VOL. 14 NO. 3 (1991) 485-496
PARTIAL DIFFERENTIAL
EQUATIONSWITH
PIECEWlSECONSTANT DELAY
JOSEPH WIENER
Department
ofMathematics TheUniversityofTexas- Pan
AmericanEdinburg,
Texas
78539and LOKENATH DEBNATH
Department
of MathematicsUniversityof Central Florida
Orlando,Florida 32816
(Received
October25,1990 andin revisedformNovember23,1990)
ABSTRACT.
The influence of certain discontinuous delays on the behavior of solutions to partialdifferentialequationsisstudied.
In
Section2,the initial valueproblems(IVP)
arediscussed for differentialequations with piecewise constantargument
(EPCA)
in partial derivatives.A
class of loaded partialdifferentialequationsthat arise insolvingcertain inverseproblemsisstudied in some detail in Section 3. Section4isdevotedtoobtain the solutionsof
IVP
forlinearpartialdifferentialequationswithpiecewiseconstantdelay by using integraltransforms. Finally,anabstractCauchy problemis discussed.
KEY
WORDS
AND
PIRASES.
Partial Differential Equation, PiecewiseConstant
Delay, Loaded Equation,Initial ValueProblem, Existence, Uniqueness, AbstractCauchy
Problem.1980
AMS
SubjectClassification Codes. 35A05, 35B25, 35L10, 34K25.1.
INTRODUCTION.
Functional differentialequations
(FDE)
withdelay provideamathematical model for aphysicalorbiological systemin whichtherateofchangeof thesystem
depends
uponiis
past history. Thetheory
ofFDE
withcontinuousargumentiswelldevelopedand has numerousapplicationsinnaturalandengineeringsciences. Thispapercontinues ourearlierwork
[1-5]
in anattempttoextend thistheory todifferentialequationswith discontinuousargumentdeviations.
In
thesepapers,
ordinarydifferentialequationswithargumentshavingintervalsofconstancyhave been studied. Such equations representahybridof continuous and discretedynamical systemsand combinepropertiesof both differential and differenceequations. They
includeasparticularcasesloaded andimpulse equations,hence theirimportancein controltheoryand in certain biomedicalmodels.
Continuity
of a solution at apoint joining anytwoconsecutive intervalsimpliesrecursion relationsfor the valuesofthe solution at suchpoints. Therefore,differential equationswith
piecewiseconstantargument
(EPCA)
areintrinsicallyclosertodifference ratherthan differentialequations.operatorsin Hilbertspaces
[7].
Here
initial valueproblems0VP)
are studied forEPCA
inpartialderivatives.A
class of loadedequationsthatarisein solvingcertain inverseproblemsisexplored
within thegeneralframeworkofdifferentialequationswithpiecewiseconstant
delay.
2.
INITIAL VALUE PROBLEMS.
It
hasbeen shown in[6]
thatpartial differential equations(PDE)
with piecewise constant timenaturallyariseintheprocessofapproximating
PDE
by simplerEPCA.
Thus,ifin theequationu,
=aEu=-bu,
(2.1)
which describes heat flow in a rodwithboth diffusion
a2u=
alongthe rod and heat loss(or
gain)across the lateral sides of therod, the lateral heatchange
ismeasuredatdiscretemomentsof time, then wegetanequationwithpiecewiseconstantargument
u,(x,t)
a2u=(x,t)
bu(x,
nh),
(2.2)
where
_
[nh,
(n
+1)h
],
n 0,1 andh >0issomeconstant. Thisequationcan bewritten in theformu,(x,t)
a2u=(x,t)
bu(x,[t/h
]h
),
(2.3)
where
[.]
designates the greatest integer function. Ordinary differential equations with argumentsIt’],
It-hi,
It
+n]
have been investigated in[1-4],
withIt
+1/2]
in[5],
and with[t/h]h
in[8-9].
Furthermore,
EPCA
have been usedrecentlyin[9]
toapproximatesolutionsofequationswith continuousdelay.
The diffusion-convectionequationu,--a2u=-cux
(2.4)
describes,forinstance,the concentration u
(x,
t)
ofapollutantcarriedalongina streammovingwithvelocityc. Theterm
a2u=
isthe diffusion contribution and-cux
isthe convectioncomponent. Ifthe convectionpartismeasuredatdiscrete timesnh,the
process
results in theequationut(x,
a2u=(x,
cu(x, [t/h ]h
).
(2.5)
We
considerthe initialvalueproblem
0VP)
Ou(x,t)+p(O)u
Ot(x,t)
Q
-x
u(x,[t/h ]h
),
(2.6)
u(x,
o)
Uo(X),
where
P
andQ
arepolynomialsof thehighest degreetnwith constantcoefficients, designatesthegreatest integer function, h const>0,and(x,t)G
-(-, ),,
[0, ).
DEFINITION
2.1.A
functionu(x,t)
iscalled a solution ofIVP
(2.6)
if it satisfies the conditions:(i)
u(x,O
is continuous inG; (ii)
Ou/dt ando4u/Ox(k
O,
1m)
existand are continuousinG,
with thepossible exceptionof thepoints (x,
nh),
where one-sided derivatives exist(n
-0,1,2..);
(iii)
u(x,O
satisfiesEq.
(2.6)
inG,
withthepossible exceptionof thepoints(x,nh),
and the initial conditionu(x,O)
Uo(X).
Let
u,(x, t)
bethe solution of thegivenproblem
on the intervalnh-: <(n
+1)h,
thenOu,(x,t)/Ot
+Pu,(x,t)
Qu,(x),
(2.7)
where
Write
whichgivestheequation
andrequirethat
Un(x,t)=wn(x,t)+vn(x),
OWn/Ot
+Pw,
+Pvn(x)
Qua(x),
(2.9)
cgw/Ot+
Pw
0,(2.10)
Pv,(x)--Qu(x).
(2.11)
Let
vn(x)
be a solutionofODE
(2.11),
thenatt=nhwehavew(x,nh
u(x)- v.(x),
(2.12)
and it remainstoconsider
Eq. (2.10)
with initial condition(2.12).
It
iswell known that the solutionE(x,O
of theproblem
Ow/Ot
+Pw
O, w(x,O)--
Wo(X
),
(2.13)
with
Wo(X)
6(x),
where6(x)
isthe Dirac delta functional,iscalleditsfundamental solution. The solution ofIVP
(2.13)
isgiven bythe convolutionw(x,t)
E(x,t)
,
Wo(X).
(2.14)
Hence,
the solutionofproblem(2.10)-(2.12)
canbe written asw.(x,t)
E(x,t
nh), wn(x,n ),
(2.15)
and the solution of
(2.7),
(2.8)
isu(x,t)
E(x,t
nh,
(u.(x) v.(x))
+v(x),
(2.16)
(nh
st<(n
+1)h).
Continuityof the solutionat
t=(n+l)h
impliesu,(x,(n
+1)h)=U,+l(x,(n
+1)h)
u,
+l(x),
that is,
u,
+(x)-
E(x,h
).
(u,(x)- v,(x))
+vn(x).
(2.17)
Formulas
(2.16),
(2.17)
successivelydeterminethesolutionofIVP
(2.6)
oneach interval nh(n
+1)h.
Indeed,fromPv0(x)
Quo(x)
wefindVo(X)
andsubstitutebothuo(x)
andVo(X)
in(2.16)
and(2.17),
toobtainuo(x,t)
andu(x).
Then we useut(x)
in(2.11)
tofindVl(X)
and substituteut(x)
andvt(x)
in(2.16)
and(2.17),
whichyieldsul(x,t)
andu2(x).
Continuingthisprocedureleadstou(x,t),
the solution of(2.6)
on[nh, (n
+1)h
].
The solutionv(x)
of(2.11)
isdefinedtowithin anarbitrary polynomialq(x)
ofdegree<m. Sinceq(x)
isa solutionofEq. (2.13)
withthe initial conditionw(x,O)=q(x),
thenq(x)
E(x,t).
q(x)
andq(x)
cancels in formulas(2.16),
(2.17).
This concludes theproof
of thefollowingassertion.THEOREM
2.1. IfEq. (2.13)with w(x,O)=uo(x)
has auniquesolutiononttE(0,oo),
then there existsauniquesolutionofIVP
(2.6)on
(0,
oo)and
it isgiven by(2.16),
for each intervalnh(n
+1)h.
COROLLARY
2.1. ThereexistuniquesolutionsofEqs. (2.3)
and(2.5),
withu(x, O)
Uo(X),
in the classof functions thatgrowtoinfinityslower thanexp(x
)
asIx I--"
oo.For
Eqs.
(2.3)
and(2.5)
wehaverespectively,and
E (x, t)
exp(-x2/4a2t)/2avr-.
Formula
(2.16)
for thesolutionof theproblemu,(x,t)
a2u=(x,t)
bu=(x,[t/h ]h
),
u(x, O)
Uo(X)
(2.18)
onnh <(n
+1)h
becomesu.(x,t)
-’
E(x,t
nh),
u,(x)
+"-
u.(x),
(2.19)
where
E(x,O
isthe same as inEqs.
(2.3)
and(2.5).
The above methodmayalso be usedtosolve
IVP
forPDE
ofany
order in withpiecewiseconstantdelay orsystems of suchequations.
In
the lattercase,P
andQ
in(2.6)
aresquare matricesof linear differentialoperatorsandu(x,O
is a vectorfunction. Thus,the solutionu.(x,t)
of theproblemuu(x,
a2u=(x,
bu=(x, [t]),
(2.20)
u(x,O)-- fo(X
),
u,(x,O)---
go(X
(2.21)
on n< <n + is sought in the
formu(x,t)
w(x,t)+v.(x)
whencea2v."(x)-bu."(x,n)
0 andO2wn/Ol
2---aEO2wn/Ox
2.
Settingu(x,n)-- f.(x),
u,(x,n)-
g.(x)
givesv(x)-
a-2bf.(x),
w(x,n)--
(1-a-b)
f.(x), wt(x,n)--
g(x),
andu(x,t)--f.(x)+(1 __)f(x-a(t-n))+
+a(t-n))
1
f
g.(s)ds.
(2.22)
+’
a’t
n)Putting n + producesthe recursion relations
_
(
bf.(x
a +f.(x
+ af.
(x
f.
(x
+ 1---a
2
(2.23)
b
af.’(x +a)-af.’(x-a)
1--
2+1/2(g.(x
+a)+
g.(x-a)).
(2.24)
3.
LOADED
EQUATIONS.
Loaded partialdifferentialequationshavepropertiessimilartothoseof equationswithpiecewise
constantdelay. The1VP for thefollowingclassof loadedequations
Ot
"x
(x,t)
+Qy
-x
u(x,
ty),
(3.1)
wasconsidered in
10],
where(x,
t)
R"
x[0, T],
the(0,
T]
aregiven,P(s)
andQl(s)
arepolynomialsins
(s
s,,),
andQl(s)1
0.Eq. (3.1)
arises insolvingcertaininverseproblemsforsystemswith elements concentrated atspecificmoments of time. The Fourier transformU(s,t)
ofu(x,t)
satisfies theequation
whence
U,(s,t)
=P(is)U(s,t)+Q,(is)U(s,ti),
(3.3)
1"1U(s,t)
Uo(s)e
e(‘+
k(P(is),t)
Ql(is)U(s,ti),
(3.4)
j-I
where
Uo(s)
isthe Fourier transform ofUo(X)
andDenote
k(P(is),t)-
i
em")dY"
A
Uo(s)e
’,
k
k(P(is),tl),B
Q(is)U(s,
ti)
1-1thenmultiply by
Qj(is)
eachof theequationsand add them.
Hence,
or
Theequation
U(s,
ti)
=A,
+k/B,
jq
(3.5)
(3.6)
B
,
AQl(is)
+B
,
kiQi(is)
(3.7)
j-t i-I
(1-,.kQ,(is))B
(3.8)
A(s)-
1-Qj(is)k(P(is),t)-O
(3.9)
iscalled the characteristicequationfor
(3.1)
andits solution setZ
iscalled the characteristicvarietyof(3.1).
It
is said[10]
that(3.1)
is absolutely non-degenerate ifZ-fi, non-degenerate of type a if a infIres
I<
,sZ
C
n,
anddegenerateifZC".
The caseZ
O implies
A(s)
const, sinceA(s)
ismeromorphic,andameromorphicfunction that is notconstantassumesevery complexvalue withat mosttwoexceptions. TheequationA(s)
C
canbe written asq q
P(is)
+,
Q(is)-
,
Q(is)e
etch’CP(is)
(3.10)
j-I j.l
and ispossiblefor
q
> only ifP(s)=const, otherwiseexp(P(is)t)
wouldgrow
faster thanany polynomial,whichbreaks the latterequality.
For
q=l,wehaveP(isq
a(s)=(P(is)+Qa(is)-Q(is)e
)/P(is),
(3.11)
and inthiscase
Z
O
isequivalenttoP(is)
+Q(is). O.On
theotherhand,A(s)
0 isequivalenttoP(is)
+Q(is)-
Q(is)e
et")t’.0,(3.12)
which implies
P(s)=const.
This establishes thefollowing propositionwhich was stated in[10]
withoutproof.
LEMMA
3.1.Eq. (3.1)
isabsolutely non-degenerateifonlyif eitherof thefollowingconditions holdstrue:(i)
P(s)
Cx,
Q,(s)k(C,tj)
C2
* 1;j=l
or
(ii)
q-- 1,P(s)+Ql(s)=O.
Eq. (3.1)
isdegenerateif andonlyifP(s)
C1,
,
Qj(s)k(C,tj)
1.Substituting
B
from(3.8)
in(3.4)
leadstotheproof
of thefollowingtheorems which were formulated in[10l.
TItEOREM
3.1. Theuniquenessclasses for thesolutionof theCauchy
problemfor anabsolutely non-degenerate equation(3.1)
arethesame asthosefor theequation(without "loads")
u,(x,t)
Pu(x,t).
THEOREM
3.2. Thehomogeneousdegenerate IVP
(3.1)
(Uo(X)
0)
has non-trivialsolutions,withcompactsupport.
THEOREM
3.3.Suppose
thatEq.
(3.1)
is of finitetypea(0
<a <oo)
and thatu(x,O
is a solution of(3.1)
withUo(X)
O. Iflu(x,t)lCell,x
g’,
tE[0,T],
(3.13)
andct<a, then u
(x,
t)
0.For
anya>athereexistsasolution u(x,
t)
4,0of(3.1)
withuo(x)
0satisfying(3.13).
Theuniquenessclassesfor thesolutionof theCauchy problemfor the equation
u,(x, t)
Pu(x,
t)were
exploredin11]
andconsistof the functions thatgrow
no faster thanexp(a
Ix
[’)
asIx
[--
0%wherea>dependsonthedegreeof
P(s).
Integraltransformations can be used also in thestudyofEPCA.
SOLUTION FORMULAS.
The
purpose
of this sectionis toshow thatintegraltransforms can besuccessfullyusedtofind the solutions ofIVP
for linearpartialdifferentialequationswithpiecewise constfint delay.u(x,O)-uo(X
(4.1)
b
THEOREM
4.1. Thesolutionof theproblemu,(x, t)
a2u=(x,
t) bu=(x, [t]),
isgiven bytheformula
-o\31 +
1-"
E(x,t +j-It])
.Uo(X),
(4.2)
where
E(x,t)
exp(-x2/4aZt)/2avr-
andE(x, O).
Uo(X)
Uo(X).
PROOF. For
n -: <n +1,Eq.
(4.1)
becomesand theFouriertransform
U(to, t)
F(u(x,t))
satisfiestheequationU,(to,
--a2ofiU(o,
+boflU(to,
n),
whence
Att=nwehave
and
At
t=n+l thisgivesand
b
U(to,
t)
Ce
-’2’2’-n)+-U(to,
n).
U(to, n)=C
+-U(to,
n), C
bU(o,n),
b
(4.4)
U(to,
n +1)
(4.5)
(4.6)
e-’2(’-n))U(to,
n).
(4.7)
U(to,
n(4.8)
U(to,
n)-
.
1--
e-’A"U(o,O)
(4.9)
Substituting
the binomialexpansionofU(o,n)
in(4.7)
yields(4.2).
THEOREM
4.2. The solution ofEq.
(2.1)
with the initial conditionu(x,O)
Uo(X)
isgiven bythe formulab b
u(x,t)
Uo(X)
,
(F=(x,t)-"
F(x,t)
+-
F(x,h[t/h
]),
(4.10)
a a
where
F(x,t)=ttl([t/h
bj
[j\
x2j+l
,.ox
j-,.o[J
(-1)’e(x’t-o-a)h}*
(Y
+,)’H{x}"
(4.11)
H(x)
=1,for x>O,
andH(x)=0,
forx<O.
PROOF. For
nh <(n
+1)h,
wehaveut(x,t
a2u=(x,t)
bu(x,
nh),
and the two-sided
Laplace
transformU(s,t) -L(u(x,t))
satisfies theequationUt(s,t)
a2s2U(s,t)-
bU.(s), U.(s)
U(s,
nhwhence
At t=(n
+1)h
thisgivesand
+
U.Cs).
+
U.(s)
(4.12)
(4.13)
Hence
and
e"22
+(1 e")
a-s
Uo(s),
Uo(
sL
Uo(X
).
U.
(s)
Uo(s),
j a
2s
,’Z’.o
k.o(
n b i_.[j(e
O%U(s,t)-
Uo(s),
j
,.ok)
(-1)*
e
+e
whichprovesthe result.
THEOEM
4.3.e
solutionof.
(2.5)
withu(x,
O)
Uo(X)
isgiven bytheformulac c
u(x,t)=
Uo(X)
.
(F=(x,t)-
F(x,t)
+F(x,h[t/h]),
a a
were
F(
0
is definedin(4.11).
THEOEM
4.4.e
solutionofproblem(2.6)
isgiven bytheformula(4.16)
(4.17)
(4.18)
(4.19)
(4.20)
+Q
,(
olox
)Ps
(x
),
(Vs(x,
hIt/h
])
Vs(x,
)),
(4.21)
where
F(x,l-.
(-lE(x,-(]-,
E(x,O
isthe fundamentalsolutionof(2.13),
andPi(x)is
the inverseLaplacetransform ofPi(s).
PROOF.
Thesolutionu(x,t)
of(2.6)
onthe intervalh <(
+1)h
satisfies(2.7)
(2.8),
and foritstwo-sidedLaplacetransform in x we obtain theequationU(s,t)
+P(s)U(s,t)
Q(s)U.(s),
U.(s)
U(, ),.
(4.23)
whence
At
t=nh we haveand
At t-(n+l)h
thisgiveshence,
U(s,t)
Ce
-e(’)(’-")+P-t(s)Q(s)U.(s).
U.
(s)
C
+P-(s)Q
(s)U.
(s)
U(s,t)
(e
-e(’)(’ -’’)+(1
e-P(s)(t-)-lQ
)Un(s).
U.
/(s)--
(e-e(’’
+(1
-e-e<’’)P-Q)U.(s),
U.(s)
(e -es
+(l
_e4"’)h)p-’Q
)"Uo(s
(4.24)
(4.25)
(4.26)
(4.27)
and
Therefore,
U.(s)
Uo(s),
j
.ok]
-o\j k
(-l)e
-P(’)k("
-
+)tt
Q,/Ip-,-
(_l),e-e,),-0-,),
(4.30)
-oj ok
whichleadsto
(4.21).
Linear differentialequationsin Banachspacewitharguments
It]
and-nit]
have been studied in[2].
Consider inaBanachspaceY
theequationu
’(t)
Au(t)
+Bu([t ])
(4.31)
with linear constant
operatorsA:
D(A
Y
andB:D(B
Y,
theirdomainsD(A
CD(B
CY,
andD(A)
iseverywheredense in
Y.
Accordingto[2],
asolutionofEq. (4.31)
on[0, oo)
is a functionu(t)
satisfyingthe conditions:
(i)
u(t)
is continuous on[0, o)
and its values lie inthe domainD(A)
for all[0, oo). (ii)
At
each point[0, oo)
there existsa strongderivativeu’(t),
with thepossible exception of thepoints[t
[0, o)
where one-sided derivativesexist.(iii)
Eq. (4.31)
issatisfied oneach interval[n,
n +1)
C[0,
with integral endpoints. The
Cauchy
problem on[0,o)
istofind asolution of the equationon[0,
satisfyingtheinitialcondition
u(O)
uD(A ).
(4.32)
Thepropertiesof solutionsto
Eq. (4.31)
withboundedoperatorsaresimilartothose of solutions tosystemsofordinarydifferentialequationswhichcan beviewedasequationsin a finite-dimensional Banachspace. Indeed, ifA,B
:Y
Y
arebounded linearoperatorsandA isbijective,thenproblem(4.31)- (4.32)on [0, oo)
hasauniquesolution
[2]
u(t)
V(t
[t])vt’l(1)Uo,
(4.33)
where
V(t)
eAt+(e
At-
I)A-1B.
(4.34)
This solutioncannotgrowtoinfinityfaster thanexponentially. If,inaddition, there exists a bounded inverse of theoperator
V(1),
then thesolutionhas auniquebackward continuationon(-,
0]
given byformula(4.33).
TheCauchy
problemu’(t) =au(t),
u(O)
uD(A
(4.35)
is correctly posed on
[0,o)
if foranyuo
D(A)
ithas a unique solution, and this solutiondepends
continuously on the initial data in the sense that ifu,,(0) O(u,(O)_D(A)),
thenun(t)-
0 for thecorrespondingsolution atevery tU
[0,
o).
If theCauchy
problem(4.35)
iscorrect,itssolution isgiven bytheformula
where
T(t)
isasemigroupofstronglycontinuousoperatorsfor >0.For
many applicationsitisnecessarytoextend the conceptof solution of the
Cauchy
problem.A
weakened solutionofEq. (4.35)
on[0, oo)
is afunctionu(O
which is continuous on[0, ),
strongly continuouslydifferentiable on(0,
oo)
and satisfies theequationthere.By
aweakenedCauchy
problemon[0, oo)
we mean theproblemoffindingaweakened solutionsatisfyingtheinitial conditionu(0)
u0. Heretheelementu may alreadynotliein thedomainof theoperator A. Thus,thedemandsonthebehaviorof thesolution att=Oarerelaxed.
On
the otherhand,werequirethecontinuityof the derivativeof thesolutionfor >0.
However,
for acorrectCauchy
problemthisrequirementisautomaticallysatisfied. Thefollowingresult has beenprovedin
[2].
THEOREM
4.5.Suppose
thatEq. (4.31)
with linear constantoperatorsA
andB
satisfies thehypotheses:
(i)
TheoperatorA
isclosedand hasatleast oneregular point,the domainD(A)
isdense inY.
(ii)
The weakenedCauchy
problemforEq. (4.35)
iscorrecton[0,
(iii)
D(B
DD(A
andeu
ED(A
),
foranyuED(A ).
Thenon
[0,
)
problem(4.31) (4.32)
hasauniquesolutionu(t)-(T(t-
[,])+
[
tT(t,l
-s)Bds)
II(
T(I,+-t,l
f,’-
1T(k-s)Bds)u
o.(4.37)
Considerthe initial valueproblem
#u
A
(O )u(x,t)
+f(t,u(x,[t])),
(4.38)
Ot
u
(x,
o)
Uo(X
),
where
u(x,O
andUo(X)
are m-vectors, x(xi,x2,
x) R
,
A(D)=
X
A,D’
(4.39)
lal
D
’=
D?..
.Dt,
Dk-iOlOx(k-1,2
N),
the coefficientsA,
are given constant matrices of order tn xm, and the m-vector]"
(EC([n,n
+1)x
L2(RC),L:’(RV)),
n 0,1, 2,.... The number r is called the order of thesystem.
It
is assumedthat u0tEL:’(R),
and the solutionssoughtare such that u(x, t)
tEL
2(R^r),
forevery
z0.Let
lal(s),
Ix2(s)
Ix,,(s)
betheeigenvaluesof the matrix
A(s).
Thesystemissaid tobeparabolic byShilov if
A
(D)u
(4.40)
Ot
RelxiCs)
-cIs
+b, j tnwhereh >0,c >0,and b areconstants.
THEOREM
4.6. Problem(4.38)
has auniquesolutiononR
ux[0, oo)
ifsystem(4.40)
isparabolic by Shilov,the indexofparabolicity hcoincides with itsorderr,andfEC([n,n
+1)L2(RU),L2(R)),
n-0,1,2PROOF.
For
afixedtwemay
considerthe solutionu(x,O
asanelement ofL2(R^’),
andf(t,u(x,[t])
du
-Au+
f(t,u([t])),
ul,.o-
Uo
_L2.
(4.41)
dt
Applyingto
(4.40),
withthe initial conditionu(x,
O)
Uo(X),
the Fourier transformationF
inx producesthe system ofordinarydifferentialequationsUt(o,) t)
A (co)U(to, ),
(4.42)
withtheinitialcondition
U(to, 0)
U0(to),
whereU(to,
t)
F(u (x,
t)), U0(to)
F(uo(X)),
andA(to)
is a matrix with polynomialentriesdependingonto(to1,
to2,-.., to,v).
The solution of(4.42)
isgiven bythe formulaU(to,
ett’)Uo(to
).
(4.43)
Parabolicityof
(4.40)
byShilovimpliesthat thesemigroupT(t)
of operators ofmultiplication bye’’tt’’),
for >0,is aninfinitelysmoothsemigroupofoperatorsbounded inL:’(R’).
Together
with therequirement h=r, this ensures that theCauchy
problemfor(4.40)
isuniformlycorrectinL
’(RN)
andall its solutions areinfinitelysmooth functionsof t, for >0. Since
f
iscontinuously differentiable, problem(4.41)
has on[0,
1)
auniquesolution.()
T(t).
+T(t
s)f(s,
uo)ds.
(4.44)
Denotingul
u(1),
wecanfindthe solution.(t)- r(t-
1).
/f
r(-
s)/(s,u)as
(4.45)
of
(4.41)
on[1, 2)
and continue thisprocedure successively.If./’(t,
u([]))
-Bu([t]),
whereB
is aconstantmatrix,the solutionof
(4.38)
for[0,)
isgivenby(4.37).
The theorem holdstrueiffincludes
also the derivativesofu(x,[t])
inx of orderless than r,providedthe initial functionu(x)
issufficientlysmooth.ACKNOWLEDGMENT.
This research was partially supported byU.S.
Army
Grant
DAAL03-89-G-0107,andbyTheUniversityof Central Florida.REFERENCES
1.
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K. L.
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MathSci. 6(4), (1983),
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