Internat. J. Math. & Math. Sci.
VOL. 12 NO.
(1989) 29-38
29
ON A GLOBAL CLASSICAL SOLUTION OF A
QUASILINEAR HYPERBOLIC EQUATION
Y. EBIHARA
Department of Applied Mathematics
Faculty of Science, Fukuoka University
Fukuoka 814-01 Japan
D.C. PEREIRA
lnstltuto de Mathematlca-UFRJ
CP 68530, CEP 21944
Rio de Janelro-Brasll
(Received June 16, 1987)
ABSTRACT.
In this paper we establish the existence and uniqueness of global classical
solutions for the equation which arises in the study of the extensional vibrations of
thin rod, or torsional vibrations of thin rod.
I.
INTRODUCTION.
In this paper we study the existence and uniqueness of global classical solutions
of the first inltlal-boundary value problem for the equation:
Utt
Au
M(
R IVuJ2dx)Autt
(I I)
f
n, T
& x]0,T[, where R(8: the boundary) is a smooth bounded domain in R
positive number, Vu is the gradient of u, A is the Laplace operator and M(I),
in Q
>
a
real
valued
function
with M(I)
p
a
O,
We
0,
0, for some p > O.
mathematical interest in solving the equation (I.I) by the following reasons.
is
is
have
the equation (I.I) with M(A)
aflses in the study of the extensional
[I], and it was studied by one of the authors in [2]
is the
Second, the equation (I) with M(I)
x, where
o’
First,
vibrations of thin rods, see Love
and [3].
o
I #2d
torslon-functlon arises in the study of the torsional vibrations of thin rods, see
Love [I].
Third, the function M(1) in (1.1) has its motivation in the mathematical
description of the vibrations of an elastic stretched string, that is, the equation:
utt-
M
IVul2dx)
Au
0
(1.2)
30
Y. EBIHARA AND D.C. PEREIRA
what
for M(A)
>
0
0 was
studied
by
[4], Nishihara [5] and Lions [6].
[7], Ebihara-Medeiros-Milla Miranda [8]
Pohozaev
When M(A) 0 was treated by Arosio-Spagnolo
and Yamada [9].
In this paper, we establish the existence and uniqueness of global classical
(1.1).
For that we use the Faedo-Galerktn method and
solutions for the equation
compactness argument with some technical idea.
NOTATIONS, ASSUMPTIONS AND MAIN RESULT.
2.
Let
(wj)
2
on H O1)
of
(wj)
then
JN
be a system of eigen functions of -A which is defined
n H (l)
0
c
V(R) the set of all finite linear combinations
f(x)g(x)dx, we set
H-(), m
1,2
2
....
("
")m"
(’"’)m
We put V
m
define an inner product on V.
the topology of norm
m
f
Putting (f,g)
JN
(.,..)m
We denote by V
((-A)m’"’)’
m-l,2,...,
V () as the closure of V by
m
Then we see that
and the norm
Hm().
.
m
is equivalent in V to the standard norm
m
"-"
We see that all the above injections
are compact.
Let T be a positive number and B a Banach space with a norm
We shall
represent by LP(0,T;B),
p < ", the Banach space of vector-valued functions
of
If’lIB"
T
LP(0,T;B)
0
and by L (0,T;B)the Banach space of vector-valued functions
L
u:]0,T[
B which are
(0,T;B)
We denote by C(O,T;B) the space of all vector-valued functions u:[O,T]
which are J-tlmes dlfferentlable in the sense of B.
Let M(A),
0 be a real valued function such that:
M()
verify M(1)
CI[o, (R))
a) I/2
and there exist constants a
+ p, /
[0,(R)),
>
0 and p
>
0 that
/
B,
SOLUTION OF A QUASILINEAR HYPERBOLIC EQUATION
tM’()I 1/2
(A.2)
cO[o,-),
(A)M(k) where ()
()
31
0, k
0, then
we have the following result:
THEOREM 2.1.
Suppose that
Vm’
(m
(2.1)
C(0,T;Vm_I).
(2.2)
Uo,Ul
f,f’ e
L2(R)
[0,T]
Then there exists a unique function u:
in the class:
(0,T;Vm)
u e
(2.3)
that verifies
u"
M(IUI2)Au
’’’
Au
u(O)
u
u’ (0)
(2.4)
f in Q
o
(2.6)
u
(2.7)
3.
PROOF OF THEOREM I.
a)
b)
c)
Approximated solutlons
A priori estimates
Passage to the llmlt
d)
Uniqueness
We divide the proof in four parts:
a)
APPROXIMATED SOLUTIONS.
Let
[Wl,...,wk]
be
the
subspace of V,
.
of
Let
generated by
the
first k eigenvectors
k
u
k
(t)
J=1
gjk(t)wj
[Wl,...,Wkl
be a solution of the system:
(Uk"
Auk
M(lUk 12 AU ,w)
u (O)
k
uOk
u
u(O)
Ulk
Ul
U0k
jffil
strongly in V
strongly in
(u0’ wj)wj’ Ulk
-j=
(u
m
as k
Vm as k
k
k
where
0
(f,w)
l’Wj)Wj
for all w-[wl,...,w
(3.1)
/
(3.2)
k]
=,
(3.3)
32
Y. EBIHARA AND D.C. PEREIRA
Uk(t)
Then we see that the solution
exists on an interval
Uk(t)
priori estimates will permit us to extend
b)
[0,Tk)
to all interval
0
TkT.
[0,T].
A PRIORI ESTIMATES
I) Putting w
(t)
in
(3.1),
we have
Uk"102 +
Thus by
+
(Uk’Uk)
2
2
M(lUktl)lUll
(A.I),
lUkl0,,2
2
JUklllUll
6
+
where C is a positive constant such that
lull
where C
(f’uk)"
>
C
2
2
+
(’-"
t
2
Ifl01u]0
Then from this and
(2.2),
2
Ill0) <
C
2
p
0 is a constant independent of t and k.
Thus
()
is bounded in L
(3.4)
(0,Tk;Vl)"
By Fundamental Theorem of Calculus, we have
(u)
is bounded in
L
(O,Tk;V I)
and
(uk) is bounded in L
(0,Tk;V1)"
The above estimates permit us to extend
II)
Making w
(-A)m-
u,
Uk(t)
to all interval
in (3.1)
we have:
d
d
2
2
2
2
+ lUk Im+ M(lUkll)lulm]
[lu Im-1
l)(Uk’%) lluIm
By (A.2) and (3.6), we have
/
(f,t)m_ 1.
[0,T].
SOLUTION OF A QUASILINEAR HYPERBOLIC EQUATION
C2
33
2
2
Then integrating from 0 to t and using (2.1) and (2.2) we obtain:
u.2
kl
+
2
2
lull 2 +"lul)lukl
, c 3 + c t [lul
C3(Uo,U I,T)
where C
3
+
"lullul.]ds
max{l,C2}.
and C
4
Thus, by Cronwall inequality and (A. 1), we have:
lul m-1 + lUklm + I.I m2 , c 5
where C
nenc e
5
>
(Ill)
0 is a constant independent of t and k.
(uk) is bounded in L
(0,T;Vm)’
(3.7)
(u)
(0,T;Vm).
(3.8)
is bounded in L
Taking the derivative of (3.1) Ith respective to t, we obtain:
2
2’(lUkl)(.
%) ,A ..)
Putting in (3.9) w
(-A) m-I
d
2 dt
+
-
( ,,,)
o
we have
[[ Uk.12
,(i.kl 2
Thus by (3.2) and (3.6),
Integrating from 0 to t, we have:
+
2
2]
lUlm2 +.(luix)l.l
2.
.
(3.9)
,, [,,,
.... ,,k I.
34
Y. EBIHARA AND D.C. PFJEIRA
(3.1o)
(-A)
Putting w
m.-i
u"(0) in (3.1) and tending t
k
0, we obtain:
lu"<o>l +m--1 + <.,+<o> <o>>m +.<luol++>l Uk"o>I m2
(0))m_l
(f(0),
Thus,
I<>I +
+-t
+
"<:l Uok 12,> I-’CO>I.,2, +l"okl.,2 + +l.co>l +
2
+ +I,,o>I 2
Ifc>l m-I
m-
+
for T
>
0.
Then, by (A.1), (2.2) and (3.2), we have
2
luco>l m-I
where C
6
>
-I- 2(p-
’,>luco>l "::6
0 is a constant independent of t and k.
Whence, taking T as 0
<
T
<
p, we obtain:
(3.11)
Thus, by (3.10) and (3.11), we have:
I12+ I"lm +.clul>l %"1
2
+
2
t
m-1
And by Gronwall lemma and
.
(A.I), we obtain:
ii m-I + lUlIm2 + +1.12 , c8
where C
8
>
0 is a constant independent of t and k.
2
SOLUTION OF A OUASILINEAR HYPERBOLIC EQUATION
35
Whence,
()
IV)
is bounded in L
(-A)
Putting in (3.9) w
(3.12)
(O,T;Vm)"
m-luk( 3)
we obtal n:
2
+
lUk (3)I m-I
(Uk’ Uk(3)
2
(3) 2
(Uk,U)
Thus by
(A.1), (A.2), (3.7), (3.8) and (3.12), we have:
(3)
C
9
12m-1 + PtUk(3) 12 ’---+
VtUk (3)
V
m
2 +
-m
-I.f’.
12
(3) 2
Then
[Uk(3)12m-1 + 2(p ")lUk (3)Ira2 C
where
CIO
>
0 is a constand independent of t and k.
(uk
C)
(3)
is bounded in L
lO
Whence we can assert that
(3.13)
(0,T;Vm),
PASSAGE TO THE LIMIT.
By estimates (3.7), (3.8), (3.12) and (3.13) there is a subsequence of
what we denote by (Uk)ke and there exists a function u, such that:
N
(Uk)ke N
(3.14)
(0,T;Vm)
u
k
u weak star in L
u
u
u’ weak star in L (O,T;Vm)
(3.15)
(O,T;Vm)
(3.16)
(3)
u
k
weak star in L
*+
u
(3)
m)
weak star in L (0,T;V
(3.17)
By (3.14) and (3.15), for m=2, and since the embedding of V is compact in V I, it
2
follows from Aubln-Lions Theorem
u
k
[I0],
u strongly in L
2
(0,T;V 1)
whence
2
strongly in L (0,T).
Now by (3.16),
(3.18)
36
Y. EBIHARA AND D.C. PEREIRA
Au
Au" weak
k’
in L
2
(O,T;Vm_2).
(3.19)
Thus, by (3.18) and (3.19) we have
(O,T,Vm_ 2
weak in
(3.20)
The above convergences permit us to pass to the limit in the approximate equation
(3.1), as k
(R).
We then get:
Au-
(u"
Since
,)
(
L2(0,T).
for each wV in the sense of
REMARK 1.
M(lul)
the solution u of
(2.4)
is
in
C2(0,T;Vm ),
and by
Sobolev’s
theorem
with k
<
n
i
H
(a)
k + I, k
ck()
0 integer,
then u satisfies (2.4) in the classic sense if we choose m large enough.
d)
UNIQUENESS
Let u,v be solutions of (I.I) in the class of Theorem I.
Then
t
u- v
satisfies:
(3.21)
(3.22)
(3.23)
Taking of scalar product in L
2
() of (3.21) by ’,
M([u[1 (v",a’)1
Now, by (A.I) and (A.2), we have
where
-
(I
-){u{ 2
2
we obtain:
0
SOLUTION OF A QUASILINEAR HYPERBOLIC EQUATION
2
Id
whence by
2
2
2
37
2
C
2
(A.I),
2
2
2
,
C
0
(II 2
2)ds.
Thus, we have
0 in [O,T]
REMARK 2.
In the forthcoming work we will try to study the equation (I.I)
M(A)
when
has zero points, that is, degenerate case.
REFERENCES
I.
LOVE, A.H.
2.
PEREIRA, D.C. and IZAGUIRRE, R.M.
A treatise on the mathematical theory of
Dover New York
elastlclt.
1944.
llneares.
3.
Sobre urea equacao de vlbracoes nao
Proceedings of 23o semlnarlo Brasilelro
de
Anallsep Camplnas, Sao
Paulo (1986), 155-158.
PEREIRA, D.C. Problema mlsto para uma equacao de vlbracoes nao llneares.
To appear.
4.
POHOZAEV, S.I.
On a class of quasillnear hyperbolic equations.
Mat USSR
(1975), 145-148.
Sbornik
5.
NISHIHARA, K.
6.
LIONS, J.L.
On
a
global
solution
of
some
quasillnear hyprbollc equation.
Tokyo Journal of Math. 7 (1984).
physics.
7.
On some questions in boundary value problems of mathematical
IM-UFKJ (1978).
AROSIO, A. and SPAGNOLLO, S.
nonlinear
8.
hyperbolic
Global solutions of the Cauchy Problem for a
equation.
Unlverslta
Mathematlca, Roma (1982).
EBIHARA, Y., MEDEIROS, L.A. and MILLA MIRANDA, M.
de
Pisa,
D partamento
Local solutions
for a nonlinear degenerate hyperbolic equation.
Nonlinear Analysis,
Methods and Applications, I0 (1986), 27-40.
9.
YAMADA, Y.
dl
Some nonlinear degenerate wave equation.
To appear in Nonlinear
Analysis, Theory, Methods and Applications.
I0. LIONS, J.L. que.lques methodes de resolution des problemes aux llmltes non
lineaires. Dunod, Paris, (1969).
Theor[