TANABE, HIROKI

Osaka City University Repository

Tanabe, H.
Osaka J. Math.
2 (1965), 191-204
NOTE ON UNIQUENESS OF SOLUTIONS OF
DIFFERENTIAL INEQUALITIES OF PARABOLIC TYPE
HIROKI TANABE
(Received March 16, 1965)
Recently A. Friedman [3] proved some result on the uniqueness
of solutions of ordinary differential inequalities
l_du
i dt
-Λ(t)u \^v\\A(t)u\\+K\\u\\ (0.1)
in a Hubert space, and as its application he showed that a certain
uniqueness theorem holds for differential inequalities of parabolic type
(0.2)
in the class of functions satisfying some type of time-independent
boundary conditions
Bj(xy Dx)u(xy t) = 0, x^dΩy j = 1, •••, m , (0. 3)
where 9Ω is the boundary of a bounded domain Ω and || \\k stands for
the usual norm of Hk(Ω) = Wϊ(Ω). In [3] A(t) is assumed to have a
constant domain, and on this account it was required that the boundary
conditions (0.3) did not depend on t. In this paper it will not be
attempted to extend the above result concerning the inequalitites (0.1)
in a Hubert space to the case in which A{t) has a variable domain;
however, it will be shown that a similar result remains valid for more
general differential inequalities of parabolic type with time-dependent
boundary conditions
\\A{xy t, Όχy A ) « l l o ^ Σ ll0ϊ-*«ll*f+#Σ1 \\Ό\-k~M\kd, (0.4)
Bj(x, ty DχyDt)u(xy 0 = 0, x^dCLy j = 1, •••, m . (0. 5)
Here A(x, t> D
χy Dt)= Σ A^k(x9 t, Dx)Dkt is a linear d( = 2m//)-parabolic
differential operator in the sense of I. G. Petrowskii, or, what amounts
to the same thing, for each t and θ with - 7 r ^ # ^ 0 A{x} ty Dχy
192 H. TANABE
is an elliptic operator of order 2m in (xy jy)eΩx (— oo, oo). Our main
result is briefly described as follows : if for each / and θ with — π^θ^O
the Complementing Condition ([2]) is satisfied by (A(x, ty Dχy eiθD$)>
{Bj(xy ty Dχy eiθD%)}) in Ox(—oo, oo) and if v is sufficiently small, then
any solution of (0.4)-(0.5) satisfying the initial conditions Dktu(fi) = 0y
k = 0y 1, •••, /— 1, identically vanishes for />0. The restriction on the
smoothess in t of the coefficients is about the same as that of [3] and
any Holder contiunity of the coefficients of A will not be assumed. In
the proof of the statement above essential use is made of an estimate
which holds for solutions of some reduced weighted elliptic inhomo-
geneous boundary value problem (Lemma 2.1 below) and which can be
proved in the same manner as Theorem 5.2 of [1]. As in [3] we
shall use Fourier transforms with respect to t in order to obtain the
necessary estimate, and in the present case we are obliged to treat such
transforms of a class of functions which do not satisfy the homogeneous
boundary conditions since the coefficients of the boundary operators are
allowed to be dependent on t. Some results similar to Theorems 2 and
3 of [3] on the decreasing and uniqueness properties of solutions of
(0. 4)-(0. 5) in an infinite domain Ωx(-oo, T~\ will also be obtained.
1. Assumptions. We denote by Ω a domain in the w-dimensional
Euclidean space and by 3Ω its boundary. For any multi-index a = (a19
. . .,#Jwe write/)£=(——Y^ YA JLY*11 and similarly Dt = — — . | α |V i dxj \ i dxj i dt
stands for the length of a: \a\=a1-\ \-an. For any integer k we
denote by Hk{Ω) the class of all complex valued functions whose dis-
tribution derivatives of order up to k are square integrable in Ω, the
norm of Hk(Ω) being denoted by || | |Λ f Q:
ll«llί.Q= Σ ( \D;u(x)\2dx.
Let / and m be natural numbers and let d=2m/L A{xy ty Dχy Dt) is a
linear differential operator of the form
A(xy ty Dχy Dt) = g A,_k(xy t9 Dx)Dkt
where for Otίk<l
A (y / n ^ " ^ n (<v +\TΛΛ
I\l-k\Λ"> *"> U
χ
) 2L-I M/-έ^Λ, l)U
χ
\Λ\^2m—kd
is a differential operator in x of order 2m —kd at the most with coef-
ficients defined in Ox(-oo, T ] and AQ=1,
DIFFERENTIAL INEQUALITIES OF PARABOLIC TYPE 193
Let mjyj = ly •••, my be non-negative integers <2m and let ίj =
\^mj/d2 = the integral part of ntj/d. For j = l, * ,m, Bj(xy ty Dχy Dt) is a
linear differential operator of the form
Bj(x, t, D
χy Dt) = Σ Bj /._*(*, t, DX)D)
where for k = 0, ~,lj
Bj,
 h(x9 t, Dx) = Σ */ /,-* β(*> OW
is a differential operator in x of order nij—kd at the most with coef-
ficients defined on 9Ωx(-oo, T] . In what follows we suppose without
restriction that the coefficients of Bj(x, t, D
x
, Dt) are defined in the whole
of Ox(-oo, T] .
Let y be an auxiliary real variable and by Γ we denote the infinite
cylinder: Γ = Ωx{.y: — oo<jy<oo}. Then for each t and θ A(x, t, D
x
,
e
iθD%) is a linear differential operator in (x9 y)^T of order 2m. Similarly
Bj(xyty Dx, eiθDy) is a linear differential operator in ( I J ) G Γ of order
m5 at the most.
We introduce the following assumptions.
(I) A(xy ty Dχy Dt) is a d-parabolic operator in Πx(— cx>, T], i.e.
for each fixed ίe(-cx), T ] and θ with -7r^6^O A(xy ty Dχy eiθDay) is
an elliptic operator in (xy jy)eΓ.
(II) For .7 = 1,—,w, Bj(xy ty Dχy eiθD*) is a differential operator in
(xyy)(=Γ of order my for each /e(— oo, T ] and (9 with — τr^(9^0.
(III) The Complementing Condition ([2]) is satisfied by (A(x, ty Dχy
e
iθD% {Bj(xy ty Dχy eiθDay)}y Γ) uniformly in t EΞ(- OO, T ] and θ with
(IV) The coefficients of A(xy ty Dχy Dt) are uniformly continuous in
Ωx(-oo, T] . As regards the coefficients of {Bj(xy ty Dχy Dt)}
Dκ
x
bjJrkβ{xyt)y \κ\^2m-mΛ
D\bJtirkf(x, t)y i = 0, ~ \
are uniformly continuous in Ox(—oo, T] .
(V) Ω is a bounded domain of class C2m.
From (I) it follows that d is an even integer. In what follows we
shall denote by C
o
, C
ίy ••• constants depending only on the assumptions
above and by M those dependent only on / and m. (IV) implies the
existence of a continuous function S(r) satisfying £(r)>0 for r>0y €(0)
==0 such that
194 H. TANABE
(#, f)-aι_kcb(xy s)\ £β(\t-
kdy k = Oy — , / - l
y k=O,-~,lj, \κ\^2m — mjy j = ly ~ymy
By a solution w of the boundary value problem of the form
A(x, ty A*> AM*, 0 =
βy(ΛΓ, f, A<> />,)«(*, 0 =
we shall always mean one such that Dktu(t) belongs to H2m_kd(fϊ) and
is strongly continuous in t there for k = 0, •••,/.
2. Preliminary lemma. We begin this section with the following
lemma.
Lemma 2.1. Suppose that the differential operators A(x, D
x
> Dt\
{Bj(xy Dχy A)} with time-independent coefficients satisfy all the assump-
tions of section 1. // the function u(x, λ) of λ with values belonging to
H2m(Ω) satisfies
A(x> D
χy \)u(x,\) = f(xy λ), Λ G Ω , I m λ ^ O ,
Bj(x, D
xi \)u(x,\) = gj(x, λ), x<=dΩ, I m λ ^ O , j = l> ~ ,m .
where f and gjy j = l, --,my are functions of λ with values in L2(Ω) and
H2m-m.(£ϊ) respectively, then there exists a constant N>0 independent of u
such that for λ with I m λ ^ O , | λ | ^
Σ
(2.1)
Σl
j—1
Proof. (2.1) is a consequence of Agmon-Douglis-Nirenberg inquality
applied to (A(xy Dχy eiQΌ% {Bj(x, Dχy eiθD$)}y Γ), -π^θ^Oy and the func-
tion φ{y)eiμyu(xy μdeiθ) where φ is a smooth function such that φ=l for
| jvl^l/2 and φ = 0 for | ^ | ^ 1 and μ is an arbitrary real number (cf.
Theorem 5. 2 of [1]).
Let u be a solution of
= f{xyt)y x^ny -oo<t<Ty (2. 2)
, ί, D
x
, A M ^ 0 = 0, * e 8 Ω , -oo<t<Ty (2. 3)
DIFFERENTIAL INEQUALITIES OF PARABOLIC TYPE 195
Given δ > 0 let ζ{t) he a smooth function satisfying ζ(t) = O if t^O or
ί^5δ, ζ(t) = l if δ ^ ί ^ 4 δ , and \Dktζ(t)\ ^K08k for * = 0 , ,/ + l and all
t. Taking c, δ such that c + 5δ<T, we write ?
ίr
(0 = δ'('-^) a n d υ(χ> 0 =
£<.(/)»(#, ί) Then # is a solution of
A(x, t, D
χy Dt)v(xy t) = F(xy /), I G Ω , - o o < / < o o , (2.4)
β,.(#, /, D
x
, DXΛΓ, /) = Gj(xy t)9 x e 9Ω, - oo < / < oo, (2.5)
.7 = 1, —,wι,
where
F = ζ
c
f+ Σ A,-άx> t, D
x
) WkΛ ζlk-»DΊu , (2. 6)
* i o\P I
Σ
, , t, D
x
)  ( J ) K*-«I>!« (2 7)
For a function A of (#, /) we denote by h(x, λ) its Fourier transform
with respect to /:
h(χy λ) = -f= \ e~iλth(xy t)dt.
Lemma 2. 2. // δ is sufficiently small and then if — σ is sufficiently
large depending on the choice of δ, we have
Σ Γ
* = 0 J-oo
e ' | |F(/)| |SΛ+ Σ (" (Iλ + iσl^-y^HόyCλ + iσ)! ! ,) 1 ^ (2.8)
o ί = l J - o o
<rt\\GJ(t)\\im-mjdt}.
o
 J
 )
Proof. Clearly υ satisfies
A(x, cy Dχy Dt)v{xy t) = H(xy ί), ΛreΩ, - o o < f < o o , (2.9)
Bj(xf cy Dx, Dt)υ{xy t) = Φj(xy t)y x^dCί, - oo < t < oo , (2.10)
where
± άx, cy Dx)-A^k(xy ty Dx))Dktv ,
Σ (β,Jrάx, cy Dx)-BjJrk{xy ty Dx))D\v .
Taking the Fourier transforms of both sides of (2. 9), (2.10) it follows
that for any complex number λ
A(x> cy Dχy \)ύ(χy λ) = ίϊ{xy λ),
196 H. TANABE
BJ(X, cy Dx, X)v(xy λ ) = Φj(x, λ ) , * < = 9 Ω , j = l, •••, m .
If λ, σ are real and σ^—Ny then by Lemma 2.1
(2. 11)
Squaring and integrating both sides of (2.11) and using Plancherel
theorem, we get
Γ
=O J-oo
^cK e2ff'||/ί(OII^/+ Σ Γ (|λ+*V|c*"-"y3/<'||Φy(λ + iσ)||0)1rfλ (2.12)
+ Σ Γ ^ΊlΦ/OII!—,*}
j = l J _oo
From the following easily derived inequalities
Σ
it follows that
Γ
(2.13)
+C4ε(5δ)2Σ Γ f'WD
A = θ J - o o
'y
 f
+ C5£(5δ)2 Σ
k = 0 J -
Let ξ(t) be a smooth real valued function such that £(0 = 0 if t< — δ or
/>6δ, ?(0 = l i f 0</<5δ, and
. (2.15)
If we write ξ
c
(t) = ξ(t — c)9 then ξc=l on the support of v, hence
Φy = G y + Σ Σ ΎJtrkβ(x,c,f)DξD)Ό (2.16)
where
TV.//-*,β(*> c ' 0 = fc(
DIFFERENTIAL INEQUALITIES OF PARABOLIC TYPE 197
As is easily seen
Γ I ij i
Γ
k β(*> c> λ) I d\ ^C 6 δ , (2.17)
| λ |<*"-mpt*Iή j l r k β(xy cy λ)IdX^Cfi1-**"-"^ . (2.18)
In view of (2.17), (2.18) as well as the following lemma :
Lemma 2. 3. // / and g are entire functions of exponential type,
decreasing sufficiently rapidly on the real line, then for any real number
σ and 7>0
!
oo / Λ oo \ 1/2
JλH/(λ)|rfλ(J Jg(X + iσ)\>d\)
we get first integrating in λ and then in x
Γ (Iλ + iσI*—^|\φ.(X + iσ)\\oγdx
J — oo
(" (Iλ + fσl1*—'y^llO/X + iV)!!,)'^ (2.19)
oo
7δ
2
Σ Γ
*-0j-o
= 1+11+111.
It is clear that // is domiated by C8δ
2
 x the left side of (2.12). Substitut-
ing
+ φ°m-mJM\ \ύ(\ + ia)\ \2m + h~mi'd\ \ϋ{\ + ίσ)||β
in /// we get
III ^ δ2 Σ Γ (I λ + iσ I c—-V/'+*| |ί(λ + iσ)| \
mj-kafdX
k=0 J-oo J
. (2.20)
198 H. TANABE
Thus we obtain
Γ (Iλ + iσI^-mJ^\\Φj{X + iσ)\|0)2dX
J — oo
^2J°°^(I λ +*σ 1 1 - — y ^ | | ^ (
λ
 + ia)\\
o
γdX
2m Λ°°
+ C13δ
2
 Σ (I λ + *σ I i2m-kvd\\ύ(\ + iσ)\\kYd\
k=0 J -oo
It follows from (2.12), (2.13), (2.14) and (2.21) that if σ^-N
Σ (I λ + fσ I «-«Ί\ύ(X + icr)\\kγd\
i=0 J-oo
^C15{(°° (?°t\\F{t)\\tdt+ Σ (" (Iλ + i σ l ^ — - y ^ l l ό / λ + i^ l l^r fλ
tJ-oo j — 1 J - o o
+ Σ J"/"l\Gtf)\\'u.-mjdt} (2.22)
+ C16(82 +£(5δ)2) Σ (I λ + iσ I <2m-*3/r f | |tf(λ + iσ)| I*)1 rfλ
Jfe-0 J-CO
+ C(δ)Γ
J — o
Thus if δ is so small that C16(δ2 + £(5δ)2)^l/2 and if -σ^max (N,
(4C(δ))1/2/), then we get (2.8) from (2.2) choosing CX-4C15.
3. Main Theorem. Let u again be a solution of (2.2)-(2. 3) and
let us continue to use the notations of section 2. Suppose that for some
constants v, K>0
^ 7 \ (3.1)
Σ
With the aid of Leibniz formula we get
f c Σ I I £ Γ ^ | | ^ Σ I | Z ) r ^ ^ (3.2)
as well as the one with I—I in place of /, where ζ
c
 and v are functions
defined in section 2. From the definition (2.6) of F it immediately
follows that
II^Ho^ell/llo + C ^ Σ Σ mk-p'D»tu\\2m_kd
hence with the aid of (3.1) and (3. 2) we get
DIFFERENTIAL INEQUALITIES OF PARABOLIC TYPE 199
(3.3)
+cl8± 'i:\\ζ?-»Diu\\lm-u.
k=l p=0
Let γ be a real number and write w = eytu. We shall use the notations
Σ
With the aid of (3.3) and eitD»tu = (Dt + ij)pw we get
r ^^wFwutscΛv'ii^ *ί'+'f»\\D\-*Ό\\\t
(3.4)
Arguing as in the proof of Lemma 2.2 we get
( I λ + ίσ + ιV I «"-^- | |0/λ + iσ + iy)\ |0)Wλ
.ud\ (3.5)
If nij/d is not an integer, we replace the last integrals in the above by
other ones which have only integral powers of |λ | so that we may
make an immediate application of Plancherel theorem. Noting
2m — mj Ί i λ nij — kd ^ij+Vfd—mj
a
\\U)\\mj-kd^C0\\
for k=0, —,lj and any w^H
a
.+1_k}d(Ω), we can easily show that for any
complex number μ and k, w as above
Applying the above formula to (3. 5) and then using Plancherel theorem
we get
200 M. TANABβ
(" (I λ + iσ + ij IC 2 m-mP l dI \OΛ\ + iσ + iy)\ \0)2dx
J-oo
J — oo
J — oo
With the aid of Leibniz formula and by the argument in the deriva-
tion of (3. 4) we find
Γ (I λ + iσ + h I 0»-yV«<| \6j(X + iσ + i7)| |0)2JλJ
- (3.4')
S oo e^σ+lf:)t\\Gj(t)\\lm-mjdt is also
dominated by the right side of (3.47) with C21 possibly replaced by
another constant. Hence if δ is sufficiently small and γ is so large that
-γ^max( iV, (4C(δ))1/2/) (cf. the end of the proof of Lemma 2.2), then
by Lemma 2.2 with σ replaced by σ + y we get for any σ^O
Σ Γ W
(C 2 3+v + KfS'21
If η is so small that 2C22v
2<l, (3.6) implies
Σ l Γ (I λ + iσ + iγ I
*=0 J-oo
2
 Γ Γ e
2cσ+γ)
'||Z>r*"^|lf
rfΛ (3. 7)
*=0 J-oo
i?(C23 + v+K)28 ~2> {e2σcN,(w) + e2^c+ίS>
The left side of (3.7) is not smaller than
Σ Γ
k=0 J - o
^7 2Σ Γ
k = 0 J
Therefore if - γ ^ m a x ( i V , (4C(δ))1/2/, 4C22K2, 1) we get from (3.7)
iσ+iryI ' " * l i ^ ( λ + i σ
DIFFERENTIAL INEQUALITIES OF PARABOLIC TYPE 201
'-•J— (3.8)
))
where C'=4/ί§(C23+5?+iϊ")2/δ2. If we apply Plancherel theorem to the
left side of (3. 8), we find that for any σ^O
-l
Σ
o
Theorem 3.1. Suppose that u satisfy
\\A(x, t, D
x
, Dt
I
k=0
(3.10)
Bj(x, t, D
χy Dt)u{xJ) = 0, xtΞdCl, 0<t^T, (3.11)
// η is sufficiently small and if Dktu(0) = 0 for k=0, - , / - l , then u(t) =
for
Proof. If we set u(x9 /) = 0 for Λ G Ω , /<0, W satisfies (3.10)-(3.13)
for - o o < ί < T . Therefore w = eytu satisfies (3.9) if δ is sufficiently
small and — γ is sufficiently large depending on δ. Choosing c=— δ,
we obtain
*"
1
"'!!?^-^) if σ ^ - o o .
Thus w(0 = 0 if δ</<2δ. Varying δ in a sufficiently small interval
0<δ<δ 0 , we find that u(t) = 0 for 0</<2δ0. We can now proceed step
by step to show that u(t) = 0 for 2 δ
o
^ ^ 4 δ
o
, etc., and the proof is
completed.
Let us again consider a function u satisfying (3.10)-(3.11) for
— 00 <t<T. Then w = eitu satisfies (3.9). Choosing σ in such a way
that the two terms on the right hand side of (3. 9) become equal, we
see that if NX{JJO)^N2{JJO) then
Using a result of [1 pp. 241-233] one derives
202 H. TANABE
Theorem 3.2. // w(=f=O) is a solution of (3.10)-(3.11) in Ωx(-oo,
T ] with v sufficiently small, then for any £>0 there exist positive con-
stants L
o
, L, ω and μ such that
Σ ['^WDl-^uisMjds^LoL-' exp (-ωμ~*)9 - oo </< T - £ .
4. Uniquness at ί = - o o . In this section we again investigate
solutions in Ωx(-oo, T] . Writing Bjlrk in such a form
Bj,
 k(x, t9 Dx)u = Σ
we assume in addition to
( i ) sup I at_k JjKy t) - a^k Jjx, s) \ ->0 as /, ^-^ - oo
X '
for \a\^2m-kd,k=0, —,l;
(ii) sup ID'J>f.tj-tAx' 0 - D'J>t,^KP(x, s)I - 0
I K I ^  max ((/y + l)ί/, 2w - kd)
s u p I Z?ίZ>5ftί
a s /, 5 - > — o o f o r \ β \ j j j
Theorem 4.1. £/wtffer ^ ^ assumptions above if u satisfies (3.10),
(3.11) for — oo<£<T w/Z/z ^ sufficiently small and if
(OILw = 0
u(t) = 0 for -oo<t<T.
Proof. In view of Theorem 3.1 it suffices to prove that u(t) = 0 if
/ is smaller than some negative number 5. Let ζ(t) be a smooth func-
tion satisfying ζ(t) = l if t<s, ζ(t) = O if t>s + l. We can proceed
similarly to the proof of Theorem 2.1. v = ζu is a solution of
il(*, ί, D,, Dt)υ(x, t) = F(Λ, 0,
βy(jc, /, Dχ9 Dt)v(xy t) = Gj(x, t),
j = 1, •-, w,
where F and Gy are functions defined similarly to (2.6) and (2.7)
respectively. Furthermore we have
A(x9 s, Dχy Dt)v(x, t) = H{x, f)9
BJ(X, s, D
x> Dt)υ{x9 t) = Φ/ΛΓ, t)y
DIFFERENTIAL INEQUALITIES OF PARABOLIC TYPE 203
with some function H and
Φj = Gj + Σ Σ Dfr>,
 k β(x, s, t) (4.1)
where
wjkβ(xy s, t) = (bftgrKβ(x, s)-bf,ί._kβ(x, t))Dktv(x, t).
Taking the Fourier transforms of both sides of (4.1) and noting the
last formula of p. 199
+^oΣ Σ
k=0 \β\^»*j-kd
+ \\\>-*\\ώ
ί
^
β
(s,\)\\0)
for any complex number λ. Hence using Plancherel theorem we get
for any real number σ
_( I λ + iσ |<"-"yV" \\Φ}{X + i
Σ ί(Γ ^"iiDr^-1^*^,
(4.2)
If we recall the assumptions here it is not difficult to verify that given
£ > 0 we can choose —5 so large that the right side of (4.2) does not
exceed
v 1/2
(|λ + fσ|
Estimating ( ( |λ + fV|c2m"my)/rf||6y(λ + ίσ-)||0)2rfλ etc. in a similar manner
J -OO
we can show that if η is sufficiently small and if - σ is sufficiently
large we have
Σ f e^m^uWMt^C"^ ΓV'IIDr^^llWί . (4. 3)
From (4.3) it immediately follows that u(t) = 0 for — oo<f<s.
204 H. TANABE
OSAKA UNIVERSITY
References
[ 1 ] S. Agmon and L. Nirenberg: Properties of solutions of ordinary differ-
ential equations in Banach space, Comm. Pure Appl. Math. 16 (1963),
121-239.
[ 2 ] S. Agmon, A. Douglis, and L. Nirenberg: Estimates near the boundary
for solutions of elliptic partial differential equations satisfying general
boundary conditions, I, Comm. Pure Appl. Math. 12 (1959), 623-727.
[ 3 ] A. Friedman : Uniqueness of solutions of ordinary differential inequalities
in Hilbert space, Arch. Rat. Mech. Anal. 17 (1964), 353-357.


NOTE ON UNIQUENESS OF SOLUTIONS OF DIFFERENTIAL INEQUALITIES OF PARABOLIC TYPE

Tanabe, Hiroki

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