I ntzrnat. J. Math. Math. S ci.
Vol. 3 No.
(1980) 103-111
103
ON THE INTITIAL VALUE PROBLEM FOR A PARTIAL
DIFFERENTIAL EQUATION WITH OPERATOR COEFFICIENTS
MAHMOUD M. EL-BORAI
Department of Mathematics
King Abdulaziz University
P.O. Box 1540 Jeddah
Saudi Arabia
(Received May 15, 1979)
ABSTRACT.
In the present work it is studied the initial value problem for an
equation of the form
k
3k u__
t
k
j=l
k-j
Lj
u
tk-J
where L is an elliptic partial differential operator and
(Lj
j
i,
k)
is a family of partial differential operators with bounded operator co-
efficients in a suitable function space.
solution.
It is found a suitable formula for
The correct formulation of the Cauchy problem for this equation is
also studied.
KEV WORDS AND PHRASES.
Pa2ial Diff6_ntial EquaZions, Ellipt Opur,
Catchy ProSlz and Gznzral Solu2ions.
104
M.M. EL=BORAI
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES:
i.
35J15.
INTRODUCTION.
Consider the equation
lql
where q
Dt u
q
2m
qn
(ql
m
Aq,j
q
(t)D D
(1.1)
u,
is an n-tuple of nonnegative integers, and
Dq=
"8xqnn
ql
8Xl
in which
q[
ql+.. .+ qn
Dt
and m, k are positive integers.
t
It is assumed in equation (1.1) that the following conditions are satisfied;
(a) The coefficients
(aq(t),
lql
(b) For each t e [0,i],
2m) are continuous functions of t in [0,I].
a
2m
(c) The coefficients
(Aq,j(t),
q
(t) D q
lql
2m
are linear bounded operators from L
2
is an elliptic operator.
j =l,...,k) for each t
(En) into itself, where
E [0,I]
L2(En)
is the
set of all square integrable functions on the n-dimenslonal Euclidean space E
n
(d) The operators
(Aq,j(t), lql
2m, j
k), are strongly continuous in
i
t[0,1].
In section 2, we shall find a solution
suitable function space so that t
(0,i),
u(x,t)
x En
of equation (i.i) in a
and the solution
u(x,.t)
satisfies
the following initial conditions
DtJ
u(x’t)
It=0
fj(x)’
j
0,1,2,...,k-l.
The uniqueness of the solution of the problem (i.I), (1.2) is also proved.
(1.2)
Under
suitable conditions ([ 3], [4]) we establish the correct formulation of the Cauchy
problem (i.i) and (1.2).
INITIAL VALUE PROBLEM FOR PARTIAL DIFFERENTIAL EQUATION
105
A GENERAL FORMULA FOR THE SOLUTION
2.
Let
Wm(En >
be the space of all functions f
Dqf
butlonal derivatives
ql
with
m all belong
L2(En) such that
to L 2(En) [S].
the dlstrl-
We shall say that u is a solution of equation (l.1) in the space
if for every t
members of
(0,i) the derivatives
m(En)
u
0,1,2
j
w2m(En ),
k exist and are
and if u satisfies equation (l.l).
We are now able to prove the following theorem.
THEOREM i.
w2m(En), j
If f.
0,1,..., k-I and if 4m > n, then there
exists a unique solution u of the initial value problem (I.i), (1.2) in the
space
m(En).
As in [6] the differential operators (D
PROOF.
q,
lql
2m) can be trans-
formed into
q
D
=DI
m(En)
2m
q
R V
f, f 6
f
n’
3
by
(n+l)
R f
(2.1)
2
i H
---)
y
x.3
.n+l.
F
f(y) dy
Ix_yl +
j
E
n
is the gamma function, i
2
[xl
and
2
x +...+
2
x
n
(see [I]).
Using (2.1) we see that equation (i.i) is formally equivalent to,
a
lql
2m
k
(t) Rq V 2m Dk u
q
t
A
j=l
lqI=2m
q,J
(t)
Rqv2mDtk-j
(2.2)
u,
Using the notations
V
2m
u=v
V
2m
fj =gj,
]q--2(t)m lq----m Aq
a (t)
q
Rq
Ho
We obtain from (2.2) in a formal way the equation
j,
(t) Rq
H
(t),j
M.M. EL-BORAI
106
k
Ho(t)
---ql 2m aq(t)
Since the operator
H
t
o
q
D
(2.3)
is elliptic, it follows that the operator
-i
into itself
(t) has a unique bounded inverse H o (t) from L 2 (E)
n
-i
[0,i]. Applying H (t) to both sides of (2.3) we get
O
D
proved that
[0,I].
H.3(t),
k
e
j
H
v
t
j=l
R.
Since the operators
t
Dktv j=l Hi(t)Dkt-Jv.
j
-I
o
(t)
Hi(t)
D
k-j
t
for each
(2 4)
v.
i,..., n are bounded in
l,...,k are bounded operators
L2(En), it can be easily
inoL2(En) for each
It is convenient to introduce the following notations in order
to complete the proof by considering the problem in a Banach space to be
defined below.
Let A(t) denote the square matrix,
H2(t)
Hi(t)
A(t)
where H.(t)
3
-i
(t)
o
H
_I (t)
I
0
0
0
0
I
0
0
0
0
I
0
H.(t),
j
1,2,..., k and I denotes the identity
operator.
Equation
(2[4)
can be written in the form
d V(t)
dt
where V is the column matrix
A(t)
V(t),
INITIAL VALUE PROBLEM FOR PARTIAL DIFFERENTIAL EQUATION
107
Vll
V
and v.
D
tk-J
v2m u.
The column vector V satisfies formally the initial condition
gk-1
gk-2
V(0)
(2.6)
G
go
Let B denote the space of column vectors V, with the norm
k
where
lfl 2L2 (En)
f2 (x)
E
dx
n
It is clear that B is a Banach space and A(t) is a linear bounded operator
from B into itself for each t
on the coefficients
[0,I].
aq(t) Aq,j (t)
According to the conditions imposed
it can be seen that
A(t) is strictly
continuous on [ 0,1 ].
Since
gj
L2(En),
j
0,i
G is an element of the space B.
k-l, we find that the column vector
The abstract Cauchy problem (2.5), (2.6)
can be solved by applying the above argument [7].
exists for each t
In other words, there
(0,i) a unique operator Q(t) bounded in the Banach space
B such that the formula
V (t)
Q(t) G,
(2.7)
108
M.M. EL-BORAI
(2.5), (2.6)
represents the unique solution of the problem
in the space B.
The
operator Q(t) can be represented in the matrix form
1
Q2.k(t)L
Qlk(t)
QI2 (t)
Q21 (t)
Q()
Q2..2
(t)
Qk (t)
where
L
(Qrs(t)
2(En)
l,...,k, s
r
(2.8)
1,..., k) are bounded operatros in the space
for each t in [0,I].
Using (2.7) and (2.8) one gets
v (x,t)
r
v2m u(x, t)
D k-r
t
k
s=l
k
Qrs (t)
gk-s
s=l
Qrs (t)
v2m fk-s
(2.9)
From (2.9) we get immediately
u(x,t)
(V2 -I is
inverse of V 2m.
where
k
(v2m) -I
v2m fk-s’
Qks (t)
a closed operator, defined on L
2(En)
(2.10)
and representing the
We prove now that the formula (2.10) which we have obtained in a formal way
is in fact the required solution of the problem
(i.i), (1.2) in the space
2m(En).
W
Since
(v2m) -I
is a closed operator from L
2
(g
n)
onto W
2m
(En)
it ollows
,,,diately from (2.10) that
u
for each t
W
2m
(En),
[0,13.
Now the differential operator
derivative with respect to
t
d
in equation
in the space
(2.5) denotes the abstract
L2(En)
i.e. if
ft L2(En)
for
INITIAL VALUE PROBLEM FOR PARTIAL DIFFERENTIAL EQUATION
d
each t .(0,i), then
d
dt
af
lim
Since
t
d
*
ft
where
*
t
ft
A--{-
and Af
is defined by
ft
ft
it follows from
(v2m)-i dd- f t’
f
L2(En)
t
(2.9) and (2.10) that
dk-r
dt
0,
L2 (En)
ft+t- ft"
(v2m)-i ft
109
k-r
u
(V
2m) -i
dk-r Qks (t) v2m
k-r
fk-s
dt
k
s=
k
(v2m)-i
qrs (t)
s=l
V
2m
fk-s.
(2.)
The last formula proves that
dk-r
dt
r
1,2,
k
and
t
k-r
m (En)
u
(0,I).
Using (2.4) and (2.11) one gets,
for each t
.
dk___u
dt
k
(O,1).
du
In [5] we have proved that if u,
ql
m (En)
2m, 4m > n, then the partial
d Dqu
w2m (En) and dL2 (En)
derivative
Dtu exists in the usual
and that it is identical to the corresponding abstract derivative.
sense
Since
these conditions are satisfied by u in (2.10), therefore the same conclusion
applies.
u,
In a similar manner we can deduce also that the partial derivatives
1,2,...
k exist in the usual sense for each
[0 I]
that they are identical to the corresponding abstract derivatives.
x
E
and
ii0
M.M. EL-BORAI
Since
Q(0) G-- G
we have
_
I
Qrs
(0)--
therefore
V
t0
u(x, 0)
u(x,0)
f (x).
o
fj(x)
If the coefficients
r
fk-s
V
2m
f (x)
o
In a similar manner we can prove
k,
0,i,
j
(Compare [2]).
which complete the proof.
D
V 2m
Qks (0)
u(x,0)
that
commute with
,r#s,
k
2m
The last formula leads to
THEOR2M 2.
r--s
lql
(Aq,j(t)
1,2,... ,k)
2m, j
1,2,..., n, then the solution of the problem (i.i), and
r
(I. 2) is given by the formula
k
u
PROOF.
Since the operators
r
2m
W
For any f
(Aq,j
(En),
(t)
(t)
Qks
fk-s
q
we have R
]ql
2m, j
V
commute with
lql
V 2m
2m
f
1,...
1,2,..., n, it follows that the operators
(Rq,
(2.12)
(Aq,j
V
2m
Rqf
(2.13)
k) commute with Dr
(t) ql
2m, j
i,..., k)
2m) and according to (2.2) and (2.13) we can write
kt
k
[Ho (t) D u-
H(t)
j--1
j
O
k-j u]
t
0
The last equation leads immediately to
t
u
o
j=l
j(t)
u.
Applying similar steps to theorem (i), we obtain the required result.
COROLLARY.
If the operators
(Aq,j
(t),
lq]
2m, j=l,..,k) commute with
INITIAL VALUE PROBLEM FOR PARTIAL DIFFERENTIAL EQUATION
Dr, r
iii
1,2,..., n, then the Cauchy problem (i.i), (1.2) is correctly
formulated.
PROOF.
The proof of this important fact can be deduced immediately by using
formula (2.12), (compare [3], [43).
REFERENCES
[1]
Calderon A.P., Zygmund A. Singular Integral Operators and Differential
Equations 11, _Ame. r, J. Math., 79, No. 3 (1957).
[2]
Ehrenpreis L., Cauchy’s Problem for Linear Differential Equations With
Constant Coefficients ii, Proc. Nat. Acad. Sci. USA .42, No. 9
(1956), 462-646.
_
auchy
[33
Problem and the
Ei-Borai M.M. On the Correct Formulation of the
Mixed Problem for a Self-Conjugate Equation, ,Vectnik .of Moscow Univ.,
[43
Ei-Borai M.M. On the Correct Formulation of the Cauchy Problem,
1968.
Vectnik of Moscgw nivesitY,
[53
Ei-Borai M.M. on the Initial Value Problem for Linear Partial Differential
Equation With Variable Coefficients, Journal of Natural Science and
Mathemati.cs, Vol. XI.V, No. i (1976).
[63
Ei-Borai M.M. Singular Integral Operators and Cauchy’s Problem for Some
Partial Differential Equations With Operator Coefficients,
Transaction of the Science Centre, Alexandria University Vol. 2, 1976.
[73
Krein, C.G.
[8]
Yosida K.
Lnear Differen_tial Equations in Banach Space, Moscow (1967).
Functional Analysis, Spreinger-Verlag, (1974).