⇒

≥ −E(0)

2

|u(t)|

gives

c₀

2c₀|u₀|

|u(t)| ≤

=

.

c₀

|u₀|

0

1

0

2

2c₀− |u₀||u₀|

− |u₀|

2

3.3

Convergence to 0 of non-vanishing bounded solutions

As in the case of the conservative equation (1) which is well known to have

hyperbolic and parabolic trajectories, (2) may have some unbounded solu-

tions if the damping is suﬃciently weak for large values of u. The next result

shows, on the other hand, that in sharp contrast with (1), (2) does not have

any periodic trajectory at all under mild conditions on f.

Theorem 2. Assuming that f is uniformly positive and bounded for u

bounded independently of (t, u⁰), for any solution u of (2) such as |u(t)|

is global, positive and bounded on R⁺, we have

lim |u(t)| = 0.

(11)

t→+∞

An improved model for atomic shrinking

70

Proof. Since u never vanishes, it is clear that u ∈ C²(R⁺, C) and we have

2

E⁰(t) = −f(t, u(t), u⁰(t))|u⁰(t)| .

(12)

In particular, E(t) is non-increasing. Then we have two possibilities

Case 1.

lim E(t) = −∞.

(13)

t→+∞

c₀

Then since

≥ −E(t) we conclude that

|u(t)|

lim |u(t)| = 0.

t→+∞

Case 2.

lim E(t) = E^∗> −∞.

(14)

t→+∞

Then E(t) is bounded and by our hypothesis on f we have for some ν > 0

Z

t

2

∀t ∈ R⁺,

E(0) − E(t) ≥ ν

|u⁰(s)| ds.

0

In particular, u⁰∈ L²(R⁺, C). We have assumed that u(t) is bounded, hence

precompact in R⁺with values in C. Therefore if (11) is not satisﬁed we may

assume that for some sequence t_ntending to +∞

lim u(t_n) = w = 0.

(15)

n→+∞

On the other hand, for u_n(s) = u(t_n+ s), we have

lim u⁰_n= 0

(16)

n→+∞

in the strong topology of L²(0, 1) and in particular, by Cauchy-Schwarz-

inequality we ﬁnd

lim u(t_n+ s) = w

(17)

n→+∞

uniformly on [0, 1]. Since w = 0, this implies

u(t_n+ s)

w

lim

=

(18)

3

n→+∞

|u(t_n+ s)|

|w|

uniformly on [0, 1]. But then by the equation

w

lim u⁰⁰= c₀

(19)

n

3

n→+∞

|w|

A. Haraux, R. Ortega

71

in the strong topology of L²(0, 1). This is contradictory with (16). Indeed,

it implies for instance

Z

1

w

lim

s(1 − s)u⁰⁰(t_n+ s)ds = c₀

s(1 − s)ds

= z = 0

2

n→+∞

|w|

0

while on the other hand

Z

1

s(1 − s)u⁰⁰(t_n+ s)ds = −

(1 − 2s)u⁰(t_n+ s)ds −→ 0.

0

This contradiction concludes the proof.

As an immediate consequence of the previous theorem and the results of

Sections 2 and 3, we obtain:

Corollary 1. Assume |u₀||u⁰₀|²< 2c₀. Then assuming that f is uniformly

positive for u bounded independently of (t, u⁰), the local solution of (2) with

initial conditions u(0) = u₀; u⁰(0) = u⁰₀either vanishes in ﬁnite time, or

tends to 0 as t tends to inﬁnity.

4 The case of equation (4)

It is not diﬃcult to ﬁnd some solutions which vanish in ﬁnite time, when

u₀and u⁰₀are collinear with opposite orientation. On the other hand this

cannot happen for u₀and u⁰₀not collinear. Indeed in such a case u(t) has

to remain bounded and as for the simpler model (3), we ﬁnd that, as a

consequence of

d

(u ∧ u⁰)

(u ∧ u⁰) = −δ

,

2

dt

a²+ |u|

the vector (u ∧ u⁰) remains bounded away from 0 on any ﬁnite interval. As

a special case of our previous corollary, we have

Corollary 2. Assume |u₀||u⁰₀|²< 2c₀. Then the local solution of (4) with

initial conditions u(0) = u₀; u⁰(0) = u⁰₀either vanishes in ﬁnite time, or

tends to 0 as t tends to inﬁnity.

Remark 1. In [3] we established that all solutions of equation (3) are

bounded for t ≥ 0, cf. also [8, 9]. For the conservative equation (1), there

are many unbounded solutions, for instance real positive solutions with ini-

2

tial positive velocities have a linear growth at inﬁnity as soon as u₀u⁰₀> 2c₀

and there are even the unbounded solutions

1

3

2

3

9c₀

u(t) = (

) (t + b)

2

An improved model for atomic shrinking

72

2

(with b > 0 arbitrary) corresponding to the limiting case u₀u⁰₀= 2c₀. Since

boundedness in (3) is produced by dissipation and the dissipation in (4) be-

comes very weak for large values of |u(t)|, it is natural to wonder whether (4)

may have unbounded positive solutions. We show in Section 5 (Proposition

5.1) that it is indeed the case.

Now we state the most important property of (4) which makes the dif-

ference with (3) and saves us from one of the big contradictions concerning

the size of atoms in the distant past.

Theorem 3. Let us consider the backward equation of (4)

v⁰

v

v⁰⁰− δ

+ c₀

= 0.

(20)

a²+ |v|

|v|

2

3

Then the size of |v(t)| grows up at most linearly for large time. More pre-

cisely, for any non-vanishing global solution v of (20) there exists a number

µ_v∈ [0, ∞) such that

|v(t)|

lim

= µ_v.

(21)

t→+∞

t

We shall rely on the identity

n

o

d

(v, v⁰)

1

|v|

c₀

|v ∧ v⁰|²

−

arctan

= −

+

(22)

(23)

2

3

dt

|v|

a

|v|

combined with

d

(v ∧ v⁰)

(v ∧ v⁰) = δ

,

2

dt

a²+ |v|

both proved in the appendix. The invariance under rotations of the equation

(20) allows to reduce one dimension in the system. In fact, if we deﬁne the

new unknowns

1

r

r = |v|, ξ = r⁰−

arctan , ` = |r ∧ r⁰|

a

the above identities lead to the equations

2

1

r

c₀

r²

`

δ

r⁰=

arctan

+ ξ, ξ⁰= −

+

, `⁰=

`.

(24)

a

r³

a²+ r²

For the proof of the theorem we shall rely on the following technical lemma

A. Haraux, R. Ortega

73

Lemma 1. Let W(t) := (r(t), ξ(t), `(t)) be a solution of (24) with r(0) > 0

and forward maximal interval [0, ω) and assume that, for some m > 0,

c₀

r⁰(0) ≥ m +

.

(25)

mr(0)

Then ω = +∞ and, for each t ∈ [0, ∞),

r⁰(t) > m.

Proof. We will prove that

r⁰(t) > m, t ∈ [0, ω).

(26)

Then a standard continuation argument implies that ω = +∞. By contra-

diction assume that the claim (26 ) does not hold. We take the ﬁrst instant

τ ∈ (0, ω) such that r⁰(τ) = m. The deﬁnition of τ together with (25) imply

that r⁰(t) > m if t ∈ [0, τ). In consequence r(t) > r(0) + mt if t ∈ (0, τ].

Also, from the second equation of (24),

c₀

r(t)²

c₀

ξ⁰(t) > −

> −

.

(r(0) + mt)²

After integrating this inequality,

Z

τ

Z

∞

c₀

ξ(τ) > ξ(0)−

ds > ξ(0)−

ds = ξ(0)−

.

(r(0) + ms)²

mr(0)

0

Since r(t) is increasing on [0, τ] the ﬁrst equation in (24 ) implies that

1

r(0)

c₀

r⁰(τ) >

arctan

+ ξ(τ) > r⁰(0) −

.

a

mr(0)

This is the searched contradiction because r⁰(τ) = m is not compatible with

the assumption (25).

We are ready for the proof of the theorem.

Proof. We distinguish two cases.

q

c₀

r(t)

Case 1: For every t ∈ [0, ∞), r⁰(t) < 2

.

q

c₀

r(t₀)

Case 2: There exists some t₀∈ [0, ∞) such that r⁰(t₀) ≥ 2

.

In the ﬁrst case we solve the diﬀerential inequality to deduce that

r(t) = O(t^2/3) as t → +∞

An improved model for atomic shrinking

74

and µ_v= 0.

Assume now that we are in the second case. We shall apply Lemma 1

q

c₀

0

to the translate solution W(t₀+ .) with m =

_{r(t )}. As a consequence, for

each t ∈ (t₀, ∞), r⁰(t) > m. Along the lines of the proof of the Lemma we

observe that the two integrals below are ﬁnite:

Z

∞

ds

r(s)²

I₁=

, I₂=

.

a²+ r(s)²

0

In particular, the third equation of (24) implies that `(t) ≤ `(0)e^δIand as

1

a consequence

Z

∞

`(s)²

r(s)³

I₃=

ds < ∞.

0

Now we can use the third and second equations to prove that `(t) and ξ(t)

have limits at inﬁnity,

`(∞) = `(0)e^δI, ξ(∞) = ξ(0) − c₀I₂+ I₃.

1

Finally, from the ﬁrst equation,

π

lim r⁰(t) =

+ ξ(∞).

t→+∞

2a

After invoking L’Hˆopital rule, µ_v= r⁰(∞).

Remark 2. The number µ_vcan be arbitrarily large. This can be shown

by an application of Lemma 1 with m → ∞. The above proof allows to

q

c₀

estimate µ_vin terms of the initial conditions if r⁰(0) ≥ 2

_r(0). Note that

q

`(∞)²

1

I₁≤

^r(0)(^π₂− arctan(^r(0))), `(∞) ≤ `(0)e^δI, I₃≤

.

2c¹₀^/2r(0)^3/2

a

c₀

a

5 Some special results for real solutions of

equation (4)

5.1

The equation (4) has unbounded solutions

Proposition 1. Let u₀, u⁰₀be both positive and such that

64πδ

c¹₀^/2

1/2

u⁰₀> max{

, 2

}.

(27)

(28)

7a

u₀

Then u is unbounded and more precisely

u⁰

∀t ≥ 0,

u(t) ≥ u₀+ ⁰t.

4

A. Haraux, R. Ortega

75

Proof. We establish by contradiction that

u⁰₀

∀t ≥ 0,

u⁰(t) ≥

.

(29)

4

Assuming the contrary, let

u⁰₀

T := inf{t > 0, u⁰(t) <

}.

(30)

(31)

4

Then by deﬁnition

u⁰₀

∀t ∈ (0, T),

u⁰(t) ≥

4

u⁰

and u⁰(T) = ⁰, hence

4

Z

T

1

E(T) =

32

c₀

u⁰²(s)

a²+ u²(s)

2

u⁰₀

−

= E(0) − δ

ds

u(T)

0

implying

Z

T

1

u⁰²(s)

a²+ u²(s)

2

E(T) > u⁰₀− δ

ds

(32)

4

0

c¹₀^/2

1/2

2

since u⁰₀> 2

implies E(0) > u⁰₀. In addition, u is increasing on (0, T)

1

4

u₀

and the inequality

1

c₀

1

2

c₀

2

E(t) = u⁰²(t) −

≤

u⁰₀

−

2

u(t)

u₀

shows that

∀t ∈ (0, T),

u⁰(t) ≤ u⁰₀.

(33)

Using this inequality in (32) we obtain

Z

T

1

4

1

2

u⁰₀

>

u⁰₀− δu⁰₀

ds.

32

a²+ u²(s)

0

2

Multiplying by 32 and simplifying by u⁰₀, we obtain

Z

T

1

7 < 32δ

ds.

a²+ u²(s)

0

u⁰

4

As a consequence of (31 ), we have u(s) > ⁰s and in particular this now

implies

Z

T

∞

1

u⁰₀

1

u⁰₀

7 < 32δ

ds < 32δ

ds.

(34)

2

a²+

s²

a²+

s²

0

16

An improved model for atomic shrinking

76

u⁰

A standard change of variable σ = ⁰s provides the value

4a

Z

∞

1

2π

au⁰₀

ds =

,

2

u⁰₀

a²+

s²

0

16

and we end up with the inequality

64πδ

7 <

64πδ

⇒ u⁰₀<

au⁰₀

7a

contradicting our assumption on u⁰₀.

5.2

Real solutions of the backward equation

For real solutions of the backward equation (20), we have the more precise

estimate given by

Proposition 2. For any real solution v and more generally if v and v’ are

collinear, we have

π

∀t ≥ 0,

|v(t)| ≤ |v₀| + t(

+ |v₀⁰|).

(35)

(36)

2a

Proof. Since v and v’ are collinear, our basic identity reduces to

n

o

d

(v, v⁰)

1

|v|

c₀

−

arctan

= −

≤ 0

2

dt

|v|

a

|v|

which implies in particular

d

(v, v⁰)

π

(v₀, v⁰)

1

|v₀|

|v₀|⁰

a

|v(t)| =

≤ +

+

−

arctan

dt

|v|

2a

and ﬁnally

ꢀ

ꢁ

π

|v(t)| ≤ |v₀| + t

+ |v₀⁰| .

2a

Remark 3. The sublinear rate of growth of solutions of (20) is likely to be

optimal. Actually, large solutions tend to behave as solutions of the simpler

equation

v⁰

v

v⁰⁰− δ

+ c₀

= 0

(37)

2

3

|v|

which has the explicit real positive solutions

c₀

v(t) =

(t + C).

δ

However, the optimality of inequality (35) itself is not clear, since as a tends

to 0, the right hand side blows-up.

A. Haraux, R. Ortega

77

6 Appendix

6.1

Proof of (23 )

Since v⁰∧ v⁰= 0 we have

d

v ∧ v⁰

(v ∧ v⁰) = v ∧ v⁰⁰= δ

,

2

dt

a²+ |v|

because v ∧ v = 0.

6.2

The derivative of |v(t)|

p

2

Since |v(t)| =

|v(t)| , we have

d

2(v(t), v⁰(t))

(v(t), v⁰(t))

p

|v(t)| =

=

.

2

dt

|v(t)|

2

|v(t)|

6.3

Proof of (22 )

We ﬁrst compute

ꢂ

ꢃ

d

|v(t)|

1

|v(t)|

(v(t), v⁰(t))

|v(t)|(a²+ |v(t)| )

arctan

=

= a

.

(38)

2

dt

a

a|v(t)|

1 +

a²

Then, dropping for convenience the variable t in all subsequent calculations,

we evaluate

ꢂ

ꢃ ꢂꢂ

ꢃ₀

ꢃ

ꢂ

ꢃ

ꢂ

ꢃ₀ꢃ

d (v(t), v⁰(t))

v

|v⁰|²

1

=

, v⁰⁰

+

, v⁰

=

, v⁰⁰

+

+ v⁰,

v .

dt

|v(t)|

|v|

Since

ꢂ

ꢃ₀

1

(|v|)⁰

(v, v⁰)

= −

2

3

|v|

we end up with

ꢂ

ꢃ

ꢂ

ꢃ

d (v(t), v⁰(t))

v

|v⁰|²

(v, v⁰)²

v

|v ∧ v⁰|²

=

, v⁰⁰

+

−

=

, v⁰⁰

+

. (39)

3

dt

|v(t)|

|v|

By replacing in (39) the vector v⁰⁰by its expression in (20), we ﬁnd, after

combining with (38), the equality

n

o

d

(v, v⁰)

1

|v|

c₀

|v ∧ v⁰|²

−

arctan

= −

+

,

2

3

dt

|v|

a

|v|

whence (22 ).

An improved model for atomic shrinking

78

7 Conclusion

Under minimal assumptions on the function f, we have obtained that all

bounded solutions of the general equation (2) converge to 0 for t large. This

applies in particular to the model (4) and at least for all suﬃciently small

initial data. Ultimately, the rate of decay should be similar to (3). On

the other hand, the linear expansion for the backward equation (20 ) makes

the model more realistic than (3). However, the exponential rate of decay

for positive time, as examined in [3] is not very realistic either, and if we

want to avoid an unrealistic apocalypse prediction in 20 000 years or so, it

is probably necessary to take account of some other phenomena precluding

that electrons approach the nucleus too closely. It is in fact quite possible

that the electro-static attraction becomes repulsive at short distances, cf [5]

for some mathematical models in this direction.

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[3] A. Haraux, On some damped 2 body problems, Evol. Equations Control

Theory 10 (2021), 657-671.

[4] A. Haraux, On carboniferous gigantism and atomic shrinking. Preprints

2020, 2020110544 (doi: 10.20944/preprints 202011.0544.v2).

[5] A. Haraux, On a variant of Newton-Coulomb’s law. Preprints 2020,

2020120741 (doi: 10.20944/preprints 202012.0741.v1).

[6] A. Haraux, The method of adapted energies for second order evolution

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[7] E. Hubble and M.L. Humason, The velocity-distance relation among

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48.

A. Haraux, R. Ortega

79

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