Finite time blow up of complex solutions of the conserved Kuramoto-Sivashinsky equation in $\mathbb{R}^d$, $d \geq 1$

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Finite time blow up of complex solutions of the

conserved Kuramoto-Sivashinsky equation in R

d

_{, d}

≥ 1

Léo Agélas

To cite this version:

Finite time blow up of complex solutions of the

conserved Kuramoto-Sivashinsky equation in R

, d

_{≥ 1}

L´eo Ag´elas

Department of Mathematics, IFP Energies Nouvelles, 1-4, avenue de Bois-Pr´eau, F-92852 Rueil-Malmaison, France

Abstract

We consider complex-valued solutions of the conserved Kuramoto-Sivashinsky equation which describes the coarsening of an unstable solid surface that con-serves mass and that is parity symmetric. This equation arises in different aspects of surface growth. Up to now, the problem of existence and smooth-ness of global solutions of such equations remained open in Rd_{, d ≥ 1. In this} article, we answer partially to this question. We prove the finite time blow up of complex-valued solutions associated to a class of large initial data, more precisely, we show that there is complex-valued initial data that exists in every Besov space (and hence in every Lebesgue and Sobolev space), such that after a finite time, the complex-valued solution is in no Besov space (and hence in no Lebesgue or Sobolev space).

Keywords: Blow up, Surface growth, conserved Kuramoto-Sivashinsky

Introduction

In this paper, we consider the conserved Kuramoto-Sivashinsky (cKS) equa-tion described by the following partial differential equaequa-tion,

∂tv + ∆2v + ∆|∇v|2= 0 (1) with initial condition

v(0) = v0 (2)

on Rd _{with solutions vanishing at infinity as |x| → ∞ or on the d−dimensional} torus Td _{≡ R}d_/(2πZ)d_{, with periodic boundary conditions and in this case we} require in addition that v0 is a periodic scalar function of period 2π with zero mean value, that isRTdv0(x) dx = 0.

5

The cKS equation models the step meandering instability on a surface charac-terized by the alternation of terraces with different properties [13]. It appeared

as a model for the boundaries of terraces in the epitaxy of Silicon [13]. It also describes the growth of an amorphous thin film by physical vapor deposition [19] and [20]-in this case, conserved dynamics are obtained by transforming to

10

a frame that translates upward with constant velocity.

For simplicity of presentation, we consider the rescaled version (1) with a dimensional length-scales. Sometimes the equation is considered with a linear instability +∆v , which leads to the formation of hills, and the Kuramoto-Shivashinky-type nonlinearity −|∇v|2 _{leading to a saturation in the coarsening}

15

of hills (see [16, 20]). Both terms are neglected here. They are lower order terms not important for questions regarding regularity and blow up. Furthermore, the equation is usually perturbed by space-time white noise referred as η (see for instance [16, 20, 19]), which we also neglect here, although many results do hold for the stochastic PDE also (see [3]).

20

Previous work shows that numerical simulations based on (1) can be well fitted to experimental data, and that (1) adequately describes the phenomena of coarsening and roughening that are characteristic for the growth of corre-sponding surfaces on intermediate time scales [16, 20, 23]. In particular, the characteristic statistical measures of the surface morphology such as the

corre-25

lation length and the surface roughness calculated from the cKS model show very good agreement with available experimental data and, therefore, support the validity of this modeling approach (see [16] for more details).

Nevertheless, without the existence of a unique solution there is no hope of guaranteeing that a numerical approximation is really an approximation in any

30

meaningful sense, since it is not clear what is being approximated.

Thus, a crucial open problem for the cKS equation (1) is the fact that ex-istence and uniqueness of global solutions is not known (see [8, 7]) even in the one dimensional case (see [7] and references therein).

For the one dimensional case, existence of global weak solutions on bounded

35

domains has been established in [5, 23]. The key point of the construction of global weak solutions lies on a L2_{−energy-type estimate deriving from the fact} that, in this case, the nonlinearity in (1) is orthogonal to the solution itself in the sense of L2_.

For the two-dimensional case, the situation seems even worse, as the

ex-40

istence of global weak solutions could only be established in H−1 _{using the} non-standard energyR₀2πev(x)_{dx (see [25]).}

However, up to now, the question of global regularity for the cKS equation (1) is still open (see [7, 8] and references therein). Only existence, uniqueness and regularity of local solutions or global strong solutions with smallness condition

45

on the initial data have been established in [4, 23] with initial values in W1,q with q ≥ 2 for d = 1 and W1,4 _{for d = 1, 2, 3 and later improved in [7, 8] for} ini-tial values in the critical Hilbert space Hd/2_{or in a critical space of BMO-type.} The main difficulties for treating problem (1) are caused by the nonlinearity term ∆|∇v|2 _{and the lack of a maximum principle. Due to its nonlinear parts,}

50

ex-istence, uniqueness and smoothness of global solutions of (1).

As in [17], in this paper, we omit the condition that ˆv is the Fourier transform of a real-valued solution v of (1) in the d−dimensional space and consider it in

55

the space of all possible complex-valued functions.

In this situation, we answer to the existence and smoothness problem for the cKS equation (1) by showing that for sufficiently large initial data, we get complex-valued solutions which blow up in finite time. More precisely, by borrowing the arguments used in [18], in our Theorem 3.1 combined with Corollary 3.1, we show that there is complex-valued initial data that exists in every Triebel-Lizorkin or Besov space (and hence in every Lebesgue and Sobolev space), such that after a finite time, the solution is in no Triebel-Lizorkin or Besov space (and hence in no Lebesgue or Sobolev space).

This finite time blow up result may suggest as it was shown in [1], that a better taking into account of the main physical phenomena and a better approximation of terms related to them in the surface growth mathematical model can help to get existence and uniqueness of global strong solutions for such equations as the ones modeling epitaxy thin film growth.

The paper is organized as follows: In section 1, we give some notations. In section 2, we introduce the Banach spaces as the Hilbert spaces Hs_{, Lebesgue} spaces Lp _{and Besov spaces B}s

p,q. In section 3, we prove our Theorem 3.1 with our Corollary 3.1.

If we set u = −v and u0= −v0, we notice that u satisfies the following equivalent Equation to (1):

∂tu + ∆2u − ∆|∇u|2= 0, (3) with initial condition

u(0) = u0. (4)

Then, without loss of generality, in what follows, we will consider Equation (3) rather than (1).

1. Some notations

For any x ∈ Rd_{, we denote by {x}}

+ the vector having for components the

60

values max{xm, 0} for 1 ≤ m ≤ d. We denote by | · | the modulus of a complex number. We denote by k · k, the euclidean norm on Cd _{defined for all x ∈ C}d_by kxk =   X 1≤m≤d |xm|2   1 2

. We denote by k · k∞, the infinity norm on Cd defined for all x ∈ Cd _{by kxk}

∞= max 1≤m≤d|xm|. For x ∈ Cd _{and r > 0, let B}

r(x) = {y ∈ Cd : ky − xk∞≤ r}. Notice, here that

65

the ball of Cd _{is defined with the norm k · k}

∞and not with the euclidean norm of Cd _{as it is usually the case. This change is made in order to deal with the} periodic case also.

For any a ∈ R and r > 0, we denote with the same notation Br(a) the ball Br(A) where A ∈ Rd is such that for all 1 ≤ m ≤ d, Am= a.

For any x ∈ Rd_{and y ∈ R}d_{, we say that x ≤ y (resp. x ≥ y) if for all 1 ≤ m ≤ d,} xm≤ ym(resp. xm≥ ym).

For any x ∈ Rd _{and a ∈ R, we say that x ≤ a (resp. x ≥ a) if for all 1 ≤ m ≤ d,} xm≤ a (resp. xm≥ a).

For any function f defined on Rd_{× R}+_{, for any t ≥ 0, for a simplicity in the}

75

notation, we denote by f (t) the function x 7−→ f(x, t) defined on Rd_.

Given an absolutely integrable function f ∈ L1_(Rd_{), we define the Fourier} transform bf : Rd_{7−→ C by the formula,}

b f (ξ) =

Z Rd

e−ix·ξf (x) dx,

and extend it to tempered distributions. For a function f which is periodic with period 1, and thus representable as a function on the torus Td _{, we define the}

80

discrete Fourier transform bf : Zd 7−→ C by the formula, b

f (k) = Z

Td

e−ix·kf (x) dx,

when f is absolutely integrable on Td_{, and extend this to more general} distri-butions on Td_.

2. Some Banach spaces

We denote by S (Rd_{) the class of complex-valued tempered Schwartz}

func-85

tions on Rd _{and by S (T}d_{) the space of complex-valued infinitely differentiable} functions on Td_{. Its dual space S}′_(Rd_{) (resp. S}′_(Td_{)) is called the space of} distributions. In particular, we recall the following two facts ([22])

• Any function f ∈ S (Td_{) can be represented as} f (x) = X

k∈Zd

akeik·x for any x ∈ Rd with (ak)k∈Zd scalars such that

90

sup

k∈Zd(1 + |k|)

m_|a

k| < ∞ for any m ∈ N.

In this case one has ak = bf (k) for each k ∈ Zd. • Any function g ∈ S′_(Td_{) can be represented as}

g(x) = X k∈Zd

akeik·x for any x ∈ Rd (5) with (ak)k∈Zd scalars such that

sup

k∈Zd(1 + |k|)

−m

2.1. Some Sobolev spaces

For s ∈ R, we define the Sobolev norm kfkHs_(Rd₎of a tempered distribution 95 f : Rd_{7−→ C by,} kfkHs_(Rd₎= Z Rd(1 + |ξ| 2₎s_{| bf (ξ)|}2_dξ 1 2 ,

and then we denote by Hs_(Rd_{) the space of tempered distributions with finite} Hs(Rd) norm, which matches when s is a non negative integer with the classical Sobolev space Hk(Rd), k ∈ N. For s > −12, we also define the homogeneous Sobolev norm, 100 kfkH˙s_(Rd₎= Z Rd|ξ| 2s | bf (ξ)|2dξ 1 2 ,

and then we denote by ˙Hs_(Rd_{) the space of tempered distributions with finite} ˙

Hs_(Rd_{) norm. Similarly, on the torus T}d_{and s ∈ R, we define the Sobolev norm} kfkHs_(Td₎of a tempered distribution f : Td7−→ C by,

kfkHs_(Td₎=  X k∈Zd (1 + |k|2)s| bf (k)|2   1 2 ,

and then we denote by Hs_(Td_{) the space of tempered distributions with finite} Hs_(Td_{) norm. On the torus T}d_{, for s > −}1

2, we also define the homogeneous

105 Sobolev norm, kfk_H˙s_(Td₎=  X k∈Zd |k|2s| bf (k)|2   1 2 ,

and then we denote by ˙Hs_(Td_{) the space of tempered distributions with finite} ˙

Hs_(Td_{) norm.}

For any p ≥ 1, we denote by Lp_(Rd_{) the space of functions f : R}d _{7−→ C such} that the norm,

110 kfkLp_(Rd₎:= Z Rd|f(x)| p_dx 1 p < +∞.

For any p ≥ 1, we denote by Lp_(Zd_{) the space of functions f : Z}d _{7−→ C such} that the norm,

2.2. Besov spaces

We introduce the usual dyadic unity partition of Littlewood-Paley decompo-sition (see [2, 9, 10, 24] for more details). To this end, we take an arbitrary

real-115

valued radial function ϕ in S (Rd_{) whose Fourier transform} b

ϕ is non-negative and is such that,

suppϕ ⊂ Bb 1(3/2) and ϕ(ξ) ≥b 1

2 for ξ ∈ B12(3/2),

and define ϕj(x) = 2jdϕ(2jx) so that ϕbj(ξ) = ϕ(2b −jξ) for j ∈ Z. We may assume,

∀ξ ∈ Rd\{0}, X j∈Z b

ϕj(ξ) = 1.

For any f ∈ S′_(Rd_{) or S}′_(Td_{), we denote by ∆}

jf , j ∈ Z, the function,

120

∆jf := ϕj⋆ f.

If f ∈ S′_(Rd_{), we notice that for all x ∈ R}d _{and for all j ∈ Z,} ∆jf (x) = F−1(ϕbjf )(x) =b

Z Rd b

ϕj(ξ) bf (ξ)eiξ·xdξ. (6) If f ∈ S′_(Td_{), using (5) we notice that for all x ∈ R}d _{and for all j ∈ Z,}

∆jf (x) = X k∈Zd

b

ϕj(k) bf (k)eik·x. (7)

Then a tempered distribution f belongs to the homogeneous Besov space ˙Bs p,q(Rd) (resp. B˙s

p,q(Td)) modulo polynomials if and only if kfk_B˙s

p,q(Rd) < ∞ (resp. kfk_B˙s p,q(Td)< ∞) where for Ωd= R d _{or T}d_, kfk_B˙s p,q(Ωd):=           X j∈Z 2jsqk∆jf kq_Lp_(Rd₎   1 q if q < ∞ sup j∈Z 2jsk∆jf kLp_(Rd₎elsewhere, (8)

and f =P_j∈Z∆jf ∈ S′/Pm where Pm is the space of polynomials of degree ≤ m and m = [s −d

p], the integer part of s − d p.

3. Blow up of complex-valued solutions of the cKS equation

We set Ωd= Rdor Ωd= Tdfor the periodic case, Fd= Rdor Fd= Zdfor the

125

We set A = ∆2_{. In either R}d_{or T}d_{, we let e}−tA_{for t > 0 be the usual biharmonic} heat semigroup associated to the biharmonic heat equation wt+ Aw = 0 (see [14], for an explicite form of its solution on Rd_).

Then, we start with the definition of mild solutions of cKS Equation (3)

130

obtained from Kato’s semigroup approach [15].

Definition 3.0.1. We say that u is a mild solution of cKS Equation (3) if u is a solution to the equation u = G(u) where G : C([0, T ], X) → C([0, T ], X), with X being a space of complex-valued tempered distributions on Rd_{: for a.e} t ∈ [0, T ]

G(u)(t) = e−tAu0+ Z t

0

e−(t−s)A∆|∇u(s)|2ds. (9) The Kato’s semigroup approach used to find a fixed point of G is to show that G is a contraction mapping on C([0, T ], X) or on some subset of C([0, T ], X). It turns out that the natural spaces in which to consider solutions are of the form C([0, T ], X), where X is a scale-invariant space (we call scale-invariant

135

space, any Banach space X satisfying kf(λ·)kX = kfkX for all λ > 0): for instance the homogenenous Sobolev space ˙Hd2, Besov spaces ˙B

d p

p,∞ for p < ∞ or the BMO-type spaces introduced in [8].

In fact, it can be shown (arguing similarly as in Frazier, Jawerth and Weiss in [11]) that all scale-invariant spaces of distributions, that also contain all

140

Schwartz functions, are contained in the Besov space ˙B0 ∞,∞.

The main ingredient of the proof of the existence of blowing-up solutions as in [18] consists in noticing that if the initial data has a positive Fourier transform, then that positivity is preserved for the solution at all further times. One can then use the Duhamel formulation of the solution and deduce a lower

145

bound for the Fourier transform that blows up in finite time.

Theorem 3.1. _{Let d ∈ N}∗_{. Let w ∈ S (Ω}d) such that w is a real-valuedb

function,w is non-negative, has Lb 1_{norm equal to 1, and has support in B1}

2(

3 2)∩ Fd(so w is in every Triebel-Lizorkin or Besov space). Then if A > 2

16

15, and if u

is a mild solution to cKS Equation (3) whose Fourier transform is a non-negative

150

real-valued function supported in Fd+, with initial data u0= 27Aw, then u(t) is

not in any Triebel-Lizorkin or Besov space when t = Td:= log(2

1 15d2).

Proof. To get the proof, we adapt the construction of [18] to our case. We

set wn = w2

n

with n ∈ N. We observe that wn+1 = w2n and then wbn+1 = b

wn⋆wbn. Sincew is non-negative, has Lb 1 norm equal to 1 and is supported in B1

2(

3

2) ∩ Fd = {ξ ∈ Fd : 1 ≤ ξ ≤ 2}, then by using an induction argument, we deduce that for all n ∈ N, bwn is also non-negative, has L1 norm equal to 1 and is supported in {ξ ∈ Fd: 2n ≤ ξ ≤ 2n+1}.

Let t ≥ 0. We will show now by induction that the proposition P(n) = {bu(t) ≥ A2n

αn(t)wbn} is true for all n ∈ N, where {αn}n∈N is the sequence of functions defined for all s ≥ 0 by αn(s) = 27e−2

n+4_d2_s

defined by t0 = 0 and tn = log(2) d2 n+1 X j=1

2−4j for all n ≥ 1. Notice that the sequence {tn}n∈N is increasing and lim

n→∞tn = log(2

1

15d2) = T_d which implies

that for all n ∈ N, tn< Td.

Let us show that the proposition P(n) is true for n = 0. Since

u = G(u) (10)

where G(u) is given by (9), then after taking the Fourier transform of Equation (10), we deduce that for all ξ ∈ Fd,

b

u(ξ, t) = e−t|ξ|4u_b0(ξ) − Z t

0

e−(t−s)|ξ|4|ξ|2( c∇u ⋆ c∇u)(ξ, s) ds. (11) We notice that c∇u(ξ, s) = iξbu(ξ, s). Sinceu(·, s) is supported in F_b d+, then we get c∇u(ξ, s) = i{ξ}+u(ξ, s) and thereforeb

−i c∇u(ξ, s) = {ξ}+u(ξ, s).b (12) Since_bu(·, s) is non-negative, from (12) we get −i c∇u(ξ, s) ≥ 0 and therefore we deduce,

( c∇u ⋆ c∇u)(ξ, s) = −((−i c∇u) ⋆ (−i c∇u))(ξ, s)

≤ 0. (13)

Therefore, from (11), we get u(ξ, t) ≥ eb −t|ξ|4_ub

0(ξ) which gives us u(ξ, t) ≥b 27_Ae−t|ξ|4_{w(ξ) and thenb u(ξ, t) ≥ 2b} 7_Ae−24d2t_{w(ξ) since |ξ| ≤b} √_dkξk

∞andw isb supported in {ξ ∈ Fd : 1 ≤ ξ ≤ 2}. Hence, we get that the proposition P(0) is true. Let us assume that the proposition P(n) is true for a given n ∈ N. Then, let us show that P(n + 1) will be also true. From (11), since bu0 = 27Aw ≥ 0,b we have, b u(ξ, t) ≥ − Z t 0 e−(t−s)|ξ|4|ξ|2( c∇u ⋆ c∇u)(ξ, s) ds. (14) Since P(n) is true, then from (12), we get −i c∇u(ξ, s) ≥ {ξ}+A2

n

αn(s)wbn(ξ). Sincew_bn is supported in {ξ ∈ Fd : 2n ≤ ξ ≤ 2n+1}, then we get −i c∇u(ξ, s) ≥ 2n_A2n

αn(s)wbn(ξ) ≥ 0 (which means that each component of the vector is greater than 2n_A2n

αn(s)wbn(ξ) ≥ 0) and therefore we deduce, ( c∇u ⋆ c∇u)(ξ, s) = −((−i c∇u) ⋆ (−i c∇u))(ξ, s)

≤ −d(2nA2nαn(s))2(wbn⋆wbn)(ξ) = −d(2n_A2n

αn(s))2wbn+1(ξ).

(15)

Using (15), from (14), we deduce b

u(ξ, t) ≥ Z t

0

Sincew_bn+1is supported in {ξ ∈ Fd: 2n+1≤ ξ ≤ 2n+2}, then we get 22(n+1)d ≤ |ξ|2_{≤ 2}2(n+2)_{d, hence from (16) we get,}

b u(ξ, t) ≥ d224n+2A2n+1w_bn+1(ξ) Z t 0 e−(t−s)24(n+2)d2αn(s)2ds = 214d224n+2A2n+1wbn+1(ξ) Z t 0 e−(t−s)24(n+2)d2e−2n+5d2s1_{tn≤s≤t}ds ≥ 214d224n+2A2n+1w_bn+1(ξ)e−2 n+5_d2_t 1{t≥tn} Z t tn e−(t−s)24(n+2)d2ds = 28A2n+1wbn+1(ξ)e−2 n+5_d2_t 1_{{t≥tn}(1 − e}−24(n+2)d2(t−tn)_).

However, for all t ≥ tn+1, we have 1 − e−2

4(n+2)_d2_(t−t n)_≥ 1 2, since tn+1− tn ≥ 155 log(2) d2 2 −4(n+2)_{. Then, we deduce,} b u(ξ, t) ≥ 27A2n+1w_bn+1(ξ)e−2 n+5_d2_t 1{t≥tn+1} = A2n+1w_bn+1(ξ)αn+1(t). Then, we deduce that the proposition P(n + 1) is true. Therefore, we deduce that for all n ∈ N, for all ξ ∈ Fd,

b

u(ξ, t) ≥ A2n

b

wn(ξ)αn(t). (17)

Thanks to (17), we have for all j ∈ N, for all ξ ∈ Fd, b

ϕj(ξ)u(ξ, Tb d) ≥ A2

j

b

ϕj(ξ)wbj(ξ)αj(Td).

From Section 2.2, we notice thatϕ_bj(ξ) ≥ 1₂ for all ξ ∈ {ζ ∈ Fd: 2j≤ ζ ≤ 2j+1} which is the support of w_bj and moreover wbj is non-negative, then we deduce that for all ξ ∈ Fd,ϕbj(ξ)wbj(ξ) ≥ 1₂wbj(ξ). Therefore, we infer that for all j ∈ N, for all ξ ∈ Fd, b ϕj(ξ)u(ξ, Tb d) ≥ 1 2A 2j b wj(ξ)αj(Td) ≥ 0. (18) Since for any j ∈ Z, bϕjbu ≥ 0 (thanks to bϕ ≥ 0 and bu ≥ 0), then thanks to (6) and (7), we deduce that for all j ∈ Z, ∆ju(0, Td) = k∆ju(Td)kL∞

(Rd₎. Moreover,

we observe also that ∆ju(0, Td) = k bϕjbu(Td)kL1_(F

However, for any j ∈ N, k bwjkL1_(F

d)= 1 and since for all j ∈ N, Td> tj, we get

αj(Td) = 27e− 2j+4 log(2) 15 = 27(e− 16 log(2) 15 )2 j

. Then, we deduce that, ku(Td)kB˙s ∞,∞(Ωd)≥ 2 6_sup j∈N 2js_(Ae−16 log(2)_{15 )}2j . Therefore, we deduce that if A > e16 log(2)15 = 21615 then ku(T_d)k_˙

Bs

∞,∞(Ωd) = ∞,

which allows us to conclude the proof.

165

Corollary 3.1. _{Let d ∈ N}∗. Let Ωd = Rd or Td for the periodic case. Let Xd = ˙H

d

2(Ω_d) or X_R the scale-invariant space of BMO-type introduced in [8]

with value in C. Let w be as in Theorem 3.1 and let A > 21615. Then there is no

mild solution u to the cKS Equation (3), with u0 = 27Aw in C([0, Td], Xd) for Td= log(2

1 15d2). 170

Proof. Here, we use the same arguments as in [18]. Notice that Xd⊂ ˙B∞,∞0 (Ωd). Suppose for a contradiction that there is a solution u ∈ C([0, Td], Xd) to Equa-tion (3). By the semigroup methods of [15], we know from [8, 7] for which their result extends straight forward to complex-valued solutions, that there is a num-ber ǫ > 0, depending only upon ku(t)kXd, such that for every t ∈ [0, Td] that

there is a mild solution v ∈ C([t, t + ǫ], Xd) to Equation (3) with v(t) = u(t) where on [t, t + ǫ], v is obtained as the fixed point of the iterated sequence {v(n)_{(· − t)}}

n∈N defined by v(0)= 0 and for all n ∈ N by v(n+1)(σ) = e−σAu(t) +

Z σ 0

e−(σ−τ )A∆|∇v(n)(τ )|2dτ, ∀σ ∈ [0, ǫ]. (19) Then, after taking the Fourier transform of (19), we obtain for all σ ∈ [0, ǫ] and ξ ∈ Rd_, \ v(n+1)_{(ξ, σ) = e}−σ|ξ|4 b u(ξ, t) − Z σ 0 e−(σ−τ )|ξ|4|ξ|2(∇v\(n)_⋆∇v\(n)_{)(ξ, τ ) dτ. (20)} Thus, ifbu(t) is a non-negative real-valued function supported in Fd+then from the equation (20) just above, by using an induction argument, we infer that for all n ∈ N, for all σ ∈ [0, ǫ], dv(n)_{(σ) is also a non-negative real-valued function} supported in Fd+ which implies that for all s ∈ [t, t + ǫ], bv(s) is a non-negative real-valued function supported in Fd+. Furthermore, by the uniqueness results

175

similar as the ones obtained in [7, 12, 15], we get that for all s ∈ [t, t + ǫ], u(s) = v(s). Therefore, we deduce that for every t ∈ [0, Td], if u(t) is a non-b negative real-valued function supported in Fd+then for all s ∈ [t, t + ǫ], bu(s) is a non-negative real-valued function supported in Fd+. Since u(0) = u0= 27Aw and w is a non-negative real-valued function supported in F_b d+, then we infer

180

that for all s ∈ [0, Td],u(s) is a non-negative real-valued function supported inb Fd+.

Acknowledgements

185

The author would like to thank Professor Dirk Bl¨omker for his helpful dis-cussions and suggestions.

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