Ordinary and Partial Differential Equations: Proceedings of the Eighth Conference held at Dundee, Scotland, June 25–29, 1984

357 Pages·1985·12.2 MB·English

by  A. M. A. Abou-El-Ela (auth.), Brian D. Sleeman, Richard J. Jarvis (eds.)

A STABILITY RESULT FOR THE SOLUTIONS @F A CERTAIN
SYSTEM DF THIRD-ORDER DIFFERENTIAL EQUATIONS
A .~ . A . Abo~-E£-E£~
The paper determines sufficient conditions under which all solutions
of (1.1) tend to zero as t + ~ .
INTRODUCTION AND STATEMENT OF THE RESULT
We consider the real non-linear third-order vector differential
equation
~" + F ( X , X ) ~ +G(X) + H(X)= P ( t , X , X , X ) ( 1 . 1 )
where X c R n, F is an nxn-matrix function, G: R n ÷ Rn,H:R n ~ R n and
p:R+xRnxRnxR n + R n.
Let the non-linear functions F,G,H and P be continuous and constru-
cted, such that the uniqueness theorem is valid and the solutions are
continuously dependent on the initial conditions.
The equation (1.1) represents a system of real third-order differen-
tial equations of the form:
... n
x i + k=~l fik(Xl,..,×n;~l, ''' ~ n)"X"k +gi(x.l '''' ~n)+hi(x l''''Xn)
=Pi(t,xl,..,Xn;~l,..,~n;Rl,..,Xn) (i=l,2,..,n
~fik ~fik ~gi ~h
Moreover, le t t he derivatives , - - , .......... and - -
~xj a~j @~j axj
Bh.
~fik
(j=l,2,..,n) exist; furthermore and i are continuous.
~xj Bxj
Special cases of the differential equation (i.I) have been treated
in Abou-E1-Ela [1] ,[2] ; Ezeilo [3],[5]; Ezeilo & Tejumola [4] and
Others. This paper generalizes Ezeilo [5; theorem 3] for the case
A=F(X,X) and also gives an n-dimensional extension for Ezeilo [3] .This
work extends further a result given by the author in [2] where P is not
necessarily identically zero.
Using Y=X and Z=Y the differential equation (1.1) will be trans-
formed to the equivalent system
= Y~
= Z,
= -F(X,Y)Z-G(Y)-H(X)÷P(E,X,Y,Z). (1 .z )
We need the following notations and definitions:
I.Xi(A) (i=l,2,..,n) are the eigenvslues of the nxn-matrix.
2.<X,Y> corresponding to any pair X,Y of vectors in R n is the usual
n
scalar product i~l xiYi'II x If2 = <x,x> for arbitrary X in R n.
3.The matrix A is said to be negative-definite, when <AX,X> < o for
all nonzero X in R n
4. The Jacobian matrices JG(Y),JH(X),J(F(X,Y)YIX) and J(F(X,Y)YIY)
are given by:
8g i ~h i
JG(Y) = ( - ~ j ) , JH ( X ) = ( ~--x-T.) ,
J
n n 8fik
J(F(X,Y)YIX)= (~x ~ fikYk )= ( ~ ~ Yk )'
j k=l k=l j
n n
J(F(X Y)YIY)= (--~--~ Z fikYk)=F(X,y)+ ( ~ fik yk) "
' ~Yj k=l k=l ~Y---~
Now let us formulate the foIlowing conditions:
(i) FiX,Y) is symmetric and ki(F(X,Y))~ 61 > o(i=l,2,...,n).
(ii) J(F(X,Y)YIY) is symmetric and J(F(X,Y)YIX) is negative-definite.
iii) G(O)=O, JG(Y) is symmetric and Xi(JG(Y)) ~ 62 > o (i=l,2,..,n).
iv) H(O)=O, JH(X) is symmetric and 65 3 Xi(JH(X)) ~ 63 > o
(i=l,2,..,n).
v) 6162 - 6 5 > o .
(vi) JG(Y) and JH(X') commute with JH(X) for all X,X',Y ¢ R n.
(vii) There exist constants ~(o ~ ~ < i), 6o ~ o and continuous
functions @l(t),O2(t) ; such that for all t ~ o and every solut-
ion X,Y,Z of (I.2) the following inequality
II P(t,X,Y,Z)II ~ 81(t)+e2(t)(ll Y 112+11ZII~ )½a+6o(II YII2 +II zIl2 )½
(1 .3)
is valid; furthermore Ol(t),O2(t) are positive and satisfy
max el(t) < ~ and ~ 8i(t)dt < ~ (i=1,2). (1 .4 )
o<t<~ 0
Our a~m is to prove the foilowing
THEOREM: Under the additional assumptions (i)-(vii) there exists a
posilive constant A , whose magnitude depends only on 61,62 and 6''3,
o
such that if 8 ° ~ A 0 then every solution X(t) of (i.i) satisfies
3
II x ( t ) I I + o, II ~ ( t ) I I + o and II R ( t ) I I + o as t + ~. (1 .5 )
Remark: In the case F(X,Y)E A, i t follows from the assumptions ( i ) and
(ii):
A is symmetric and li(A) ~ 61 > o (i=l,2,..,n).
The assumptions (i)-(vii) are then exactly those of Ezeilo [5;theorem 5].
A FUNCTION T(X,Y,Z)
The proof of the theorem depends on a scaler differentiable compar-
ison function ~(X,Y,Z). This function and its total time derivative
satisfy fundamental inequalities. We define ~ as follows
~(X,Y,Z)= ofl<H(oX),X > do + ~of~ G(oY),Y > do
+ ofl< F(X, oY)Y,Y>do+ ½ ~ < Z,Z>
+ ~ < Y,H(X)> + <Y,Z> (2.1)
where 62 ~ - i > 6 > 6; 1 .
Let (X(t),Y(t),Z(t))be an arbitrary solution of (1.2), we define then
@(t)= ~(X(t) , Y ( t ) , Z ( t ) ) . (2.2)
The following two lemmas are important for the proof of the theorem
LEMMA 1: There exists a posit ive constant 67, such that
T(X,Y,Z) Z 67(11 xlP +11 YIP +II zll 2) (2. ; )
is valid for every solution of (1.2).
LEMHA 2: There exists a positive constant Ao=8o(~1,62,6~) such that if
6 o _< 8 o then for t _> 0
~(t) d - ~8P2(t)+ 69 {81( t )p ( t )+ez( t )pa+ l ( t ) } (2.4)
where P2(t)~ II Y(t)II 2 +II Z(t)II 2 and ~8,~9 are posit ive constants,
dependent on ~i,~2, ~ .
For the proofs of the two lemmas see Abou-E1-Ela [ i ] .
SOME PRELIMINARY RESULTS
We shall require the following two algebraic results
LEMMA 3: Let A be a reel symmetric nxn-matrix. If A a ~ xi(A) ~ 6a > o
(i=l,2,..,n), then
Aa il x IP z <AX,X>z 6a II X II= ( 3 . 1 )
for any X c R n.
LEMMA 4: Let A,B be any two real nxn commuting symmetric matrices, then
the eigenvalues of AB are all real; and if ~ Z ~(A) ~ ~a > o ,
Z ki(B) Z ~b > o (i=l,2,..,n), then
AaAb Z Xi(AB)~ ~a 6b" (3.2)
For the proofs of the two lemmas see Abou-E1-Ela [2].
The actual proof of the theorem depends on the following two proper-
t i es of H
LEMMA 5: If H(O)=O, JH(X) symmetric and Xi(JH(X)) <~' (i=1,2 . n)
then
]I H(X)-H(X' ) II _<~ II x-x, II (3.3)
for a l l X,X' in R n.
Proof: See Ezeilo [5;Lemma 9]
LEHMA 6" If H(O)=O, JH(X) symmetric and commutes with JH(X') for all
X,X' in Rn . If Xi(JH(X)) > 63 > o , then
II H(X) II > ~3 II x II (3.4)
for all X ¢ Rn .
Proof: Since
8 < H(OlX),H(olX)> = 2 < JH(OlX)X, H(olX) >
8a 1
by integrating both sides from oi=o to ~i=i , and because of H(O)=O,
we obtain
1
II H(X)II = =ol 2 < JH(OlX)X,H(alX) > da I
but H(~IX) = ° f l " ~ 2 H ( a l ° 2 X ) d a 2
1
= o f a 1 J H ( a l e 2 X ) X da 2
therefore
1
<JH(OlX)X,H(~IX) >= ~< JH(alX)X,OlJH(alO2X)X > do 2
o
consequently we get
II H(X)II 2 =2oS~f I ~i < JH(~IX)X'JH(°IO2X)X > do2 do1
11
= 2o f o f < 3 H ( a l o 2 X ) ~ H ( a l X ) X , X > a 1 do 2 da 1. ( 3 . 5 )
Since JH(OlX), JH(OlO2 X) commute and are symmetric and because of
Xi(JH(X)) _ > 63 for any X c R n , then it follows
X i ( JH ( O l O 2 X ) J H ( O l X ) ) ~ 632 ( 3 . 6 )
by Lemma 4.
Hence, according to Lemma 3, we have from (3.5) and (3.6)
II H(X)II~ _> ~3211 x ii~ .
THE PROOF OF THE THEOREM
L e t ( X ( t ) , Y ( t ) , Z ( t ) ) be any s o l u t i o n o f the e q u i v a l e n t sys tem ( 1 . 2 ) .
We s h a l l show t h a t t he r e s u l t ( 1 . 5 ) o f t he theorem h o l d s f o r any s o l u t -
ion X(t) of (1.1), if the constant 6 ° in (1.3) satisfies 6o -<- & o where
4 o is the constant in Lemma 2.
The proof will be in two stages: first we show that II Y(t)II + o and
II z(t) ll ÷o as t ÷ ~ , and then, as a consequence we verify that
l lx ( t ) ll + o as t +
From Lemma 2 we have seen t h a t 9 ( t ) s a t i s f i e s the i n e q u a l i t y ( 2 . 4 )
f o r t > o . By i n t e g r a t i n g bo th s i d e s o f ( 2 , 8 ) we o b t a i n t he i n e q u a l i t y
t t
9 ( t ) 2 4 ( o ) - 88 f p2 (T )dT+ 69 f { e t ( ~ ) p ( ~ ) + e 2 ( ~ ) p ~ + l ( ~ ) } d T ( 4 . 1 )
o o
From ( 2 . 3 ) i n Lemma i and ( 2 . 2 ) we c o n c l u d e t h a t
9 ( t ) 2 o for t ~ o . ( 4 . 2 )
Combining (4.1) and (4.2) leads to
t t
~8 fP2 ( ~ ) d T ~ 4 ( 0 ) + 6 9 ~ { 8 . 1 ( T ) p ( T ) + e 2 ( T ) p a + I ( T ) } dT. ( 4 . 3 )
o O
The a u t h o r p r o v e d i n [ 1 ] a boundedness t heo rem f o r t he s o l u t i o n s
o f ( i . 1 ) w i t h t he same h y p o t h e s e s e x c e p t f o r e i , e 2 s a t i s f y i n g
o I 8i(t)dt < ~ and o 7 ~ e 2K / ( 1 - ~ ) ( t ) d t < ~ ( 4 . 4 )
where i 2 ~ 2 2.
But since (1.4) implies (4.4) the boundedness result holds also here
under the new strong conditions on el,e 2. Hence there is a finite
positive constant &l such that
II x ( t ) l l = +ll Y ( t ) I P + I I z ( t ) l l 2 ~ A~, for t Z o . (4 .5 )
Thus in particular, p(t) ~ &l for t Z o and t h e n i t f o l l o w s f r om
(4.3)
(4 .6 )
o ;~p 2 ( T ) d T <
by (1.4).
Since Tdt p 2 ( t ) = 2 < Y, Z > + 2 < Z, ~ >
=2 < Y,Z> -2 < Z,F(X,Y)Z + G(Y)+H(X)-P(t,X,Y,Z)> we obtain
by using (4.5), (3.1) and the hypotheses on F,G,H and P
dp2(t) is bounded as t + (4.7)
It is quite an elementary matter (see for example [6;P. 273]) to show
from (4.6) and (4.7) that
p2 ( t ) + 0 as t +
t h a t i s , i n v i e w o f t h e d e f i n i t i o n o f p
I[ Y ( t ) H + o and U z ( t ) I I + o a6 t + ~ . ( 4 . 8 )
Thus we h a v e t h e f i r s t p a r t o f o u r r e s u l t .
To c o m p l e t e t h e p r o o f o f ( i , 5 ) i t r e m a i n s t o v e r i f y t h a t
tl X ( t ) ] ] ÷ o as t ~ ~ ( 4 . 9 )
I t w i l l be s u f f i c i e n t t o p r o v e t h a t
II H ( X ( t ) ) l l ÷ o as t ÷ ~ ( 4 . 1 0 )
s i n c e , by Lemma 6 , 1 1 H ( X ( t ) ) I I ~ 63 I[ X ( t ) l l , so t h a t ( 4 . 1 0 ) n e c e s s a r i l y
i m p l i e s ( 4 . 9 ) .
The m e t h o d o f t h e p r o o f o f ( 4 . 1 0 ) i s d e r i v e d f r om an a d a p t a t i o n o f
an i d e a i n [ 7 ; §2 7 .
Now by ( 1 . 2 )
Z= - F ( X , Y ) Z - G ( Y ) - H ( X ) + P ( t , X , Y , Z )
and on i n t e g r a t i n g b o t h s i d e s f r om t t o t + l ( t > o) we have
t + l
H ( X ( t ) ) = - [ Z ( t + I ) - Z ( t ) ] - f F ( X ( % ) , Y ( 7 ) ) Z ( % ) d % -
t
t + l t + l
- I G(Y(T) )d% + f { H ( X ( t ) ) - H ( X ( T ) ) } d~+
t t
t + l
+ I P( T,X(T) ,Y(T) ,Z(T) ) tiT. (4.ii)
t
I t is evident from (4.8) that
l l z ( t + i ) -Z ( t ) II ÷ 0 as t ÷ ~o. ( 4 . t 2 )
Now s i n c e
t + l t + l
11 f F(X(~) ,Y(T))Z(~)d~l l 2{ f II F ( X ( T ) , Y ( T ) ) Z ( T ) I I 2 t iT} ½
t t
< max II F ( X ( T ) , Y ( T ) ) Z ( T ) I I
t<~<t+l
max II F(X,Y)II . max llz(~)II
II x II2 41Y ll~a~ tE~t+l
= ~0 max II z ( ~ ) II ,
t<T< t + l
t + l
it follows l[ f F(X(T),Y(T))Z(~)d~ll ÷o as t ÷ ~ , (4.13)
t
by (4.8).
According to (4.8) also and the fact that G is continuous in Y and
satisfies G(O)=O, we conclude that
t+l
I I I G ( Y ( T ) ) d T l l ÷ 0 as t ÷ ~ . (4.i4)
t
By Lemma 5
II H (X ( t ) ) - H ( X ( T ) ) H ~ 6~ II X ( t ) - X ( T ) II
bu t
t+l t+l
II X ( t + l ) - X ( t ) l l = II I Y(T )dT I I 2{ I II Y (~ ) I I 2 dT} ½
t t
< max II Ye)ll,
t<T_< t + l
then from
II x(t)- x(~ ) I I _< max f lY(s)I I < msx llY(s)ll ,
t <s<T t < s < t + l
we have
t + l
I lf { H ( X ( t ) ) - H ( X ( T ) ) } d~lIda~ max
II y ( ~ ) I I ,
t t<T< t + l
therefore
t+ l
I I f { H ( X ( t ) ) - H ( X ( T ) ) } dT II -~ o as t + ~. (4.15)
t
We have a t last from (1 .3 )
t + l t + l
II S P(T,X(T),Y(T),Z(T))dTII2±I {e l (T )+ e 2 ( T ) p ~ ( ~ ) + 6 p (~ ) }2 d~
0
t t
t+ l
±
t
But , i n v iew of ( 1 . 4 ) and ( 4 . 5 ) , there are cons tan ts 811,612 such t h a t
e i ( t ) ~ $ i i ( i =1 ,2 ) , p2(t) ! 612 , t ! o,
Consequently
t + l t + l
~ (T) } +~2-2(T) ]dT
II P(T,X(T),Y(T),Z(T))NTII2~ 3 ; [811{01 (T)+ 81202 0 p
t
t+l
5 613 7 [e I (T )+O2(T )+p2 (T ) ] dT,
t
where 613 = 3 max (611,611612 , o
Because of the integrability conditions on 0 i and p2in (1.4) and (4.6)
respectively, it is obvious that the last integral tends to zero as
Hence
t + l
i [ I P(T,X(T),Y(T),Z(~))dTII + o as t ÷ =. (4.16)
t
From (4.11) and owing to (4.12)-(4.16) it follows (4.10) and the proof
of the theorem is now complete.
REFERENCES
[l]Abou-Ei-Ela, A.M.A., VII. Internationale Konferenz Ober Nichtlineare
Schwingungen,Band I, I. pp. 17-23. Abh. Akad. Uiss., Akademie-Verlag
Berlin 1977.
[2]Abou-Ei-Ela, A.M.A., Math. Nachr. 81(1978), 201-208.
[3 ]Eze i l o , J.O.C., Ann. mat. Pura AppI. , IV Vol. 66 (1964),233-249.
[ 4 ]Eze i l o , 3.0.C. & Tejumola, H.O., Ann. mat. Pura Appl . , IV Vol. 74
(1966), 283-316.
[5 ]Eze i lo , J.O.C., Journal of math. Analysis and appl . , 18 (1967),
395-416.
[6]hefschetz,S.; Differential equations: geometric theory (interscience,
New York, 1957).
[7]Levin 3.J. & Nohel 3.A., Arch. Rational Mech. Anal. 5 (1960),194-211.
A LOCAL EXISTENCE THEOREM FOR THE QUASILINEAR WAVE
EQUATION WITH INITIAL VALUES OF BOUNDED VARIATION
H.D. Alber
i. Introduction
We consider the quasilinear hyperbolic initial value problem
u t : f(u) x x E ~, t ~ O, (i.I)
u(x,0) = U(x), (1.2)
with u = (Ul,U 2) : ~ × ~+o ÷ ~2, U = (UI,U 2) : ~ ÷ ~2 f(u)': (u2,~(ui)).
We assume~.that ~ E C2(~, ~) satisfies ~'(~) ) 0 for all T £ ~, which
implies that the system is hyperbolic. This system is genuinly non-
linear if a"(<) $ 0 for all ~ £ ~, and we assume without restriction of
generality that ~"(T) > O. For i:i,2 the i-Riemann invariants
R i : R 2 + ~ are defined by
_ . , )112
RL(UL,U2 ) = u2 ~1 ~ (~ d~, (1.3)
O
R2(Ul,U2 ) = U2 + ~i c,(T)i/2 d~, (1.4)
O
and we assume that the following condition holds:
Condition C. Every level curve of R i intersects with any level curve of
R 2 •
Under these assumptions the following result holds:
Theorem i.i: If the initial data U have compact support and are of
bounded variation, then there exists T > 0 such that a weak solution u
of (1.1), (1.2) exists in the domain ~ x [O,T) with
var u(.,t) ~ C , 0 ! t < T.
This weak solution satisfies the entropy condition.
A bounded measurable function u : ~ × [O,T) ÷ ~2 is called weak solutiol
of (1.1), (1.2) if
F ~ (u¢ t - f(U)~x)dX dt + ~ U(x)~(x,O)dx : 0
O -~ -~
for all smooth functions ¢ vanishing on the line t : T. A function
n : ~2 ÷ ~ is called entropy, if there exists a function q : ~2 ÷
10
such that for every continuously differentiable solution u of (i.l) the
additional conservation law
n(u(x,t)) t + q(u(x,t)) x = 0
holds. We say that a weak solution u of (1.1), (1.2) satisfies the
entropy condition, if for every convex entropy the relation
nt + qx ~ 0 (1.5)
holds in the distributional sense, cf. [6]. An example for a convex
entropy is the energy density
1 2 ~i
n(Ul,U 2) : ~ u 2 + c(~)dT, q(Ul,U 2) : ~ U 2 q(Ul).
O
Condition C) is necessary and sufficient for the result in theorem I.i
to hold. For, if this condition is not satisfied, then it is possible
to construct initial data U for which a local solution satisfying the
entropy condition does not exist.
Glimm and Lax proved in [4] that a global weak solution of (i.i),
(1.2) exists under essentially the above assumptions for c, provided
that U has bounded variation and, in addition, small oscillation. This
means that U(x+) - U(x-)il must be less than a given small constant for
all x E ~. Recently DiPerna proved in [2] that to all bounded initial
data a global weak solution exists if ~" < 0 for x < 0 and ~" > 0 for
x > O. The proof uses the results on compensated compactness in [7].
This method can be applied if a-priori L -estimates are known, and for
the systems considered by DiPerna these estimates follow from the exist-
ence of invariant subregions. However, under the above assumptions the
system (1.1) does not have invariant subregions, in general. We there-
fore cannot show in this simple way that a-priori estimates hold. In-
stead, for the proof of theorem i.i we proceed as follows. A sequence
{u(n)}n~ of approximate solutions to problem (1.1), (1.2) is construc-
ted out of solutions to the Riemann initial value problem, which entirely
consist out of shock waves. Of course, we have to admit shocks violating
the entropy condition (1.5). However, we construct u (n) such that the
strength of the shocks violating the entropy condition tands to zero for
large n, and approximate in this way rarefaction waves. It will be seen
that the construction is similar to the one given by DiPerna in [I]. We
prove that these approximate solutions are uniformly of bounded varia-
tion, and therefore contain a subsequence converging to a solution of
(1.1), (i.2). The proof of the theorem is too long to reproduce it here
completely. So we only give a sketch. ~n section 2 we introduce special
solutions of the Riemann initial value problem and define the approxi-
mating sequence, and in section 3 we give some of the essential steps