599
I nt. J. Mh. Math. Sci.
No. 3 (7980] 599-603
Vol.
ON THE BACKWARD HEAT PROBLEM:
EVALUATION OF THE NORM OF
YVES BIOLLAY
Department of Mathematics
Cornell University
Ithaca, New York
14850
U.S.A.
(Received July 19, 1979)
BSTRCT.
W how
.n
h. pape ha
II ,u II
I ull
bounded
(o)
<Tif
one imposes on u (solution of the backward heat equation) the condition
}I u<x,t> II
{l ut(x,T) l{
-
A HIder type of inequality is also given if one supposes
M.
K.
KEY WORDS AD PHRASES: Pabolic equations, Impropy posed problems.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES:
i.
35R0, 35R25.
INTRODUCTION.
A lot of authors have dealt the backward heat problem and considered equations
of various kind.
It is known that this problem is an improperly posed problem
and the dependence of the solution as function of the initial data is an important
aspect of it.
The a priori inequalities
eral informations.
(see Sigillito Ill) give
immediately sev-
Among the methods of investigation, that of the logarithmic
convexity is relatively simple when one is able to define- the difficulty
is
600
Y. BIOLLAY
the functional which leads us to the required result (see,
there
e.g. Knops [2]). But in the particular case of the hereafter equation, the functional is, for reasons of analogy, rather easy to determine.
2.
llu(x,t) ll
EVALUATION OF
llut(x,t)II
AND
WITH u e
CaB.
Let us consider the backward heat equation:
Au + u
t
O
in
D[O,T)
(2.1)
u
0
on
8D[O,T)
(2.2)
and the initial data
u(x,O)
f(x)
D is a bounded open domain in
Let
Ca8
(2.3)
n with
smooth boundary ZD.
be the class of the functions 8(x,t), continuous in
[O,T],
so that
for fixed t e(O,T), 8 is twice continuously differentiable in the
x-variable and for xeD, 8 is continuously differentiable in t
[te(O,T)]
8)
ID
I8(x,T)I 2 dx
z_ K 2
Then, if the solution u of (2.1-3) is subjected to belong to C
t-ZT,
one has:
where, by definition
llutll- KTII ut(x,O)II l-t/T
llv(x’t)ll2 ID v2(x’t) dx
Let us set g(x)
u t(x,O); then, the evaluation of
an upper bound of
llg(x)II
llutl
is
found,
if
is known.
To prove (2.4), let us use the property of the logarithmic convexity
[3],
applied to the functional (see
(t)
One has:
’
(t)
ID
2
ut(x,t)
p. 11-12)
(2.5)
dx
2J" D
u
+
dx
ututtdx =-2J" D utAutdx =-2D
2
2
2/D gradutl dx 2/D gradutl
t.
--u t
dc +
(2.6)
BACKWARD HEAT PROBLEM
_ -
since (2.2) implies that
"(t)
u
4I D
gradut.gradutt
41 D
utt
2
dx
601
O on D when te(O,T). It follows that
dx
4D utt---ut
4J" D
dc
uttAu t
dx
Using Schwarz’s inequality one finds:
,,
L
Thus,
,2
(In)"
or
0
(T)t/T P(O)I-t/T that is
flu t112 _z K2t/Tllg112(1-t/T)
O
to say
On the other hand, one has:
ID
with l
I
uu
=-fD
dx
t
gradul
ID
uAu dx
2dx- ll/D u2dx
A+I#= O in D ]
O on D
first eigenvalue of the problem
Then, for t=T
i12 (fD u2dx)
2 _z
J’D u2dx" J’D
and consequently (see
llull
_z
[3],
u2t
/D u2dx
dx
._11
z 1
=
2t
I D [0(x,t)I
and let
Cag
2
)2
i D u dx z
m
2
1
p. 13)
(l)t/T I fll
l-t/T
(2.8)
Now, let us substitute the conctition ) of the class
8’)
(2.7)
Cag
by
dx _z K ,2
be the class of functions which satisfy a) and ’).
Let us show why one can find one bound of
llutl
if
uC8
-
llut(x,t)II WITH u e C8
LEMMA:
(3.1)
if
ueC8 llut(x,t)II My for t-T- O<T-T
2
PROOF: We remark first that #(t)=:
ID u (x,t)dx is an increasing function since
’ 2I D uutdx 2I D gradul 2dx > O. Thus,
3.
EVALUATION OF
(t) _z K ,2
for
t-ZT
Let h(t) be a function of class C
2
(3.2)
and let us define
Y. BIOLLAY
60
/t=b
t=a /D
I
2
h(t) ut(x,t) dxdt
b
-/a[/D
utAu
dx]h(t)dt
u do I gradu.gradu dx]h(t)dt
_ib[
a D ut"
t
D
2
1
2
b
t=b
i/ D [/ba h Igradul d t]dx I D (hlgradu 2 t=
h’dt)dx
a -/algradul
1 b
1
1
h’ (/D Igradul 2dx)dt
hlgradul 2dx be ]- Ia
[’’’]
[/D
1
1 b
1
I 2
+ 1 I D iba h" u 2 dxdt
2/a h’fD u dxdt [
I D h’u2dx It=b
t=a
l[h(b) I D Igradu(x,b) 2dx h(a)I D gradu(x,a) 12dx]
(3 3)
l[h’ (b)/D u 2 (x b)dx h’ (a)/D u 2 (x a)dx] +
b
1
+ fDfa h"(t)u 2(x t)dxdt
I
TI
Then,
T
fO IID
u
< T
2
dxdt
t
L
I TI
I
0 D
2
dxdt +
ut
T-t
T
2
ITI f D T-TI .u t dxdt
and using (3.3) for each term of the sum, we obtain
2
[/D Igradu(x’T) 12dx ID Igradu(x’O) 12dx] +
1
Igradu(x’T)12dx] + 1 [ID u2 (x,T1)dx- ID u2 (x,O)dx]
[-/D
1
2
1
,K’ 2 =: M 2
by (3.2)
4 (T-T)ID u (x,T)dx
4(T-T)
1
dxdt z_
IOTfD ut
+
z
Also,
_
M2
But,
-
fosfD ut2 dxdt
Clearly,
s
IOI D
I
oTIID ut2
and
’
’T-T---I
-
is an increasing function of s. Thus,
u
dxdt
-
dxdt z_ M
s
/ToTID ut2 dxdt /ToT
Setting finally
To
So, we can evaluate
Let us write T-T
finds similarly
(3.4)
(O-T0<T)
A_
(T0) (T-T0)
K’2/4(T-T)(T-T0)
T-T, TI
(t) _z (T-T) z_
< T
(t)dt
M2
are positive; this leads to
P(T0)
T
K,2/T 2
llAull
T-T/2, one arrives to
=: M
2
tZ-T-T
with a bound of
T’ and
2
fD utt
flu t(x,T’)ll
2
dx z M,,
llu(x,T)ll-
_x M’. Then,
/
q.e.d.
t-T"
if
utt
since
exists, one
A(ut)
+
(ut) t
BACKWARD HEAT PROBLEM
(u t)
and
in D[O,T)
0
And generally
ID
k)ku’2
(t
dx
-
[M(k)] 2
t-zT(k)<T,
603
0
on
D[O,T)
k q
(3.5)
REMARK: A result analogous to (3.1) can be obtained if we consi)t t
where u are the eigena u (x)e n
der the serie u(x,t)
n
n n
n=l
a
Fourier coeff+/-the
and
e+/-genvalues
of
the
(2.7), X n
functions
n
cients of f(x). But, because of the necessity of the uniform conver-
gence, the developments are long and rather complicated.
ACKNOWLEDGMENT
This work was supported by the Swiss National Science Founda-
tion. I thank also cordially Professor L. E. Payne for his advice.
REFERENCES
[i]
Sigillito, V.G., Explicit a priori inequalities with applications to boundary value problems, Research Notes in Mathematics 13, Pitman (1977).
[2]
Knops, R.J., .Logarithmic convexity and other techniques applied
to problems in continuum mechanics, Lecture Notes in Mathematics 316, Springer Verlag (1973).
[3]
Payne, L.E., Improperly posed problems in partial differential_
equations, Regional Conference Series in Applied Mathematics 22, Philadelphia (SIAM 1975).