Trends in the Theory and Practice of Non-Linear Analysis
V. Lakshmikantham (Editor)
0 Elsevier Science Publishers B.V. (North-Holland), 1985
42 1
CAUCHY PROBLEM FOR HYPER-PARABOLIC
PARTIAL DIFFERENTIAL EQUATIONS
Showalter
R.E.
Department o f Mathematics, RLM 8.100
U n i v e r s i t y o f Texas a t A u s t i n
Austin, Texas
U.S.A.
INTRODUCTION
We s h a l l consider i n i t i a l - b o u n d a r y - v a l u e problems f o r p a r t i a l d i f f e r e n t i a l equat i o n s o f the form
Ut= Uxx- Uyy.
Such equations a r i s e i n a discussion o f c l a s s i f i c a t i o n . S p e c i f i c a l l y , every
l i n e a r second-order c o n s t a n t - c o e f f i c i e n t p a r t i a l d i f f e r e n t i a l equation i n n
ables can be reduced t o t h e form
k
i L:
= l uxixi
-
j
Z
i=l
u
YiYi
vari-
+ a u t + bu = f ( x , y , t ,...).
The general p a r a b o l i c case i s k + j + 1 = n; we a r e i n t e r e s t e d i n the non-normal
case of k j # 0 which we c a l l hyper-parabolic.
Equations o f t h i s type have a r i s e n i n d i v e r s e non-standard a p p l i c a t i o n s [1,4,7-11,
151, most of which r e q u i r e a s o l u t i o n s u b j e c t t o c l a s s i c a l i n i t i a l and boundary
c o n d i t i o n s on a space-time c y l i n d e r . However such a problem i s n o t w e l l posed b u t
i s "hyper-sensitive'' t o v a r i a t i o n s i n t h e data [121. I t i s c l e a r t h a t n e i t h e r t h e
i n i t i a l - v a l u e nor the f i n a l - v a l u e problem i s w e l l posed f o r ( 1 ) and, moreover, t h e
boundary-value problem f o r s t a t i o n a r y s o l u t i o n s i s ill posed.
Our p l a n i s t o develop some elementary n o t i o n s o f generalized s o l u t i o n o f an abs t r a c t model of ( 1 ) as an e v o l u t i o n equation i n H i l b e r t space. The Cauchy problem
can be approximated by a q u a s i - r e v e r s i b i l i t y method 151. Then we present some
w e l l posed problems f o r t h i s equation, and these r e s u l t s suggest a more n a t u r a l
method of approximating s o l u t i o n s o f t h e ill posed Cauchy problem. This new
approximation scheme we c a l l t h e quasi-boundary-value method.
INITIAL-VALUE PROBLEM
Hereafter A and B denote s e l f - a d j o i n t non-negative operators on a H i l b e r t
space H, and we assume t h e i r r e s o l v e n t s commute. Thus -A generates a (holomorphic) semigroup o f c o n t r a c t i o n s (exp(-At) : t 2 03 on H; t h e i r inverses a r e
unboundedloperators
exp(At) which could a1 so be obtained by t h e s p e c t r a l theorem.
If u E C
i s a s o l u t i o n o f t h e e v o l u t i o n equation
u ' ( t ) + A u ( t ) = 0,
then
d
e x p ( - A ( t - s ) ) u ( s ) = 0,
hence,
t(t)
r
<t
= exp(-A(t-
s))u(s), r
5 s 5 t,
is
independent o f s. Thus one o b t a i n s t h e semi-group r e p r e s e n t a t i o n
u ( t ) = e x p ( - A ( t - s ) ) u ( s ) , s 5 t; t h e operators e x p ( - A ( t - s ) ) a r e t h e propagators f o r t h e e v o l u t i o n equation and t h e i r c o n t i n u i t y i m p l i e s t h e i n i t i a l - v a l u e
R. E. Showalter
422
problem i s w e l l posed.
Proceeding s i m i l a r l y f o r t h e hyper-parabolic equation
(2)
U ' ( t ) + A u ( t ) - Bu(t) = 0,
u E C1 i s a s o l u t i o n on t h e i n t e r v a l
_rl_ e x p ( - A ( t - s ) ) e x p ( - E ( s - 7 ) ) u ( s ) = 0, 7 < s < t, so
we see t h a t i f
then
[7,tl,
ds
(3)
t(t,T)
= exp(-A(t- s))exp(-B(s-T))u(s),
7
<s
5t
i s independent o f s . Thus, we a r e l e d t o d e f i n e a weak s o l u t i o n o f ( 2 ) on [r,tl
as a continuous H-valued funct-ion f o r which ( 3 ) holds, i.e., t h e r i g h t s i d e of
( 3 ) i s independent o f s E [ 7 , t ] .
Lemma.
[71't11
I f u i s a weak s o l u t i o n on 17,t.I
then i t i s a weak s o l u t i o n on each
c k,tl and then E ( t , r ) = e x P ( - A ( t - t,))exp(-B(7,-.))E(t1,Tl),
and
u(t) = E(t,t-).
If u
i s a continuous H-valued f u n c t i o n on [0,11,
then u i s a weak s o l u t i o n
e x p ( - B t ) u ( t ) = exp(-At)u(O), 0 _< t 5 1. Thus t h e i n i t i a l - v a l u e problem o f
f i n d i n g a weak s o l u t i o n o f ( 2 ) on [0,11 w i t h u ( 0 ) = f g i v e n i n H i s equivalent to
iff
0
u E C ([0,11 ,H)
with
exp(-Bt)u(t) = exp(-At)f,
0
5
t
5
1.
Since each exp(-Bt) i s one-to-one, t h e r e i s a t most one s o l u t i o n o f t h e i n i t i a l value problem. Also, t h e r e p r e s e n t a t i o n v i a unbounded o p e r a t o r s as u ( t )
= exp(Bt)exp(-At)f
shows t h e i n i t i a l - v a l u e problem i s n o t w e l l posed. Consideri n g existence, we see t h a t i f f E RgIexp(-B)} = domIexp(B)l,
then
u(t)
=
e x p ( - A t ) . e x p ( - B ( l - t ) ) . exp(B)f
(C")
defines a strong solution
have t h e f o l l o w i n g
o f t h e i n i t i a l - v a l u e problem.
More g e n e r a l l y we
Pro o s i t i o n 1. There e x i s t s a weak s o l u t i o n o f t h e i n i t i a l - v a l u e problem i f and
o n l y if exp - A ) f = exp(-B)g f o r some g E H, i . e . ,
f E domIexp(B- A)}.
QUASI -REVERSIBI L ITV METHOD
Since t h e l a c k of well-posedness o f t h e i n i t i a l - v a l u e problem f o r ( 2 ) i s due t o
t o o b t a i n an apt h e unboundedness of B, we use a Q - R method [3,5,13,141
proximate s o l u t i o n . F i r s t r e p l a c e B by i t s bounded Vosida approximation
BE
=
B ( I + EB)-',
The f i n a l - v a l u e
> 0,
E
and s o l v e t h e equation ( 2 ) f o r
exp(BE- A)f
belongs t o
RgIexp(-A)}
exp((BE- A ) t ) f , O < t _ < l .
so we o b t a i n a ( s t r o n g )
s o l u t i o n o f ( 2 ) backward from here,
u,(t)
= exp((A- B ) ( 1 - t))exp(BE- A ) f
= exp(-At)exp(-B(l-
and i t s a t i s f i e s
following
Theorem 1.
~ ~ ( =0 exp(BE)
B)f.
For any
(weak) s o l u t i o n
u
f E H, , l i m o
u (0)
t))exp(BE)f,
0
5
t
5
1,
Using r e s u l t s from 1141 we o b t a i n t h e
=
f
and
Ilu,(O)ll
5
of t h e i n i t i a l - v a l u e problem f o r ( 2 ) on
llfll.
[O,ll
There e x i s t s a
i f and o n l y i f
423
Cauchy Problem
lim
E + O
i u E ( t ) l exists i n
H
for all
t E [0,1],
and t h e n
lim
E-+O
u,(t)
= u(t).
T h i s QR-method i s t h e o r e t i c a l l y i n c i s i v e : t h e r e e x i s t s a s o l u t i o n i f and o n l y i f
i t converges. I t i s s l i g h t l y more s u b t l e t h a n t h e u s u a l case (A = 0) i n [141
s i n c e t h e method must g i v e a u,(l) E RgIexp(-A)} i n o r d e r t h a t t h e backward problem have a s t r o n g s o l u t i o n . The method i s always s t a b l e and convergent a t t = 0,
b u t only a t t = 0. F o r example, Ilu,(l)ll
may grow l i k e a e x p ( l / E ) ) ,
so as a
numerical method i t i s e s s e n t i a l l y w o r t h l e s s . Even i f one uses l o g - c o n v e x i t y
e s t i m a t e s t o s t a b i l i z e t h e method [2,61, t h e use o f i n i t i a l - v a l u e o r f i n a l - v a l u e
problems i n t h e procedure i s n o t n a t u r a l f o r ( 2 ) .
BOUNDARY-VALUE PROBLEM
From t h e Lemma above i t f o l l o w s
Suppose t h e r e i s a weak s o l u t i o n o f ( 2 ) on [0,11.
t h a t €(0,1) and hence t h e s o l u t i o n u w i l l depend on b o t h u ( 0 ) and u ( 1 ) .
One need o n l y d e t e r m i n e t h e domain o f i n f l u e n c e o f u on € t h r o u g h t h e f o r m u l a e
o f t h e Lemma. T h i s suggests t h a t a boundary-value problem on t h e i n t e r v a l [0,1]
i s more a p p r o p r i a t e t h a n an i n i t i a l - v a l u e problem.
We can s u b s t a n t i a t e t h i s o b s e r v a t i o n as f o l l o w s . F i r s t l e t C be a s e l f - a d j o i n t
o p e r a t o r whose spectrum i s unbounded i n b o t h p o s i t i v e and n e g a t i v e r e a l numbers,
t h u s C = A - B as above where A and B a r e t h e p o s i t i v e and n e g a t i v e p a r t s o f
C, r e s p e c t i v e l y . We seek a r e p r e s e n t a t i o n o f a s o l u t i o n o f
(4)
u ' ( t ) + C u ( t ) = 0,
i n t h e form
so
u(t) =
I,
{eXp(CZ) : z E PI
exp(Cz)U(t,z)dz.
u(t) =
< t < 1,
I n o r d e r t o choose t h e c o n t o u r
z = ir,
i s bounded, we t a k e
(5)
0
7
E
r
in
$
IR, so we have
IIR
exp(irC)U(t,T)dT.
Substitution o f (5) i n t o (4) y i e l d s
RI exp( ir C) (Ut + i U 7 )d7 + ( I/ i)exp( i 7 C)U( t ,7
t o s a t i s f y t h e Cauchy-Riemann e q u a t i o n i n
so we need r e q u i r e t h e k e r n e l U(t,7)
t h e s l a b 0 < t < 1 and t o v a n i s h a t T + f-. Thus U w i l l be determined by i t s
These i n t u r n a r e der e m a i n i n g boundary-values,
u ( 0 , ~ ) . U(1,7), -- < T < +-.
t e r m i n e d by u ( 0 ) and u ( 1 ) t h r o u g h ( 5 ) . These f o r m a l c a l c u l a t i o n s can (and
w i l l ) be made p r e c i s e elsewhere b u t t h e v a l r e a d v suqqest t h a t t h e e a u a t i o n ( 1 ) i s
e l l i p t i c and t h a t t h e f o l l o w i n g problem- i s w e l l - p o s e i . The boundary-value problem
i s t o f i n d a weak s o l u t i o n u o f ( 2 ) on [ O , l l
f o r w h i c h a u ( O ) + bu(1) = f .
Here f E H and a,b E I R a r e g i v e n .
P r o p o s i t i o n 2.
If
u
i s a s o l u t i o n o f t h e boundary-value problem t h e n
(aexp(-B)+ bexp(-A))u(t) = exp(-At)exp(-B(1-
>
t))f,
0
5t5
1
I f a l s o a,b
0 and n o t b o t h a r e zero, t h e n t h e r e i s a t most one s o l u t i o n .
If
b o t h a and b a r e s t r i c t l y p o s i t i v e t h e n t h e r e e x i s t s a s o l u t i o n u f o r each
f E H and i t s a t i s f i e s
u(t)
5
Ilfll/al-tbt
,
O < t < l .
QUASI-BOUNDARY-VALUE METHOD
Consider a g a i n t h e i n i t i a l - v a l u e problem f o r ( 2 ) on t h e i n t e r v a l
[0,1]
with
424
R. E. Showalter
u ( 0 ) = f . The q u a s i - r e v e r s i b i l i t y method o f a p p r o x i m a t i o n was t o r e g u l a r i z e t h e
problem by p e r t u r b i n g t h e e q u a t i o n , i . e . , r e p l a c e B by B.,
The method suggested
by P r o p o s i t i o n 2 i s t o r e g u l a r i z e t h e problem by p e r t u r b i n g t h e i n i t i a l c o n d i t i o n ,
i . e . , r e p l a c e i t by t h e boundary c o n d i t i o n
(6)
u ( O ) + EU(1) = f .
Thus f o r each
(21, ( 6 ) .
Theorem 2.
E
>O
F o r any
we l e t
f E ti,
uE be t h e s o l u t i o n o f t h e boundary-value problem
l ~ m u,(O)
0
i n i t i a l - v a l u e problem f o r ( 2 ) on
[0,1]
f o r a l l t E [0,11,
and t h e n
(6) s a t i s f y t h e estimates
,li+!mo
ti
tlUE(t)ll
(7)
5
Ilfll/Et,
= f.
There e x i s t s a s o l u t i o n
i f and o n l y i f
u,(t)
0
= u(t).
5 t 5 1,
,l-i+m~~
{uE(t)I
The s o l u t i o n s
€
u
o f the
exists i n
uE o f ( Z ) ,
> 0.
The r e g u l a r i z a t i o n procedure o f Theorem 2, t h e QB-method, and t h e QR-method o f
Theorem 1 b o t h approximate w i t h a w e l l posed problem f o r each E > 0. Moreover,
t h e estimate (7) i s a 1 / ~ ) a t t = 1 i n c o n t r a s t t o (9lexp(l/E))
i n t h e QRmethod, so t h e QB-method i s r e a s o n a b l e f o r n u m e r i c a l implementation. However t h e
r e g u l a r i z e d problems i n t h e QB-method a r e g l o b a l i n t, so marching methods and
t h e i r r e s u l t a n t sparse m a t r i c e s and reduced s t o r a g e r e q u i r e m e n t s a r e n o t d i r e c t l y
a v a i l a b l e i n n u m e r i c a l work. Our p r e c e d i n g remarks on t h e " e l l i p t i c " n a t u r e o f
t h e s e e q u a t i o n s suggest t h a t such d i f f i c u l t i e s may be i m p l i c i t i n t h e problem, n o t
j u s t t h i s method.
There i s a fundamental d e f i c i e n c y i n t h e use o f Theorem 2 t o a c t u a l l y f i n d a solut i o n u f r o m d a t a f; namely, t h e d a t a i s never measured e x a c t l y . T h i s measurement e r r o r can be handled i f we s t a b i l i z e t h e problem by c o n s i d e r i n g o n l y t h o s e
s o l u t i o n s which s a t i s f y a p r e s c r i b e d g l o b a l bound. Whereas Theorem 2 m e r e l y guara n t e e s a good a p p r o x i m a t i o n a t t h e i n i t i a l t i m e t = 0, we s h a l l g e t a g l o b a l
a p p r o x i m a t i o n on t E [ O , l ] .
Theorem 3 , L e t u be a weak s o l u t i o n o f ( 2 ) on [0,11.
L e t M 2 1, 6 > 0, and
f E ti be g i v e n such t h a t Ilu(0) - fll < 6 and
u ( 1 ) 5 M. Choose E
M/S
and
l e t uE be t h e s o l u t i o n o f t h e boundary-value problem ( Z ) , ( 6 ) . Then we have t h e
estimate
f o r the error.
The procedure above i s t h e s t a b i l i z e d quasi-boundary-value method. I t i s a p p r o p r i a t e i n a p p l i e d problems where one knows f r o m p h y s i c a l c o n s i d e r a t i o n s t h e r e i s a
s o l u t i o n w i t h a bound b u t t h e d a t a u ( 0 ) i s n o t known e x a c t l y .
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The f i n a l ( d e t a i l e d ) v e r s i o n o f t h i s paper w i l l be s u b m i t t e d f o r p u b l i c a t i o n e l s e where.