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On the quasilinear wave equation: utt − Δu + f(u, ut) = 0 associated with a mixed nonhomogeneous condition
Nonlrneor Anolysrs, Theory, Merhods & App,pl;mfions, Vol. 19, No. 7. pp. 613-623, Printed in Great Britam.
0362-546X192 $5.00 + 00 4:~’1992 Pergamon Press Ltd
1992.
ON THE QUASILINEAR WAVE EQUATION: u,, - Au + f(u, u,) = 0 ASSOCIATED WITH A MIXED NONHOMOGENEOUS CONDITION NGUYEN THANH LONG? and ALAIN PHAM Ncoc DINH$ t Ecole Polytechnique, Dkpartement de Mathematiques, Ho-Chi-Minh Ville, Vietnam and $ DCpartement de Mathkmatiques, Universite d’OrlCans, BP 6759, 45067 Orleans Cedex, France (Received
15 October
1990; received in revised form
Key words and phrases:
12 September
1991; received for publication
8 November
1991)
Nonlinear Volterra integral equation, local and global existence.
1. INTRODUCTION WE STUDY the following
initial
and boundary
value (i.b.v.)
utt - Au + f(u, u,) = 0;
problem:
O
(1.1)
h constant
u,(O, t) - hu(O, t) = g(t);
> 0 (1.2)
u(1, t) = 0
4x, 0) = u,(x); In [3], Ficken equation
and
Fleishman
4(x, 0) = u,(x).
established
unique
global
llXX- u,, - 2cyiu, - (YOU= eu3 + b, Rabinowitz
[5] has proved
the existence
of periodic
solutions
u,, - UXX+ 2cYiu, =
(1.3)
existence
and
stability
E > 0 small.
for the (1.4)
for
&f,
(1.5)
where E is a small parameter and f is periodic in time. Also relevant to our considerations is the paper by Caughey and Ellison [ 11, in which a unified approach to these previous cases is presented to discuss the existence, uniqueness and asymptotic stability of classical solutions for a class of nonlinear continuous dynamical systems. In [2], Ang and Pham established unique global existence for the i.b.v. problem (l.l)-(1.3) with f(u,u,)= ju,la sgn(u,), (0 < CY< 1) and h = 0. In this latter case this problem governs the motion of a linear viscoelastic bar. The aim of this paper is to give a generalization of [2] and to prove for convenience assumptions on f a local existence theorem for h > 0. If we call u,,(x, t) such a solution, this solution depends continuously on the parameter h and we give sufficient conditions for global existence. 2. EXISTENCE
THEOREM
We first set some notations: Lq = L4(sz),
N’ = ffyn),
fi = (0, I),
Q = C2 x (0, T)
where H’ is the usual Sobolev space on 0. Let ( *, * > denote either the L* inner product or the pairing of a continuous linear functional with an element of a function space. The L2 norm will 613
614
NGUYEN THANH LONG and A. PHAM NGOC DINH
be denoted We denote
by I/ * 11.Let II - Jlx b e a norm on a Banach space X and let X* be its dual space. by Lp(O, T; X), 1 i p _ < co, the space of real functions f: (0, T) + X with
ilfIILpcO,T;Xj = (~oTiifWkGdr>‘lp < 03
for 1 5 P < ~0
Ilf IIL-(O,T$)= e;; ;.tp llf(~)llx
forp
= co.
Let V = {V-E H’ / v(l) = 0)
‘I a(u, u) = = V is a subvector
III 0
au/ax.
space of H’ and on I/ (a(v, v))“‘~ is a norm
au/ax do.
(2.1)
equivalent
to llvllH~. We will set
II4H1 = (4v, 4Y2. We shall make the following
assumptions: O
uo g E
E
(2.2)
H’,
ff’(O, 7-1
u1
(2.3)
EL2
vT>O;
(2.4)
g(0) exists
f: R2 + R satisfies the following conditions: (i) f(0, 0) = 0 (ii) (f(u, ~7) - f(4 3)) * (u? - @I 2 0 v 4 v, U, E R there exists 01,/3 constants
(2.5)
0
0 < 01 < 1,
(2.6)
1
nondecreasing and two functions B, , B,: R, + R, continuous, (i) B,(luj) E L2’(‘-“) (Q), vu E L”(0, T; V), v T > 0
and such that:
E L'(Q) v v?E L2(Q) v) - f(u, @)I5 B,(ld)b - @I”,vu, P, ulE R (iv) If@, u?) - f(& @I 5 B2(ld)lu - tile, vu, 6, v,E R. We write u’(t) := adat = u,; u”(t) := a2u/at2 = utf. We then have the following (ii) B2(ld)
(iii) If@,
1. Let (2.3)-(2.6)hold. Then there exists T > 0 such (1 .l)-( 1.3) has at least a weak solution u on (0, T) such that THEOREM
u E L”(0, T; I’), If p = 1, then the problem Proof.
The proof
Step 1. The Galerkin
consists
by the eigenfunctions
the i.b.v.
u, E L”(0, T; L’).
has exactly one solution.
of several steps.
approximation.
Wj(X) = J2/(1 formed
(1 .I)-(1.3)
that
+ /1/‘) .
Consider COS
a special orthonormal
(JLJX))
of the Laplacian
Aj
A.
=
(2j - 1)X/2,
basis on V’. jEN*
theorem. problem
Quasilinear
(wt , w2, . . . , w,) be generated
Let the subspace of I/. Put
u,(t)
=
615
wave equation
by the distinct
i
w, , w2, . . . , w,
basis elements
(2.7)
5j,(flwj>
j=l
where tj,, satisfy the following
wj>
+
a(un(r),
system: wj)
+
thu,tO,
t,
+
gCt))
.
wj(“)
+
(f(“n(t)3
uA(t))f
j
u,(O) = %,1 =
~ j=l
~j~
Wj
j
UO
in H’
+
u1
in L2.
wj>
=
=
O3
1,2, . . ..n (2.8)
II
u;(o) =
Uln
=
C PjnWj j=l
It follows from the hypotheses of the theorem that system (2.8) has a solution interval [0, T,]. The following estimates allow one to make T, = T for all n. Step 2. A priori estimates. We then have
Multiply
+ d/dt S,(t)
each equation
+ (f(u,,(t),
= -g(t)u;(O, S,(f) By assumptions to t we have:
in (2.8) by
u;(t))
0 - (f(u,(t),
where = IlUt)l12
+
on an
- f(u,(t),
0), u;(t)> (2.9)
O), u;(t)>
11%1(~)11: + ~l~,(O9 o12.
(2.4) and (2.5) and after integrating
(2.10)
with respect to the time variable
from 0
‘I S, 5 S,(O) + 2g(ON,,(O) - 2 ”
- %(t)u,(O,
0 + 2
I0 I_
g’(W,(O,
4 dt (2.11)
01, u,‘Js)) ds.
Since -u(x, t) = ji u&s, t) dt for every u E I/, then the embedding can be taken as 1. Thus, from (2.11) we get
constant
Co for V G e’(Q)
-1 s,(t)
5 g(t) +
I -0
%,(@ dt + 4 “I 11.%(7), !0
O)ll
. iI&,,d
dt
(2.12)
where
C, being the constant
g(t) = 2C, + 4g2(t) + 4 “’ ]g’(r)l’ds. ! ,O defined by S,(O) +
In deriving
21g(0) * ~OAW5
c,.
(2.1 l), we have made use of the inequality
2ab 5 a2/cr + cxb2
for a, b 2 0
V n! >
0.
(2.13)
616
NGUYEN THANH LONG and A. PHAM Ncor
Hypothesis
(2.6iv) implies
DINH
that
l.W#),
0)l 5 &(O)l~,~(W
(2.14)
5 &(0)IWW2.
Since H’(0, T) G C’[O, T] the function g(t) is bounded a.e. on [0, T] by a constant depending on T. Therefore it follows from (2.12) and (2.14) that S,,(r) I M, + “&S,(r)) ,,I 0
Ost<7;,sT
dr,
M,
(2.15)
where k”(s) = s + 4B2(0)sl+3’2. The function
&s) is nondecreasing s,(t)
for s 2 0, hence v t E [0, T] for each T > 0.
5 s(t)
S(t) being the maximal solution of the nonlinear Volterra kernel [4] on an interval [0, T], equation given by S(r) = M, Now we need an estimate
integral
equation
(2.16) with nondecreasing
I“&S(r))dr.
+
(2.17)
on
I”
s12ds.
lu;(O,
.O
lj,(t)
The coefficient
of u,(t) satisfies
the system of ordinary
=
=
p/n.
to the following:
~j~ COS(A,t)
’
(2.18)
ajn
rJx3 =
rj,(t)
equations
-141wjI12[+ thunto,f) + g(t)) . wj(“)l ljn(O)
System (2.18) is equivalent
differential
+ @jn(Sin(Itjt)/Jj)
I”
(sin(Aj(t
-
-
1//l~jl(2
U;(T)),
$/3Lj)[(f(Un(f),
wj)
+ (hu,(O, T) + g(r))Wj(O)] ds.
(2.19)
.O
Put K,,(f) = c
sin JLJt/jLJ
(2.20)
j=l
Y,(t) = i
j=l -
wjm J2
i j=l
CYJn COS(~jt)
+ /Ij,(Sill(2jt)/Itj)]
i’ (sin Aj)(f - r)/Ajzj(f(u,(r),
u;(r)),
W,/I(WjII> ds
(2.21)
.O
d,(l)
= 2h
K,(t
-
r)u,(O, t) dr.
(2.22)
Quasilinear
Then ~~(0, f) can be rewritten
as
~~(0, t) = y,(t) We shall require
the following
LEMMA 1. There Cz such that
exists a positive
~1 tvMi2
617
wave equation
+ s,(t)
(2.23)
K,,(t - r)g(r) dr.
- 2
lemmas. continuous
dr 5 G + o(t) 1:
D(t) independent
function
of n and a constant
v t E [0, T] for any T > 0.
lIf(u,(7),~~(5))ll~d~
(2.24)
i
The proof
of lemma
LEMMA 2.
1 can be found
,af ns
in [2].
2
K;(s
-
T)g(T)
dr
v t E [0, T] for any T > 0
ds 5 A4;
(2.25)
.0 1 II..o M: indicating Proof.
a constant
Integration
depending
on T.
by parts gives ‘>t K,‘,(t - r)g(r) dr = g(O)K,(t) I,I 0
I’ f K,(t
+
(2.26)
- r)g’(t) dr.
1
Next, by (2.26) ‘1 ‘s K;(s
- r)g(r)dr,‘ds
5 21;K&9)d0Y
,g2(0)
+ Tj:g”(i)dr].
(2.27)
.0 ! I!.0 Lemma
2 is proved
since K,(t)
converges
strongly
to a function
K(t) in L2(0, T) for any T > 0.
LEMMA 3. There exists two constants M$ and M+ depending on T such that sf ”t V t E [0, T], for any T > 0. IdA( dr 5 M; + h2TM; (u;(O, r)12 dr ! I .0 .O Proof.
By (2.22) we get 6;(t)
From
= 2hu,,(O)K,,(t)
(2.29) we easily derive,
+ 2h
K,(t
- r)u;(O, t) dr.
(2.29)
+ T,[; (u;(O, r)/‘drj.
(2.30)
’ ju;(O, r)12 dr
(2.31)
after some rearrangements,
” (S;(s)(2 ds I 8h2 ,~K:(B)d& I ! .0 Finally
(2.28)
[u&(O)
we obtain ’ lS;(s)/2 ds 5 8C, hM; i ,0
+ 8h2TM; .0 i
since h&(O) I S,(O) I C, V n and ScKi(@ d0 5 M: constant ready to state an a priori estimate for S,’ luA(O, 7)12 ds.
depending
on T. Now we are
NGCJYENTHANH LONG and
618
PHAM NGOC DINH
A.
LEMMA 4. There exists T > 0 such that ji’ lu;(O, s)12 dr I MS. Proof.
By (2.23) we have uA(O, I) = y:,(t) + s;(t)
(2.32)
- 2 \“K&@ - s)g(r) ds. .lo
From (2.32) we get “f I 3 ” /Y:,(s)12ds + 3 ) I&(s)/2ds II 0 I0
aI I ,0
lu:,(0,s)/2ds
It follows from lemmas l-3 and (2.33) that ,I I( l&(0, s)/‘ds 5 3C2 + 3W) .o
I
I’f I .O
as
II
+ 12 1’ K;(s I 0 .0
T such that 3h’TM:
” l&(0, r)12 dt. \ ,0
by assumption
IlfGM)~ UO)l12 Note that
(2.35)
(2.6) we have
5 W(Il~,(f)llv)
* ll~:,(m~~“(*, + 2~2mwl/$~.
(2.36)
I/. IIL~crcRj 5 1)* Ilt(i(n,. Hence IlfoM),
Finally
(2.34)
5 4, then (2.34) yields
” I&,(0, s)12 ds S M; + 6D(f) [’ 11 f(u,,(T),uL(t))l12 ds. I I0 ,O On the other hand,
(2.33)
llf(U4, &(~)ll~ ds
f 3M; + 12M; + 3h’TM; If we choose
- s)g(r) dr 2 ds.
4W)l12 5 mJwww
from (2.35) and (2.37) it follows
process.
(2.37)
that there exists T > 0 such that
“l~;~(0, ,,\ 0 Step 3. The limiting (u,) such that
+ wm9?a
t)12 dr 5 M;.
By (2.16), (2.37) and (2.38) (u,) has a subsequence
(2.38) still denoted
u, --* u
in L”(0, T; V) weak*
(2.39)
U:, + U’
in L”(0, T; L2) weak*
(2.40)
&(O, 0 + u(O, f)
in L”(0, T) weak*
(2.41)
Uk(O, t) + U’(0, t)
in L2(0, T) weak
(2.42)
in L”(0, T; L2) weak*.
(2.43)
f@,> UA)+ x
By (2.16) and (2.38) on the one hand, and by (2.39) and (2.40) on the other, we can extract from {u,) a subsequence still denoted by (u,) such that u,(O, r) + u(O, 0 u, + U
uniformly strongly
in CO([O, T]) in L2(Q).
(2.44) (2.45)
619
Quasilinear wave equation If we pass to the limit in the equation (2.8) we find without (2.43) and (2.44) that u(t) satisfies the equation d/dt(u’(t),
u> + a(u(t),
u) + (hu(0, t) + g(0)@)
+ Q(t),
Since U, U, E C’(O, T; L’), we have u,(O) + u(0) strongly
difficulty
from (2.39),
(2.40),
for any u E V. (2.46)
U> = 0,
in L’. Thus, (2.47)
U(0) = I_&). On the other hand, the functions (u;(t), wj) and (u’(r), (u;(O) - u’(O), wj> + 0 for n + co. Hence, U’(0) = We shall now require
the following
LEMMA 5. Let u be the solution
Ur
wj) belong
to C’(O, T). Therefore,
(2.48)
.
lemma.
of the following
problem
UU -Au+X=O u,(O, 0 - WO,
U(0) = U(J;
(2.49)
U(1, I) = 0
f) = g(t);
4(O) = UI
with u E L”(0, T; V) and U, E L”(0, T; L’), then we have $l]tlf(t)~12 + +IIu(t)II$ + i’ (hu(O,T) + g(s))Ut(O, 5)dr + [’ (X(T), U,(T)) dr I,0 .I0 1 The proof
+A2 + tll&.
of lemma
(2.50)
5 can be found
in [2].
Remark.
If u. = U, = 0 there is equality Next, we claim that
in (2.50). (2.5 1)
x = f(u, u,). It follows
from (2.8) that
”f I ,O
u;(r)),
U;(T)) dT = ill~~n~~2 + tll%,ll; + th&(O)
By lemma
-
- +lb;(f)il’
-
[kbJ:@. .O
T) dT
+llMllF - w47(0, f)12.
(2.52)
5 we have
]im_szp ,‘I (.IGM),
!
u;(r)),
G(r))
dr 5
- *hz2(0,t) HullI + +llu& + +hu,2(0) ‘f
_
!
g(@u’@,r>dT - +llu(t>ll$ - +llu’(t)l12
.O
(X(T), u’(T))
dT
a.e. t E (0, T).
(2.53)
NGUYEN THANH LONG and A. PHAM NGOC DINH
620
By (2.45) (2.54)
a.e. in Q.
u,, + u By (2.6) and (2.54) we get
f(u, 16) + AU>4) Hence,
by the dominated
convergence
(2.55)
a.e. in Q, V C$E L’(Q).
theorem,
we obtain (2.56)
From
(2.56) we derive that ”
lim
”
I n-m .,o
(f(u,(t),
ad),
U;(T) - 6(r)) ch =
r
I .O
4441, U’(T) - +(d) dr
v 4 E L2(Q). (2.57)
Next, consider
It follows
from (2.43), (2.53), (2.57) and (2.58) that
#(r)), u’(r) - cb(r)> dr.
(2.59)
L’(Q).
(2.60)
0 5 limsupX,(r) n-m
ZS
<,: (X(r) - f(u(r), i
4(r)), u’(r) - 4(r)) dr 2 0
/ .O
(X(r) - f(u(r),
Thus
Let d(t) = u’(t) - lw(t),
/1 > 0, w E L’(Q).
V ~5 E
Then we obtain
‘I
I
(X(T) - f(U(T), U’(T) - AW(T)),W(T)))dt 2 0
v w
E
L'(Q).
(2.61)
.O
We have
‘f lim x-0,
(f(u(r),
U’(r) - A”‘(r)) - f(u(r),
u’(r)), W(r)) dr = 0
(2.62)
since Il&, Finally,
it follows
u, - Aw) - f(‘,
u,)lI&(2) S
(2.63)
‘allB,(1UI)IIL2’(1-0)tQ,Ilwll~Z~Q,.
from (2.61) and (2.62) that f
(X(T) - f(U(T), U’(d), W(T)> dT 2 0
v w
i .O Therefore x = f(u, u,)
a.e. in Q, as claimed.
E
L2(Q).
(2.64)
Quasilinear
621
wave equation
Uniqueness
Assume now that p = 1 in (2.6). Let ur, u2 be two solutions of the i.b.v. problem and let u = ur - u2. Then u is the solution of the following i.b.v. problem Utt -Au+x=O,
o
1,
(1.1)~(1.3)
O
hu(0, t) = 0
(2.65) u(1, t) = u(0) = u,(O) = 0 u E L”(0,
By using lemma
T; I’);
5 with u. =
311~‘(t)l12 + tllw>ll2,+h
u, E L”(0, u1
=
T; L’);
2 = _I-(4 9 u;) - f(uz 9 W.
g(t) = 0, we have
‘f
t
(J?(r), u'(7))dr = 0 I ~(0, s)u'(O,7)dr + JO Jo
a.e. t E (0, T). (2.66)
By (2.66)
lIu’(t)l12 + Mm + hu2(0, T) 5 2 since the function
f(ur , -) is monotone. lI.m, 94)
li.f(~1(7>, U;(T))- _f(U7),ui(Q1ll llu’(7)lidr .O I (2.67)
From the hypothesis
(2.6) we deduce that
- .mz 3G)ll 5 Il~z(l& I)11* Ilull v.
(2.68)
Let a(t) By (2.67)-(2.69)
it follows
= Ilu’(t)l12 + Ilu(t)112, + hu?O,
that
II&(Iu;(d)ll~(7)d7
o(t) 5 2 i.e. o = 0 by Gronwall’s
(2.69)
0.
(2.70)
lemma.
3. CONTINUOUS
DEPENDENCE
OF THE
SOLUTION
In this part we assume that j3 = 1. The problem (1 . l)-( 1.3) according to theorem 1 admits a unique solution u(x, t) = u,(x, t), h > 0. We make the following supplementary assumption on the function B,(e): B,: L2 + L2, takes
Then we have the following
bounded
sets into bounded
sets.
(3.1)
theorem.
2. Let /3 = 1. Let (2.3)-(2.6) and (3.1) hold. Then there exists T > 0 such that the i.b.v. problem (l.l)-(1.3) with h = 0 has a unique solution C E L”(0, T; V) and such that Qt E L”(0, T; L2). Furthermore THEOREM
lim(Ilh -
h-0
&~O,~o
+ II4
- ~‘lIL-cO,T;L~j) = 0.
NGUYEN THANH LONG and A. PHAM NGOC DINH
622
Proof. Let u,, (resp. u,,) be the solution of the i.b.v. h(resp. h’). Let w = u,, - uh,. Then w satisfies
problem
(1 .l)-(1.3)
with the parameter
O
Wrt -Aw+%,=O
w,(O, I) = h . w(0, t) +
hz+(O,t) (3.2)
w( 1) f) = w(x, 0) = w,(x, 0) = 0
w, E L”(0, T: L”)
w E L”(0, T; V); where
x, =f(u,
4) - f(u, 4) (3.3)
t’i = h - h'. Proceeding
as in the proof
of theorem
u (resp. u,) is bounded
1, we deduce
independently
of h in L”(0, T; V) [resp. in L”(0, T; L’)].
Let o(t) = IIw’(t)l12 + As before
we can derive the following a(t) I
Assumptions
that (3.4)
Ilw(dF.
inequality:
c,lhl +
I
” /1~2(lG,4~)l)lldd
(3.5)
dr.
(3.1) and (3.4) yield that lI~,tl~~4~)I)II
5 CA constant
Next, by (3.5) and (3.6) and Gronwall’s a(t) 5 C+li
+ C:
depending
only on T.
(3.6)
lemma,
)I a(s)dr 5 C’$/$ I .0
for any t E [0, T].
(3.7)
5 C;lh
(3.8)
Thus lluh - ~,&o,~;V)+l14 Denote
by W the Banach
- G/~&,,T;Lz)
- h’l.
space W = (u E L”(0, T; V)lu, E L-(0, T; L’)]
(3.9)
with the norm II4Il.v = (Il4Zm(0,7;Y)
+ II&(O,r;&.
Let (h,]be a sequence such that h, > 0, h, -+ 0 as n + 00. It follows Cauchy sequence in W. Thus there exists u” E W such that uh,
By passing to the limits the equation d/dt(W,,
+
u
from (3.8) that (u,,?) is a
in W strongly.
such as in the proof
of theorem
u) + a(fi(t), u) + g(t)u(O)+ (f(u”,tit), u> = 0
(3.10) 1, we conclude
that
~2 satisfies
for any u E V’, a.e. t E (0, T) (3.11)
623
Quasilinear wave equation
and the initial
conditions G(O) =
240
u”,(O) =
241.
(3.12) Uniqueness is proved h’ -+ 0 in (3.8) gives
in a standard
manner,
11% - G=,(O,T;V) +
such as in the proof
of theorem
II& - Q’llsyO,T;L2) 22a.
1. Hence,
m
letting (3.13)
THEOREM 3. Let /3 = 1. Let (2.3)-(2.6) and (3.1) hold. (i) Let h = 0. Then for each T > 0, the i.b.v. problem
(l.l)-(1.3) has a unique solution U* E L-(0, T; V) such that u*’ E L”(0, T; L2). (ii) v E > 0, v T > 0, there exists h = h(&, T) > 0 such that if 0 < h < K then the i.b.v. problem (1 . l)-( 1.3) has a unique solution uh such that u;, E L”(0,
uh E L”(0, T; V),
T; L2)
and satisfying (3.14)
Proof (if following
In the case h = 0 we proceed a priori bounds:
as in the proof
11~~(~)112+ ~l~~(~)ll~~5 MT i‘T
lu,(O,
t)12dt i Mr
of theorem
1 and we derive
the
vr~[O,T],vT>0
(3.15)
v f E [0, T], v T > 0.
(3.16)
\ ,0 Then we can prove in a similar manner, such as in the proof of theorem 1, that the limit ZP of the sequence lu,] defined by (2.8) satisfies equation (1.1) associated with the boundary conditions ~~(0, t) = g(t), ~(1, t) = 0. (ii) In the proof of theorem 1, if we choose k = Min(e/C:, (M; defined
lld6T.
in (2.28) and C+ in (3.13)),
VE>O,VT>O
M:)
then for 0 < h < k we get the following:
I/% - ~*GmfO,T;V) + llut, - ~*‘l&o,T;L2> The proof
(3.17)
< 8.
(3.18)
is complete.
Acknowledgemenl-The
authors would like to thank the referee
for
his corrections and suggestions.
REFERENCES T. & ELLISONJ., Existence, uniqueness and stability of solutions of a class of nonlinear differential equations, J. m&h. Analysis Appiic. 51, 1-32 (1975). 2. DANG D. ANG & PHAM N. DINH, Mixed problem for semi-linear wave equation with a nonhomogeneous condition, I.
CAUCHEY
Nonlinear Analysis 12, 581-592 (1988).
3.
FICKEN F. & FLEISHMANB.,
Initial value problems and time periodic solutions for a nonlinear wave equation,
Communspure
appl. Math. 10, 331-356 (i957). V. 81 LEELAS.. Differential and Intenrul heauaiities. 4. LAKSHMIKANTHAM
5.
RABINOWITZP.
H., Periodic solutions of nonlinear hyperbolichifferential 20, 145-205 (1967).
Vol. I. Academic Press, New York (1969). equations, Communs pure app/. Mark
0362-546X192 $5.00 + 00 4:~’1992 Pergamon Press Ltd
1992.
ON THE QUASILINEAR WAVE EQUATION: u,, - Au + f(u, u,) = 0 ASSOCIATED WITH A MIXED NONHOMOGENEOUS CONDITION NGUYEN THANH LONG? and ALAIN PHAM Ncoc DINH$ t Ecole Polytechnique, Dkpartement de Mathematiques, Ho-Chi-Minh Ville, Vietnam and $ DCpartement de Mathkmatiques, Universite d’OrlCans, BP 6759, 45067 Orleans Cedex, France (Received
15 October
1990; received in revised form
Key words and phrases:
12 September
1991; received for publication
8 November
1991)
Nonlinear Volterra integral equation, local and global existence.
1. INTRODUCTION WE STUDY the following
initial
and boundary
value (i.b.v.)
utt - Au + f(u, u,) = 0;
problem:
O
(1.1)
h constant
u,(O, t) - hu(O, t) = g(t);
> 0 (1.2)
u(1, t) = 0
4x, 0) = u,(x); In [3], Ficken equation
and
Fleishman
4(x, 0) = u,(x).
established
unique
global
llXX- u,, - 2cyiu, - (YOU= eu3 + b, Rabinowitz
[5] has proved
the existence
of periodic
solutions
u,, - UXX+ 2cYiu, =
(1.3)
existence
and
stability
E > 0 small.
for the (1.4)
for
&f,
(1.5)
where E is a small parameter and f is periodic in time. Also relevant to our considerations is the paper by Caughey and Ellison [ 11, in which a unified approach to these previous cases is presented to discuss the existence, uniqueness and asymptotic stability of classical solutions for a class of nonlinear continuous dynamical systems. In [2], Ang and Pham established unique global existence for the i.b.v. problem (l.l)-(1.3) with f(u,u,)= ju,la sgn(u,), (0 < CY< 1) and h = 0. In this latter case this problem governs the motion of a linear viscoelastic bar. The aim of this paper is to give a generalization of [2] and to prove for convenience assumptions on f a local existence theorem for h > 0. If we call u,,(x, t) such a solution, this solution depends continuously on the parameter h and we give sufficient conditions for global existence. 2. EXISTENCE
THEOREM
We first set some notations: Lq = L4(sz),
N’ = ffyn),
fi = (0, I),
Q = C2 x (0, T)
where H’ is the usual Sobolev space on 0. Let ( *, * > denote either the L* inner product or the pairing of a continuous linear functional with an element of a function space. The L2 norm will 613
614
NGUYEN THANH LONG and A. PHAM NGOC DINH
be denoted We denote
by I/ * 11.Let II - Jlx b e a norm on a Banach space X and let X* be its dual space. by Lp(O, T; X), 1 i p _ < co, the space of real functions f: (0, T) + X with
ilfIILpcO,T;Xj = (~oTiifWkGdr>‘lp < 03
for 1 5 P < ~0
Ilf IIL-(O,T$)= e;; ;.tp llf(~)llx
forp
= co.
Let V = {V-E H’ / v(l) = 0)
‘I a(u, u) =
III 0
au/ax.
space of H’ and on I/ (a(v, v))“‘~ is a norm
au/ax do.
(2.1)
equivalent
to llvllH~. We will set
II4H1 = (4v, 4Y2. We shall make the following
assumptions: O
uo g E
E
(2.2)
H’,
ff’(O, 7-1
u1
(2.3)
EL2
vT>O;
(2.4)
g(0) exists
f: R2 + R satisfies the following conditions: (i) f(0, 0) = 0 (ii) (f(u, ~7) - f(4 3)) * (u? - @I 2 0 v 4 v, U, E R there exists 01,/3 constants
(2.5)
0
0 < 01 < 1,
(2.6)
1
nondecreasing and two functions B, , B,: R, + R, continuous, (i) B,(luj) E L2’(‘-“) (Q), vu E L”(0, T; V), v T > 0
and such that:
E L'(Q) v v?E L2(Q) v) - f(u, @)I5 B,(ld)b - @I”,vu, P, ulE R (iv) If@, u?) - f(& @I 5 B2(ld)lu - tile, vu, 6, v,E R. We write u’(t) := adat = u,; u”(t) := a2u/at2 = utf. We then have the following (ii) B2(ld)
(iii) If@,
1. Let (2.3)-(2.6)hold. Then there exists T > 0 such (1 .l)-( 1.3) has at least a weak solution u on (0, T) such that THEOREM
u E L”(0, T; I’), If p = 1, then the problem Proof.
The proof
Step 1. The Galerkin
consists
by the eigenfunctions
the i.b.v.
u, E L”(0, T; L’).
has exactly one solution.
of several steps.
approximation.
Wj(X) = J2/(1 formed
(1 .I)-(1.3)
that
+ /1/‘) .
Consider COS
a special orthonormal
(JLJX))
of the Laplacian
Aj
A.
=
(2j - 1)X/2,
basis on V’. jEN*
theorem. problem
Quasilinear
(wt , w2, . . . , w,) be generated
Let the subspace of I/. Put
u,(t)
=
615
wave equation
by the distinct
i
w, , w2, . . . , w,
basis elements
(2.7)
5j,(flwj>
j=l
where tj,, satisfy the following
wj>
+
a(un(r),
system: wj)
+
thu,tO,
t,
+
gCt))
.
wj(“)
+
(f(“n(t)3
uA(t))f
j
u,(O) = %,1 =
~ j=l
~j~
Wj
j
UO
in H’
+
u1
in L2.
wj>
=
=
O3
1,2, . . ..n (2.8)
II
u;(o) =
Uln
=
C PjnWj j=l
It follows from the hypotheses of the theorem that system (2.8) has a solution interval [0, T,]. The following estimates allow one to make T, = T for all n. Step 2. A priori estimates. We then have
Multiply
+ d/dt S,(t)
each equation
+ (f(u,,(t),
= -g(t)u;(O, S,(f) By assumptions to t we have:
in (2.8) by
u;(t))
0 - (f(u,(t),
where = IlUt)l12
+
on an
- f(u,(t),
0), u;(t)> (2.9)
O), u;(t)>
11%1(~)11: + ~l~,(O9 o12.
(2.4) and (2.5) and after integrating
(2.10)
with respect to the time variable
from 0
‘I S, 5 S,(O) + 2g(ON,,(O) - 2 ”
- %(t)u,(O,
0 + 2
I0 I_
g’(W,(O,
4 dt (2.11)
01, u,‘Js)) ds.
Since -u(x, t) = ji u&s, t) dt for every u E I/, then the embedding can be taken as 1. Thus, from (2.11) we get
constant
Co for V G e’(Q)
-1 s,(t)
5 g(t) +
I -0
%,(@ dt + 4 “I 11.%(7), !0
O)ll
. iI&,,d
dt
(2.12)
where
C, being the constant
g(t) = 2C, + 4g2(t) + 4 “’ ]g’(r)l’ds. ! ,O defined by S,(O) +
In deriving
21g(0) * ~OAW5
c,.
(2.1 l), we have made use of the inequality
2ab 5 a2/cr + cxb2
for a, b 2 0
V n! >
0.
(2.13)
616
NGUYEN THANH LONG and A. PHAM Ncor
Hypothesis
(2.6iv) implies
DINH
that
l.W#),
0)l 5 &(O)l~,~(W
(2.14)
5 &(0)IWW2.
Since H’(0, T) G C’[O, T] the function g(t) is bounded a.e. on [0, T] by a constant depending on T. Therefore it follows from (2.12) and (2.14) that S,,(r) I M, + “&S,(r)) ,,I 0
Ost<7;,sT
dr,
M,
(2.15)
where k”(s) = s + 4B2(0)sl+3’2. The function
&s) is nondecreasing s,(t)
for s 2 0, hence v t E [0, T] for each T > 0.
5 s(t)
S(t) being the maximal solution of the nonlinear Volterra kernel [4] on an interval [0, T], equation given by S(r) = M, Now we need an estimate
integral
equation
(2.16) with nondecreasing
I“&S(r))dr.
+
(2.17)
on
I”
s12ds.
lu;(O,
.O
lj,(t)
The coefficient
of u,(t) satisfies
the system of ordinary
=
=
p/n.
to the following:
~j~ COS(A,t)
’
(2.18)
ajn
rJx3 =
rj,(t)
equations
-141wjI12[
System (2.18) is equivalent
differential
+ @jn(Sin(Itjt)/Jj)
I”
(sin(Aj(t
-
-
1//l~jl(2
U;(T)),
$/3Lj)[(f(Un(f),
wj)
+ (hu,(O, T) + g(r))Wj(O)] ds.
(2.19)
.O
Put K,,(f) = c
sin JLJt/jLJ
(2.20)
j=l
Y,(t) = i
j=l -
wjm J2
i j=l
CYJn COS(~jt)
+ /Ij,(Sill(2jt)/Itj)]
i’ (sin Aj)(f - r)/Ajzj(f(u,(r),
u;(r)),
W,/I(WjII> ds
(2.21)
.O
d,(l)
= 2h
K,(t
-
r)u,(O, t) dr.
(2.22)
Quasilinear
Then ~~(0, f) can be rewritten
as
~~(0, t) = y,(t) We shall require
the following
LEMMA 1. There Cz such that
exists a positive
~1 tvMi2
617
wave equation
+ s,(t)
(2.23)
K,,(t - r)g(r) dr.
- 2
lemmas. continuous
dr 5 G + o(t) 1:
D(t) independent
function
of n and a constant
v t E [0, T] for any T > 0.
lIf(u,(7),~~(5))ll~d~
(2.24)
i
The proof
of lemma
LEMMA 2.
1 can be found
,af ns
in [2].
2
K;(s
-
T)g(T)
dr
v t E [0, T] for any T > 0
ds 5 A4;
(2.25)
.0 1 II..o M: indicating Proof.
a constant
Integration
depending
on T.
by parts gives ‘>t K,‘,(t - r)g(r) dr = g(O)K,(t) I,I 0
I’ f K,(t
+
(2.26)
- r)g’(t) dr.
1
Next, by (2.26) ‘1 ‘s K;(s
- r)g(r)dr,‘ds
5 21;K&9)d0Y
,g2(0)
+ Tj:g”(i)dr].
(2.27)
.0 ! I!.0 Lemma
2 is proved
since K,(t)
converges
strongly
to a function
K(t) in L2(0, T) for any T > 0.
LEMMA 3. There exists two constants M$ and M+ depending on T such that sf ”t V t E [0, T], for any T > 0. IdA( dr 5 M; + h2TM; (u;(O, r)12 dr ! I .0 .O Proof.
By (2.22) we get 6;(t)
From
= 2hu,,(O)K,,(t)
(2.29) we easily derive,
+ 2h
K,(t
- r)u;(O, t) dr.
(2.29)
+ T,[; (u;(O, r)/‘drj.
(2.30)
’ ju;(O, r)12 dr
(2.31)
after some rearrangements,
” (S;(s)(2 ds I 8h2 ,~K:(B)d& I ! .0 Finally
(2.28)
[u&(O)
we obtain ’ lS;(s)/2 ds 5 8C, hM; i ,0
+ 8h2TM; .0 i
since h&(O) I S,(O) I C, V n and ScKi(@ d0 5 M: constant ready to state an a priori estimate for S,’ luA(O, 7)12 ds.
depending
on T. Now we are
NGCJYENTHANH LONG and
618
PHAM NGOC DINH
A.
LEMMA 4. There exists T > 0 such that ji’ lu;(O, s)12 dr I MS. Proof.
By (2.23) we have uA(O, I) = y:,(t) + s;(t)
(2.32)
- 2 \“K&@ - s)g(r) ds. .lo
From (2.32) we get “f I 3 ” /Y:,(s)12ds + 3 ) I&(s)/2ds II 0 I0
aI I ,0
lu:,(0,s)/2ds
It follows from lemmas l-3 and (2.33) that ,I I( l&(0, s)/‘ds 5 3C2 + 3W) .o
I
I’f I .O
as
II
+ 12 1’ K;(s I 0 .0
T such that 3h’TM:
” l&(0, r)12 dt. \ ,0
by assumption
IlfGM)~ UO)l12 Note that
(2.35)
(2.6) we have
5 W(Il~,(f)llv)
* ll~:,(m~~“(*, + 2~2mwl/$~.
(2.36)
I/. IIL~crcRj 5 1)* Ilt(i(n,. Hence IlfoM),
Finally
(2.34)
5 4, then (2.34) yields
” I&,(0, s)12 ds S M; + 6D(f) [’ 11 f(u,,(T),uL(t))l12 ds. I I0 ,O On the other hand,
(2.33)
llf(U4, &(~)ll~ ds
f 3M; + 12M; + 3h’TM; If we choose
- s)g(r) dr 2 ds.
4W)l12 5 mJwww
from (2.35) and (2.37) it follows
process.
(2.37)
that there exists T > 0 such that
“l~;~(0, ,,\ 0 Step 3. The limiting (u,) such that
+ wm9?a
t)12 dr 5 M;.
By (2.16), (2.37) and (2.38) (u,) has a subsequence
(2.38) still denoted
u, --* u
in L”(0, T; V) weak*
(2.39)
U:, + U’
in L”(0, T; L2) weak*
(2.40)
&(O, 0 + u(O, f)
in L”(0, T) weak*
(2.41)
Uk(O, t) + U’(0, t)
in L2(0, T) weak
(2.42)
in L”(0, T; L2) weak*.
(2.43)
f@,> UA)+ x
By (2.16) and (2.38) on the one hand, and by (2.39) and (2.40) on the other, we can extract from {u,) a subsequence still denoted by (u,) such that u,(O, r) + u(O, 0 u, + U
uniformly strongly
in CO([O, T]) in L2(Q).
(2.44) (2.45)
619
Quasilinear wave equation If we pass to the limit in the equation (2.8) we find without (2.43) and (2.44) that u(t) satisfies the equation d/dt(u’(t),
u> + a(u(t),
u) + (hu(0, t) + g(0)@)
+ Q(t),
Since U, U, E C’(O, T; L’), we have u,(O) + u(0) strongly
difficulty
from (2.39),
(2.40),
for any u E V. (2.46)
U> = 0,
in L’. Thus, (2.47)
U(0) = I_&). On the other hand, the functions (u;(t), wj) and (u’(r), (u;(O) - u’(O), wj> + 0 for n + co. Hence, U’(0) = We shall now require
the following
LEMMA 5. Let u be the solution
Ur
wj) belong
to C’(O, T). Therefore,
(2.48)
.
lemma.
of the following
problem
UU -Au+X=O u,(O, 0 - WO,
U(0) = U(J;
(2.49)
U(1, I) = 0
f) = g(t);
4(O) = UI
with u E L”(0, T; V) and U, E L”(0, T; L’), then we have $l]tlf(t)~12 + +IIu(t)II$ + i’ (hu(O,T) + g(s))Ut(O, 5)dr + [’ (X(T), U,(T)) dr I,0 .I0 1 The proof
+A2 + tll&.
of lemma
(2.50)
5 can be found
in [2].
Remark.
If u. = U, = 0 there is equality Next, we claim that
in (2.50). (2.5 1)
x = f(u, u,). It follows
from (2.8) that
”f I ,O
U;(T)) dT = ill~~n~~2 + tll%,ll; + th&(O)
By lemma
-
- +lb;(f)il’
-
[kbJ:@. .O
T) dT
+llMllF - w47(0, f)12.
(2.52)
5 we have
]im_szp ,‘I (.IGM),
!
u;(r)),
G(r))
dr 5
- *hz2(0,t) HullI + +llu& + +hu,2(0) ‘f
_
!
g(@u’@,r>dT - +llu(t>ll$ - +llu’(t)l12
.O
(X(T), u’(T))
dT
a.e. t E (0, T).
(2.53)
NGUYEN THANH LONG and A. PHAM NGOC DINH
620
By (2.45) (2.54)
a.e. in Q.
u,, + u By (2.6) and (2.54) we get
f(u, 16) + AU>4) Hence,
by the dominated
convergence
(2.55)
a.e. in Q, V C$E L’(Q).
theorem,
we obtain (2.56)
From
(2.56) we derive that ”
lim
”
I n-m .,o
(f(u,(t),
ad),
U;(T) - 6(r)) ch =
r
I .O
v 4 E L2(Q). (2.57)
Next, consider
It follows
from (2.43), (2.53), (2.57) and (2.58) that
#(r)), u’(r) - cb(r)> dr.
(2.59)
L’(Q).
(2.60)
0 5 limsupX,(r) n-m
ZS
<,: (X(r) - f(u(r), i
4(r)), u’(r) - 4(r)) dr 2 0
/ .O
(X(r) - f(u(r),
Thus
Let d(t) = u’(t) - lw(t),
/1 > 0, w E L’(Q).
V ~5 E
Then we obtain
‘I
I
(X(T) - f(U(T), U’(T) - AW(T)),W(T)))dt 2 0
v w
E
L'(Q).
(2.61)
.O
We have
‘f lim x-0,
(f(u(r),
U’(r) - A”‘(r)) - f(u(r),
u’(r)), W(r)) dr = 0
(2.62)
since Il&, Finally,
it follows
u, - Aw) - f(‘,
u,)lI&(2) S
(2.63)
‘allB,(1UI)IIL2’(1-0)tQ,Ilwll~Z~Q,.
from (2.61) and (2.62) that f
(X(T) - f(U(T), U’(d), W(T)> dT 2 0
v w
i .O Therefore x = f(u, u,)
a.e. in Q, as claimed.
E
L2(Q).
(2.64)
Quasilinear
621
wave equation
Uniqueness
Assume now that p = 1 in (2.6). Let ur, u2 be two solutions of the i.b.v. problem and let u = ur - u2. Then u is the solution of the following i.b.v. problem Utt -Au+x=O,
o
1,
(1.1)~(1.3)
O
hu(0, t) = 0
(2.65) u(1, t) = u(0) = u,(O) = 0 u E L”(0,
By using lemma
T; I’);
5 with u. =
311~‘(t)l12 + tllw>ll2,+h
u, E L”(0, u1
=
T; L’);
2 = _I-(4 9 u;) - f(uz 9 W.
g(t) = 0, we have
‘f
t
(J?(r), u'(7))dr = 0 I ~(0, s)u'(O,7)dr + JO Jo
a.e. t E (0, T). (2.66)
By (2.66)
lIu’(t)l12 + Mm + hu2(0, T) 5 2 since the function
f(ur , -) is monotone. lI.m, 94)
li.f(~1(7>, U;(T))- _f(U7),ui(Q1ll llu’(7)lidr .O I (2.67)
From the hypothesis
(2.6) we deduce that
- .mz 3G)ll 5 Il~z(l& I)11* Ilull v.
(2.68)
Let a(t) By (2.67)-(2.69)
it follows
= Ilu’(t)l12 + Ilu(t)112, + hu?O,
that
II&(Iu;(d)ll~(7)d7
o(t) 5 2 i.e. o = 0 by Gronwall’s
(2.69)
0.
(2.70)
lemma.
3. CONTINUOUS
DEPENDENCE
OF THE
SOLUTION
In this part we assume that j3 = 1. The problem (1 . l)-( 1.3) according to theorem 1 admits a unique solution u(x, t) = u,(x, t), h > 0. We make the following supplementary assumption on the function B,(e): B,: L2 + L2, takes
Then we have the following
bounded
sets into bounded
sets.
(3.1)
theorem.
2. Let /3 = 1. Let (2.3)-(2.6) and (3.1) hold. Then there exists T > 0 such that the i.b.v. problem (l.l)-(1.3) with h = 0 has a unique solution C E L”(0, T; V) and such that Qt E L”(0, T; L2). Furthermore THEOREM
lim(Ilh -
h-0
&~O,~o
+ II4
- ~‘lIL-cO,T;L~j) = 0.
NGUYEN THANH LONG and A. PHAM NGOC DINH
622
Proof. Let u,, (resp. u,,) be the solution of the i.b.v. h(resp. h’). Let w = u,, - uh,. Then w satisfies
problem
(1 .l)-(1.3)
with the parameter
O
Wrt -Aw+%,=O
w,(O, I) = h . w(0, t) +
hz+(O,t) (3.2)
w( 1) f) = w(x, 0) = w,(x, 0) = 0
w, E L”(0, T: L”)
w E L”(0, T; V); where
x, =f(u,
4) - f(u, 4) (3.3)
t’i = h - h'. Proceeding
as in the proof
of theorem
u (resp. u,) is bounded
1, we deduce
independently
of h in L”(0, T; V) [resp. in L”(0, T; L’)].
Let o(t) = IIw’(t)l12 + As before
we can derive the following a(t) I
Assumptions
that (3.4)
Ilw(dF.
inequality:
c,lhl +
I
” /1~2(lG,4~)l)lldd
(3.5)
dr.
(3.1) and (3.4) yield that lI~,tl~~4~)I)II
5 CA constant
Next, by (3.5) and (3.6) and Gronwall’s a(t) 5 C+li
+ C:
depending
only on T.
(3.6)
lemma,
)I a(s)dr 5 C’$/$ I .0
for any t E [0, T].
(3.7)
5 C;lh
(3.8)
Thus lluh - ~,&o,~;V)+l14 Denote
by W the Banach
- G/~&,,T;Lz)
- h’l.
space W = (u E L”(0, T; V)lu, E L-(0, T; L’)]
(3.9)
with the norm II4Il.v = (Il4Zm(0,7;Y)
+ II&(O,r;&.
Let (h,]be a sequence such that h, > 0, h, -+ 0 as n + 00. It follows Cauchy sequence in W. Thus there exists u” E W such that uh,
By passing to the limits the equation d/dt(W,,
+
u
from (3.8) that (u,,?) is a
in W strongly.
such as in the proof
of theorem
u) + a(fi(t), u) + g(t)u(O)+ (f(u”,tit), u> = 0
(3.10) 1, we conclude
that
~2 satisfies
for any u E V’, a.e. t E (0, T) (3.11)
623
Quasilinear wave equation
and the initial
conditions G(O) =
240
u”,(O) =
241.
(3.12) Uniqueness is proved h’ -+ 0 in (3.8) gives
in a standard
manner,
11% - G=,(O,T;V) +
such as in the proof
of theorem
II& - Q’llsyO,T;L2) 22a.
1. Hence,
m
letting (3.13)
THEOREM 3. Let /3 = 1. Let (2.3)-(2.6) and (3.1) hold. (i) Let h = 0. Then for each T > 0, the i.b.v. problem
(l.l)-(1.3) has a unique solution U* E L-(0, T; V) such that u*’ E L”(0, T; L2). (ii) v E > 0, v T > 0, there exists h = h(&, T) > 0 such that if 0 < h < K then the i.b.v. problem (1 . l)-( 1.3) has a unique solution uh such that u;, E L”(0,
uh E L”(0, T; V),
T; L2)
and satisfying (3.14)
Proof (if following
In the case h = 0 we proceed a priori bounds:
as in the proof
11~~(~)112+ ~l~~(~)ll~~5 MT i‘T
lu,(O,
t)12dt i Mr
of theorem
1 and we derive
the
vr~[O,T],vT>0
(3.15)
v f E [0, T], v T > 0.
(3.16)
\ ,0 Then we can prove in a similar manner, such as in the proof of theorem 1, that the limit ZP of the sequence lu,] defined by (2.8) satisfies equation (1.1) associated with the boundary conditions ~~(0, t) = g(t), ~(1, t) = 0. (ii) In the proof of theorem 1, if we choose k = Min(e/C:, (M; defined
lld6T.
in (2.28) and C+ in (3.13)),
VE>O,VT>O
M:)
then for 0 < h < k we get the following:
I/% - ~*GmfO,T;V) + llut, - ~*‘l&o,T;L2> The proof
(3.17)
< 8.
(3.18)
is complete.
Acknowledgemenl-The
authors would like to thank the referee
for
his corrections and suggestions.
REFERENCES T. & ELLISONJ., Existence, uniqueness and stability of solutions of a class of nonlinear differential equations, J. m&h. Analysis Appiic. 51, 1-32 (1975). 2. DANG D. ANG & PHAM N. DINH, Mixed problem for semi-linear wave equation with a nonhomogeneous condition, I.
CAUCHEY
Nonlinear Analysis 12, 581-592 (1988).
3.
FICKEN F. & FLEISHMANB.,
Initial value problems and time periodic solutions for a nonlinear wave equation,
Communspure
appl. Math. 10, 331-356 (i957). V. 81 LEELAS.. Differential and Intenrul heauaiities. 4. LAKSHMIKANTHAM
5.
RABINOWITZP.
H., Periodic solutions of nonlinear hyperbolichifferential 20, 145-205 (1967).
Vol. I. Academic Press, New York (1969). equations, Communs pure app/. Mark