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Summability of solutions of the heat equation with inhomogeneous thermal conductivity in two variables
Werner Balser, Michèle Loday-Richaud
To cite this version:
Werner Balser, Michèle Loday-Richaud. Summability of solutions of the heat equation with inhomo- geneous thermal conductivity in two variables. Advances in Dynamical Systems and Applications, Dehli : Research India Publ., 2009, 4 (2), pp.159-177. �hal-00364722�
inhomogeneous thermal ondutivity in two variables
Werner BALSER
Abteilung Angewandte Analysis
Universitat Ulm, D-89069 ULM, Germany
Email: balsermathematik.uni-ulm.de
Mihele LODAY-RICHAUD
LAREMA, Universite d'Angers, 2 boulevard Lavoisier
49 045 ANGERS edex 01, Frane
Email: mihele.lodayuniv-angers.fr
February 26, 2009
Abstrat
WeinvestigateGevreyorderand1-summabilitypropertiesoftheformalsolution
of a general heat equation in two variables. In partiular, we give neessary and
suÆientonditionsforthe1-summabilityofthesolutioninagivendiretion. When
restritedto thease of onstantsoeÆients, theseonditions oinide withthose
givenbyD.A.Lutz,M.Miyake,R.Shafkeina1999 artile([LMS99 ℄),andwethus
providea newproofof theirresult.
Keywords: Heatequation, Gevreyseries, 1-summability.
AMS lassiation: 35C10, 35C20, 35K05,40-99, 40B05.
Contents
1 The problem 2
2 Gevrey properties 4
3 1-summability 7
4 Initial onditions 15
4.1 Casea(z)=a2C
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.2 Casea(z)=bz; b2C
. . . . . . . . . . . . . . . . . . . . . . . . . . 17
1 The problem
A formalsolutionofthe lassialheatinitialonditions problem
(1)
(
t
u
2
z u=0
u(0;z)='(z)
inone dimensionalspatialvariablez reads intheform
b
u(t;z) = exp t 2
z t
'(z)
= X
j0 t
j
j!
' (2j)
(z)
providedthatall derivatives' (2j)
exist 1
. When'2O(D
p
) isholomorphiinadis
D
with enter 0 and radius and hene satises, for any r < , estimates of the
form
' (2j)
(z)
CK 2j
(1+2j)!;
forall j 0 and positive onstantsC and K, on D
r
then, u(t;b z) 2O(D
)[[t℄℄ is a
seriesofGevreytypeoforder1intforallz2D
(inshort,a1-Gevrey series). The
Gevreyestimates areloally uniformwithrespettoz inD
. These onditionsare
optimalasshown bythe followingexample: Letonsider'(z)= 1
1 z
= X
n0 z
n
so
that ' (2j)
(0) =(2j)!. The orresponding solution bu(t;z) is of exat Gevrey order
1 and, in partiular, is divergent. It turns out that it is atually 1-summable in
all diretion but R +
in the sense of Denition3.1 below, that is, 1-summable in t
uniformallywithrespetto znear 0.
In 1999, D. Lutz, M. Miyake and R. Shafke ([LMS99 ℄) gave neessary and
suÆient onditions on ' for ub to be 1-summable in a given diretion argt = .
Various works have been done towards the summability of divergent solutions of
partialdierential equations with onstant oeÆients ( [Bal99 ℄, [Miy99 ℄, [BM99℄,
[Bal04 ℄,...) or variable oeÆients ([H99℄, [Ou02 ℄, [PZ97 ℄, [Mk08℄, [Mk09℄,...)
1
Wedenotebu,withahat,toemphasizethepossibledivergeneof theseriesu.b
in two variables. In [Mk05 ℄, S. Malek has investigated the ase of linear partial
dierentialequationswithonstant oeÆientsinmore variables.
Inthisartileweareinterestedintheverygeneralheatinitialonditionsproblem
withinhomogeneous thermalondutivityand internalheat generation
(2)
(
t
u a(z) 2
z
u=q(t;z) a(z)2O(D
)
u(0;z)='(z) 2O(D
):
The heat equation desribes heat propagation under thermodynamis and Fourier
laws. The oeÆient a(z), named thermal diusivity, is related to the thermal
ondutivity by the formula a =
where is the apaity and the density
of the medium. We assume that a(z) and '(z) are analyti on a neighborhood
of z = 0. The internal heat input q may be smooth or not. An important ase
is the ase with no internal heat generationorresponding to a homogeneous heat
equation:
(3)
(
t
u a(z) 2
z
u=0 a(z)2O(D
)
u(0;z)='(z)2O(D
):
Inaseofan isotropiandhomogeneousmedium,;; andheneaareonstants.
An adequatehoie of unitsallows thento assume a=1and the equation redues
to thereferene heatequation
t
u
2
z u=0.
Atually,fornotationalonveniene, weonsidertheproblemintheform
(4) 1 a(z) 1
t
2
z
b u=
b
f(t;z) ; a(z)2O(D
) and
b
f(t;z)2O(D
)[[t℄℄
where 1
t b
u standsforthe anti-derivative R
t
0 b
u(s;z)ds of ubwith respetto t whih
vanishesat t=0.
Problem (4) isequivalent to
(
t b
u a(z) 2
z b u=
t b
f(t;z)
b
u(0;z) = b
f(0;z):
and heneto Problem (2)byhoosingq(t;z)=
t b
f(t;z) and '(z)= b
f(0;z).
Moreover, Problem (4) redues to the homogeneous ase (3) if and only if the
b
Fromnow, we denoteD=1 a(z) 1
t
2
z
and, given a seriesub2O(D
)[[t℄℄, we
denote
b
u(t;z)= X
j0 t
j
j!
u
j;
(z)= X
n0 b u
;n (t)
z n
n!
= X
j;n0 b u
j;n t
j
j!
z n
n!
Sine O(D
)[[t℄℄;
t
;
z
isadierentialalgebraanda(z)2O(D
) theoperator
Dats insideO(D
)[[t℄℄. More preisely,we an state:
Proposition 1.1 The map
D: O(D
)[[t℄℄ !O(D
)[[t℄℄
isa linear isomorphism.
Proof. The operator D is linear. A seriesu(t;b z) = X
j0 t
j
j!
b u
j;
(z) is a solutionof
Problem (4)ifand onlyif
(5) ub
j;
(z)= b
f
j;
(z)+a(z)ub 00
j 1;
(z) forallj 0 startingfromub
1;
(z)0:
Consequently,toany b
f(t;z)2O(D
[[t℄℄thereisauniquesolutionbu(t;z)2O(D
[[t℄℄,
whih provesthat Dis bijetive. 2
In Setion 2 we show that the inhomogenuity b
f(t;z) and the unique solution
b
u(t;z) aretogether 1-Gevrey.
InSetion3weproveneessaryandsuÆientonditionsforubtobe1-summable
in a given diretion argt = . The onditions are valid in the ase when either
a(0) 6= 0 or a 0
(0) 6= 0. When a(z) = O(z 2
) an easy ounter-example shows that
even therationalityof b
f(t;z) isinsuÆient.
InSetion4wedisusstheaessibilityofourneessaryandsuÆientonditions.
Indeed, the onditions aregiven not onlyin terms of the data b
f butalso interms
ofthe rst twoterms ub
;0 and bu
;1
of thesolutionubitself.
Inthepartiularasea=1ouronditionsoinidewiththoseof[LMS99 ℄. Wethus
providea newproofof theresult of[LMS99℄.
2 Gevrey properties
In this artile, we onsider t as the variable and z as a parameter. The lassial
Denition 2.1 (1-Gevrey series) Aseries bu(t;z)= X
j0 t
j
j!
b u
j;
(z)2O(D
)[[t℄℄ is
of Gevrey type of order 1 if there exist 0 <r ; C >0;K >0 suh that for all
j0 and jzjr we have
jbu
j;
(z)jCK j
(1+2j):
In other words, u(t;b z) is 1-Gevrey in t, uniformally in z on a neighbourhood of
z=0.
We denote O(D
)[[t℄℄
1
the subsetof O(D
)[[t℄℄ made of the series whih are of
Gevreytypeof order 1.
Proposition 2.2 O(D
)[[t℄℄
1
;
t
;
z
isa dierentialalgebra stableunder 1
t and
1
z .
Proof. The proof is similarto theone withoutparameter. Stabilityunder
z is
provedusingtheCauhyIntegralFormulaandisguarantedbytheondition\there
exist r :::"inDenition 2.1. 2
It resultsfromthisPropositionthattheoperatorD=1 a(z) 1
t
2
z
atsinside
thespae O(D
)[[t℄℄
1 .
Beause the mainresult of this setion (Theorem 2.5) is set up using Nagumo
normson O(D
) we beginwitha reallof theirdenitionandmainpropertiesand
we refer to [N42 ℄orto [CRSS00℄formore details.
Denition 2.3 (Nagumo norms)
Let f 2 O(D
), p 0; 0 < r and let d
r
(z) = jzj r denote the eulidian
distane of z to the boundary of the dis D
r .
The Nagumo norm kfk
p;r
of f isdened by
kfk
p;r
= sup
jzj<r
f(z)d
r (z)
p
:
Proposition 2.4 (Properties of Nagumo norms)
(i) k:k
p;r
is a norm on O(D
);
(ii) For all z2D ; jf(z)jkfk d(z) p
;
(iii) kfk
0;r
=sup
z2D
r
jf(z)j is the usualsup-norm on D
r
;
(iv)kfgk
p+q;r kfk
p;r kgk
q;r
;
(v)(most important) kf 0
k
p+1;r
e(p+1)kfk
p;r .
Notethatthesameindexr oursonbothsidesoftheinequality(v). Onegetsthus
an estimateof thederivativef 0
intermsof f withouthavingto shrinkthe domain
D
r .
Theorem 2.5 Themap
D: (
O(D
)[[t℄℄
1
! O(D
)[[t℄℄
1
b
u(t;z) 7!
b
f(t;z)=Dbu(t;z)
isa linear isomorphism.
Proof. It results from Proposition 2.2 that D O(D
)[[t℄℄
1
O(D
)[[t℄℄
1 and
from Proposition1.1 that Dis linear and injetive. We areleft to prove that Dis
also surjetive.
Let b
f(t;z)= X
j0 t
j
j!
b
f
j;
(z)2O(D
)[[t℄℄
1
. The oeÆients b
f
j;
(z) satisfy
8
>
>
<
>
>
:
b
f
j;
(z)2O(D
)forall j0:
There exist0<r; C>0; K>0 suh thatforall j0and jzjr
j b
f
j;
(z)jCK j
(1+2j)!
and we look forward to similar onditions on the oeÆients bu
j;
(z) of bu(t;z) =
X
j0 t
j
j!
b u
j;
(z).
Fromthe reurrenerelation(5)therelation
b u
j;
(z)
(1+2j)
= b
f
j;
(z)
(1+2j)
+a(z) b u 00
j 1;
(z)
(1+2j)
starting from ub
1;
(z) 0 holds for all j 0. Applying the Nagumo norms of
indies(2j;r) and properties (iv)and (v)of Proposition2.4weget
kbu
j;
(z)k
2j;r
(1+2j)
k b
f
j;
(z)k
2j;r
(1+2j)
+ ka(z)k
0;r kbu
00
j 1;
(z)k
2j;r
(1+2j)
00
+ ka(z)k
0;r e
2 kbu
j 1;
(z)k
2j 2;r
Denote g
j
= k
b
f
j;
(z)k
2j;r
(1+2j)
and=ka(z)k
0;r e
2
andonsiderthenumerialsequene
(
v
1
=0
v
j
=g
j +v
j 1
forall j0:
By onstrution, kbu
j;
(z)k
2j;r
(1+2j) v
j
forallj 0.
Letusboundv
j
asfollows. Byassumption,0g
j
CK j
(1+2j)
(1+2j) r
2j
=C(Kr 2
) j
for all j and the series g(X) = P
j0 g
j X
j
is onvergent. Due to the reurrene
relationdeningthev
j
'stheseriesv(X)= P
j0 v
j X
j
satisfy(1 X)v(X) =g(X).
ItisthenonvergentandthereexistonstantsC 0
>0;K 0
>0suhthatv
j C
0
K 0
j
forall j. Hene,
kbu
j;
(z)k
2j;r C
0
K 0
j
(1+2j) forallj 0:
Wededueasimilarestimateonthesup-normbyshrinkingthedomainD
r
. Indeed,
let 0<r 0
<r. Forall j0and z2D
r 0
,
jbu
j;
(z)j =
bu
j;
(z)d
r (z)
2j 1
d
r (z)
2j
1
(r r 0
) 2j
b u
j;
(z)d
r (z)
2j
Hene,
sup
z2D
r 0
jbu
j;
(z)j
1
(r r 0
) 2j
kbu
j;
k
2j;r
C
0
K 0
(r r 0
) 2
j
(1+2j)
2
3 1-summability
Stillonsidering t as the variable and z as a parameter, one extends the lassial
notionsofsummabilitytofamiliesparameterizedbyzinrequiringsimilaronditions,
theestimates beinghoweveruniformwithrespettotheparameterz. Forageneral
studyof serieswithoeÆients ina Banah spae we refer to [Bal00 ℄. Among the
we hoose here a generalization of Ramis denitionwhih states that a series b
f is
1-summablein the diretion if there exists a holomorphifuntion f whih is 1-
Gevreyasymptoti to b
f onan open setor
;>
bisetedby withopeninglarger
than (f. [R80 ℄ Def 3.1). There are various equivalent ways of expressing the
1-Gevrey asymptotiity. We hooseto extend theone whih sets onditionson the
suessive derivativesof f (see [Mal95 ℄ p. 171 or[R80 ℄ Thm2.4, forinstane).
Denition 3.1 (1-summability) A series u(t;b z) 2 O(D
)[[t℄℄ is 1-summable in
the diretion argt = if there exist a setor
;>
, a radius 0 < r and a
funtionu(t;z) alled 1-sumof bu(t;z) inthediretion suh that
1. u isdened and holomorphi on
;>
D
r
;
2. For any z 2 D
r
the map t 7! u(t;z) has u(t;b z) = X
j0 t
j
j!
b u
j;
(z) as Taylor
series at0 on
;>
;
3. For any proper 2
subsetor
;>
there exist onstants C > 0;K > 0
suh that for all `0, all t2 andz2D
r
`
t u(t;z)
CK
`
(1+2`)
:
We denoteO(D
)fftgg
1;
thesubsetofO(D
)[[t℄℄made ofall 1-summableseriesin
thediretionargt=. Atually,O(D
)fftgg
1;
is inludedinO(D
)[[t℄℄
1 .
For any xed z 2 D
r
, the 1-summabilty of the series u(t;b z) is the lassial
1-summabilityand Watson Lemma impliestheuniityof its1-sum, ifany.
Proposition 3.2 O(D
)fftgg
1;
;
t
;
z
is a dierential C-al gebra stable under
1
t
and 1
z .
Proof. Letbu(t;z)andbv(t;z)betwo1-summableseriesindiretion. InDenition
3.1weanhoosethesameonstantsr;C ;Kbothforubandv.b Theprodutw(t;z)=
2
Inthis ontexta subsetorof asetor 0
is saidapropersubsetor andonedenotes 0
if
itslosureinC isontainedin 0
[f0g.
u(t;z)v(t;z) satisesonditions1 and 2 ofDenition3.1. Moreover,
`
t w(t;z)
=
`
X
p=0
`
p
p
t
u(t;z)
` p
t
v(t;z)
C
2
K
`
(1+2`)
`
X
p=0
(1+`)
(1+2`)
(1+2p)
(1+p)
1+2(` p)
1+(` p)
C
2
K
`
(`+1) (1+2`)
C
0
K 0
`
(1+2`) for adequateC 0
;K 0
>0:
Thisprovesondition3ofDenition3.1forw(t;z),thatis,stabilityofO(D
)fftgg
1;
undermultipliation.
Stability under
t ,
1
t
or 1
z
is straightforward. Stability under
z
is obtained
usingtheIntegral Cauhy Formula on adisD
r 0
withr 0
<r. 2
Wemaynotiethatthe1-sumu(t;z)ofa1-summableseriesu(t;b z) 2O(D
)fftgg
1;
may be analytiwith respet to z on a disD
r
smaller than theommon dis D
of analytiity of the oeÆients ub
j;
(z) of u(t;b z) = X
j0 t
j
j!
b u
j;
(z). With respet to
t, the1-sumu(t;z) isanalytiona setorsupposedlyopenand ontaininga losed
setor
;
biseted bywithopening;thereisnoontrolon theangularopening
exept that it must be larger than and no ontrol on the radius of this setor
exeptthat itmustbe positive. Thus,the 1-sumu(t;z) iswelldened asasetion
of the sheaf of analyti funtions in (t;z) on a germ of losed setor of opening
(i.e., alosed intervalI
;
of length on theirle S 1
ofdiretions issuingfrom 0,
f. [MalR92 ℄ 1.1 or[L-R94 ℄ I.2)times f0g C
z
. We denote O
I
; f0g
thespae of
suh setions.
Corollary 3.3 The operator of 1-summation
S: (
O(D
)fftgg
1;
! O
I
; f0g
b
u(t;z) 7! u(t;z)
is a homomorphism of dierential C-al gebras for the derivations
t and
z
and it
ommutes with 1
t
and 1
z .
Theorem 3.4
Let a diretion argt= issuing from 0 and a series b
f(t;z)2O(D
)[[t℄℄ begiven.
ReallD=1 a(z) 1
t
2
z
andassumethateithera(0)6=0ora(0)=0anda 0
(0)6=0.
Then, the unique solution bu(t;z) of Dbu= b
f in O(D
)[[t℄℄ is 1-summable in the
diretion if and only if bu
;0 (t);ub
;1
(t) and b
f(t;z) are1-summable in the diretion
.
Moreover, the 1-sum u(t;z), if any, satises equation (4) in whih b
f(t;z) is
replaed by the 1-sum f(t;z) of b
f(t;z) in diretion .
Proof. Werst plae ourselvesinthe asea(0)6=0.
Denote a(z)= X
n0 a
n z
n
.
As a preliminary remark we notie that, by identiation of equal powers of z in
Equation
(4) 1 a(z)
1
t
2
z
X
n0 b u
;n (t)
z n
n!
= X
n0 b
f
;n (t)
z n
n!
;
we get
8
>
<
>
: b u
;0 (t) a
0
1
t b u
;2 (t)=
b
f
;0 (t)
b u
;1 (t) a
1
1
t b u
;2 (t) a
0
1
t b u
;3 (t)=
b
f
;1 (t)
and soon:::
so that eah ub
;n
(t) is uniquely and linearly determined from ub
;0 (t); ub
;1
(t) and
b
f(t;z).
TheonditionisneessarybyProposition3.2. Indeed,ifubis1-summablethen
so arebu
;0
(t)=bu(t;0); bu
;1 (t)=
1
z b
u(t;z) bu
;0 (t)
z=0 and
b
f =Du.
Prove thattheonditionis suÆient. Assume thatub
;0 (t); ub
;1
(t) and b
f(t;z)
are 1-summableindiretion.
Set bu(t;z)=ub
;0
(t)+zub
;1 (t)+
2
z b
v(t;z) and wb= 1
t b v.
With these notations Equation(4) beomes
(6)
1 1
a(z)
t
2
z
b
w(t;z)=bg(t;z) where bg= 1
a(z) (bu
;0 +zbu
;1 b
f)
and itsuÆesto prove thatwbis1-summableindiretion whenbg is. Tothis
end, we proeedthrougha xed pointmethod asfollows.
