Internat. J. Math. & Math. Sci.
VOL. 14 NO. 4 (1991) 731-736
731
REMARKS ON QUASILINEAR EVOLUTIONS EQUATIONS
JAMES E. MUNOZ RIVERA
NATIONAL LABORATORY OF SCIENTIFIC COHPUTATION
RuA LAURO MULLER 55
(LNCC-CNPo) BOTAFOGO. CEP. 22290. Ro DE JANEIRO. R.J.
BRA$1L.
AND
IMUFRJ. P. O. Box B8530. Ro
DE
JANEIRO. R.J.
BRA$1L
(Received June 6, 1990 and in revised form July 21, 1990)
ABSTRACT. In
solutions
with
this
for
i ni ti al
o classzcal
Jtt-aJ-MC$(IVjlzc,cDAjtt
paper we study the existence result
the
quasilinear
data
60D=,
equation
utCOD=u
and
i
homogeneous
boundary
condi ti onE.
KEY WORDS.-
Partial
di ferenti a!
equation,
quasi ! i near
evo!
equati on, boundary problem.
AMS(MO): Subject classl fication, 35B85,
n,
be an open and bounded set of
with smooth
1. INTRODUCTION: Let
boundary F. Let’s denote by O the cylinder
f’x10. T{ and by Z its
lateral boundary. r notations and function sces are standart and
follo the same pattern as Lions’s book [2].
Ebihara et al [I] was proved that there est only one classical
solution for
a selinear
del,
given
by following initial-boundary
value pr oblem
COD
.
COD
u
En
C1 D
J.E. MUNOZ RIVERA
732
when the followlng
CiD
hotheses
CICO,+oa.),
PICA_) E
hold.
arid tkere eZst pos[t)e constants a,
-
p such that
d
ere M
and for 9 we are denoting the do--in of the
e in result of thxs paper is to prove the exxstence
operator M
resu!t of c!assical solutions for ystem C 1.1-C 1.3] when
s.
HI. H is a continuos /urction such that.
HCk9 >_ m
0
>
0
A(l+s)/29,
0
2. THE MAIN RESULT: Let’s
denote by
w and by k,
k the
%
m firts oD(honorZ eigen functions and eigen value of the Laplacan
respectively.
Let’s denote by V
generated
by the fit
operator on V
that iE:
m
It is easy to see that
en
the finite dimensional
function
eigen
and
by
P
vector
the
space
pro3ecLor
=
PmA s in AS). Moreover’we have lhat
ASP
the aproted problem i defined a follow.
utmt
Au(m
MC
nlVu(mIzdx>Au
/m
0
whet e
u(m)(’tD
Before
to
prove
the
E--’m(t_)w.
um
rezult
of
main
0
P u0
this
u"i
paper
P u
we
will
fol ! owl ng Lemmas-
LEMMA 2.1.-
Le t’ s suppose
a
u,
u,
uu.
C(’O, T- izC’fl.)_) (:znd
show
the
733
QUASILINEAR EVOLUTION EQUATIONS
O.-
Since
X, t
ClCx,lZ */z
ICx,l
+ Ix,0l
Fom here t ollo
pIying the re!ation above to
have"
t
From the two Iast inequaIities we concIude:
t)IZ
Finally,
from
the
hotheses,
the
last
nd
inequality
Gronwll "s
inumlity the reEult of Le .i follo
P
00F.- By
CC[O,TI;L=CD
strong
the polntwise convergence of P u in
,
it*s sufficient
to Ehow thmt P
i m Cmuchy Eequence in C6{0,71;tz6). Let’ tmke
> 0 Euch that
we have that thee eEt
>0, by the continuity of
It.
s
<
J’nluCx,
e
By the compacity of [O,T], there exist s I,
[O,r] c
rand from the
.
ux.s31Zdx
t.)
1-6,s
sz’
I IPm’’s,3
N such that
P,..sIZ <
Fi hal I y by C 2.1 ), C
. .
3), (
,
satisfying
+6[
intwiEe convergence of Pm
sitive nur
N’
Ca. 33
<
conclude thmt there
m,,
N.
4) and the fol I owl ng
,
N
C.
I nequal i ty
/*
IP x. -P x. zlt7
[ IP (x,t-x,s3}{=2
the result of Le
THEOREH 2.3.-
a.
/=
+
[ IP x,s=-P x,s)lZ2
=/z
follo
Le’s supse hL
n
e u=Ld. TAe Aee
+
J.E. MUNOZ RIVERA
734
C1.1), (I.E) and CI.33. Remains to how that
Let’s note that u
in order
is
,
rom
is a classical solution.
uence
,
CCO,T;A*I/}} for all
CZCO,TCkC,
wll how that
LCO,T;A(t+i/z}, or all
long to
to prove that
a Cauchy’
let
m}
in
.
then
In act
then
.) and the
mve
equation
brave:
whet e
stem
Hultiplying the
(p)}
AtCu(mi-u
t
tt
above by
and
integrating
in
we
have
ACu
tt
tt
tt
tt
Fom hch it ollo that:
La.
t’C} IAs
.’m> .’.
Sdx
/nl.
_
L+t
z
From Le C3.13 and the last inequality
L,.et
L,.ed.
(( IA2Gm 12dx
IdXo
o
2f} IA2 (u-u
+
Finally from Le .E and since
+
,
4t2(
IA (-uH)
I)Exz
i
A<L*i>/zD
we have that
AL/ZG
en
the
hve
pr
Ch
Cu{
a
Cauchy
sequence
in
LmCO,T;A<L:/z}
nd
is now complete a
UNIQUENESS: If M is 1 ocaz z y Lipschitz, then we have
and
In fact, let
be two solutions, putting w
u- we
REMARK 2..uniqueness.
have
,
,’,w-
Multi pl yi ng by
Mc’.FcIV,alZd.x.),t
Att
appl yi ng HI
cMc’J"
IVulZd.x.)
Me j"
IVulZd.x),,,u
and t he Li pschi t z condi t i on
have that there exists a positive constant c
such that:
on
H we
735
QUASILINEAR EVOLUTION EQUATIONS
only one cLczsEc solution
POOF’.-
o
system
DACL*sv5
CkC
,
+
>
olution of ytem CI.13, C1.3
ufficient to show that there ests
CzC[O,l;A(t*Ivz. In order to prove
nd (1.3) atifyng
Since
tm
tt
let’ multiply (.) by
By
rand
c
c
nd integrating in
the lst equality
t.
e hve"
come:
L,t
L,i
if
L*i
from here iL ollo ha:
By Le .1 and the above inequality
obtain:
tt
L.
L+i
subsequence of
still denoting of the same way and a function u
we conclude that
there
exists
a
Cu(m).) me
which
we
sati sfyi ng
From the last convergences and the Lions-Aubin’s theorem Csee Lions’s
[-], theorem 5.1, chap 1) we conclude in particular that-
u
u
sLron[y
n CCEO,2"1;
o
By .tandard methods we can prove that u is a strong solution of system
736
J.E. MUNOZ RIVERA
from where it follows that there exists c
.I, since x,O
By
rom
this it ollo that
(x,O
O,
O, that is u
such that-
obtain that A
O, and
m
REFERENCES
1.
Y. Ebihara & D.C. Pereira.- On Global Classical solution of a
Internat. J__ Nath & Nth. Sci. Vol.
No. 1 C108g -0-38.
J. L. Lions. Ouel ques _thodes de resol uti on de probl @melimites no___n lineares. Dunod Gauthier Villars, Paris lgO
linear hyperbolic equation.
-.
aux