I nternat. J. Mah.
Mh. Si.
Vol. 3 No. 3 (1980) 505-520
5O5
A NON-LINEAR HYPERBOLIC EQUATION
ELIANA HENRIQUES de BRITO
Instituto de Matemtica
UFRJ
Caixa Postal 1835, ZC-00
Rio de Jane iro
RJ
BRAS IL
(Received January i0, 1979)
ABSTRACT.
In this paper the following Cauchy problem, in a Hilbert space H, is
considered:
(I + XA)u" +
u(0)
u
u’(0)
A2u + [ + M(IA1/2ul 2)]Au
f
O
u
i
M and f are given functions, A an operator in H, satisfying convenient
hypothesis, X >_ 0 and
For
when
Uo
is a real number.
in the domain of A and u
I
1/2,
in the domain of A
if
> 0, and u
I
in H,
0, a theorem of existence and uniqueness of weak solution is proved.
KEY WORDS AND PHRASES.
Non,near Wave Equan, Cauchy Problem, Existence and
Uniquess
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
34G20, 35L05.
506
I.
E.H. de BRITO
INTRODUCTION.
The physical origin of the problem here considered lies in the theory of
vibrations of an extensible beam of length
,
whose ends are held a fixed distance
apart, hinged or clamped, and is either stretched or compressed by an axial force,
taking into account the fact that, during vibration, the elements of a beam
perform not only a translatory motion, but also rotate; see Timoshenko [9].
A mathematical model for this problem is an initial-boundary value problem
for the non-linear hyperbolic equation
2u
t
2
4u
t22
4u
+------
4
[
+
I[
u
(s t) ]2 ds3
o
2u
2
0
(I.I)
where u(,t) is the deflection of point l at time t, a is a real constant, proportional to the axial force acting on the beam when it is constrained to lle
along the
axis, and
is a nonnegative constant
(y%=
0 means neglecting the
The non-linearity of the
rotatory inertia, while % > 0 means considering it).
equation is due to considering the extensibility of the beam.
This model, when %
0, was treated by Dickey [2], Ball [I] and, in a Hilbert
space formulation, by Medelros [5].
Lions
For related problems, see Pohozaev [?],
[4], Menzala [6] and Rivera [B].
In this paper, a theorem of existence and uniqueness of weak solution for a
Cauchy problem in a Hilbert space H, is proved for the equation
(I + kA)u" +
A2u + [ + M(IA1/2uI2)]
(1.2)
Au-- f,
with suitable conditions on the operator A and the given functions M and f.
This paper is divided in three parts.
existence of a weak solution is proved.
In Part i, the theorem is taed and
In Part 2, its uniqueness is established.
Finally, an application is given, in Part 3, when H is
n,
set with regular boundary in R
L2(),
and A is the Laplace operator
a bounded open
A.
A NON-LINEAR HYPERBOLIC EQUATION
2.
507
EXISTENCE OF WEAK SOLUTION.
Let H be a real Hilbert space, with inner product (
Let A be a linear operator in H, with domain D(A)
graph norm of A, denoted
II II,
and norm
V dense in H.
With the
i.e.
for v
V,
V is a real Hilbert space and its injection in H is continuous.
We assume this
injection compact.
Suppose A self-adjoint and positive, i.e., there is a constant k > 0 such that
klvl 2
(Av,v) >
for v in V.
(2.1)
Let V’ be the dual of V and <,> denote the pairing between V’ and V.
fylng H and H’, it follows that V
H
c
c
V’.
Injections being continuous and
dense, it is known that, for h in H and v in V, <h,v>
Define A
<
It follows that A
2
A2u,
2:
(h,v).
V’ by
V +
(Au,Av), for u, v in V.
v >
(2.2)
V’.
is a bounded linear operator from V into
1/2)
Let a(u,v) denote the bilinear form in D(A
(Au, A1/2v),
a(u,v)
Identl-
associated to
A, i.e.,
)
for u, v in D(A
a(u) means a(u,u).
Given X > 0, consider in W
IX,
D((IA)
1/2)
the graph norm of (IA)
1/2,
denoted
i-e.,
for w in W
Note that W
Let
H, if I
D(A),
0, and W
be a real number, M a real C
I
> 0; hence V is dense in W.
if
function, with M’(s)>0, for s > 0.
Assume the existence of positive constants
mo
I such that M(s)
and m
>mo + mls’
508
E.H. de BRITO
for s > 0.
by (
+
Notice that, should M be the identity function, replacement of
mo
+ (m + s),
with arbitrary m
s
> 0, ensures the fulfilment of the
above condition on M.
The theorem can now be stated.
THEOREM.
u
u(t), 0
Given f in
L2(0,T;H)
u
in V, u
[a
+
O
in W
I
there is a unique function
< t < T, such that:
a) u L (0,T;V)
b)
u’. L(0,T;W)
c) u is a weak solution of
(I + kA)u" +
A2u +
i.e., for every v in V, u satisfies in
d
[ (u’ (t),v)
dt
M([A1/2u
2
)U Au
(2.3a)
f,
D’ (0,T):
+ %a(u’ (t) v) + (Au(t),Av) +
+ [a + M(a(u(t)))] a(u(t),v)
(f(t),v)
(2.3b)
d) The following initial conditions hold:
u(0)
Uo
u’(0)
u
(2.4ab)
I
Before proving the theorem, some remarks are pertinent.
Equation (2.3a) makes sense, because (a) and (b) above imply that u,
Au, u’, (kA) u’ belong
to L
Au,
(0,T;H).
Initial condition (2.4a) makes sense, because it is
known,(see
Lions
that if u and u’ are in L (O,T;H), then
u belongs to C
(O,T;H),
(2.5)
Now, initial condition (2.4b) must be understood.
Remember u’
L(O,T;W)
implies that (l+kA)u’
<(I + A)u’,v>
L(O,T;V’),
(u’,v) + %a(u’ v)
because
for v in V
A NON-LINEAR HYPERBOLIC EQUATION
From (2.3a), it follows that (I
L2(O,T;V’).
L2(O,T;V ’) ensures
+ kA)u"
(I + A)u’ and (I + A)u" belong to
+ A)u’ (0)
(I + %A)ul, in V’.
is defined.
Given u
It follows that u’ (0)
(I + A)w
I
in W, set (I
0, for w in W, implies w
lw-v l
2
+ kA)u’ (0)
uI, because, it will be proved below,
lw-v l
+ Xa(w-
(2.7)
0.
(vj)j
,
W, i.e., as j +
that
(2.6)
Indeed, V being dense in W, there is a sequence
to w in
The fact that both
C(O,T;V ’)
(I + A)u’
Therefore (I
509
vj)
N
in V that converges
--+ 0
and
<(I +
0
kA)w,vj>
(w,vj)
+
%a(w,vj)
tends to
lw’12
(w,w) + %a (w)
Hence w
0.
Proof of Existence:
It will follow Galerkin method.
Let, then,
wI,
,Wm
be a sequence in V such that, for each m, the set
is linearly independent and the finite linear combinations of w
I, w2,
are dense in V.
(i)
(wj) jN
Let
Vm denote
Approximate Solutions
m
Search for u (t)
m
j =i
((I +
Suppose, for simplicity, v separable.
%A)u(t),v)
+
the finite subspace of V, spanned by
gjm(t)wj
(AUm(t),Av)
u (0)= u
om
m
in V
m
+
such that
Wl,... ,Wm.
for all v in V
m’
[a+M(a(Um(t))](Aum(t),v)=(f(t),v)
u’(O)=
m
Ulm’
(2.8)
(2 9)
510
E.H. de BRITO
where
Uom
converges to
Uo
in V and
Ulm
to u
I
in W.
This system of ordinary differential equations with initial conditions has
a solution u (t), defined for 0 < t < t m < T.
m
that the matrix
((l+lA)wj,wl),
It is convenient to emphasize
,m, is invertible, for otherwise the
i,j--i
homogeneous system of linear equations
,
m
[
((+
j=l
0
A)wj,w+/-)x.
would have a non-trivial solution
al,...,sm,
i--
hence
m
m
m
((I + %A)
X
j--I
m,
7. mjwj,
j--I
I
miwi
i
0,
a contradiction to the linear independence of w.,...,w
m
(ii)
A Priori Estimates
For v
2u(t),
(2.8) becomes:
+
dt
+
[(o)
Set
M(a(Um(t)))
t a(Um(t))
a
d a(Um(t)) +
2(f(t)’um’(t))
(2.10)
M(s)ds
o
We integrate (2.10) from 0 to t < t
Ju(t)
2
+
m
Xa(u(t))
and obtain:
+
JAum(t)
<Km + JaJa(Um(t))
where K
m
Ulm j2 + la(Ulm)
By choice,
(remember that
+
2
+
o
(a(um(t)))
Ju(s) 12
I
JAUom 12 + M(a(Uom)) +
Uom and Ulm converge respectively
Ulm j2 + Xa(Ulm))
Juzml2
+
to
T
o
Uo
(2.
d,
Jf(s) J2ds"
in V and to u
I
in W
511
A NON-LINEAR HYPERBOLIC EQUATION
Therefore, there is a constant C o > 0, independent of m and greater than Km
such that (2.11) still holds, with K replaced by C
o
m
mo + s
_>
Now, M(s)
a(Um(t))
For
M()
implies
_>
mo
+
(2.12)
lels
from (2.11), (2.12) and
__Isl
<
2m
+
I
m
I o2
2
one obtains
2
lu’(t)
m
where C
C
+
%a(u(t))
+ 2m
1
o
2
IAUm(t)
+
+
moa(Um(t))
< C
+
o
lu(s) 12d s
(2.13)
a constant independent of m.
In particular,
lu’(t)
12
m
< C
+
o
lug(s) 12ds.
Hence, applying Gronwall inequality
lu’(t)
12
m
< C e
T
(2.14)
It follows from (2.13) and (2.14) that
lU’m(t) 12 + la(u(t))
where K
TeT),
C(I +
In particular, as
bounded hence
+
IAUm(t) 12 + moa(Um(t))
for all t in [0,t
2 _<
klUm(t)
m
(2.15)
K,
and all m.
a(Um(t)),
it can be extended to
<
[0,T-I.
it follows that
urn(t)
remains
Therefore, (2.15) holds, in fact,
for all m and t in [0,T].
(iii)
Passase
to the Limit
It follows that there is a sub-sequence of
which, as m
/
,,
(urn),
still dentoed
(urn),
for
the following is true, in the weak star convergence of L (0,T;H):
u
m
a(u
m
--+
u,
(2.16)
a(u),
(2.17)
512
E.H. de BRITO
Au
--+
--
m
U
m
%a(ul)
-+
Au,
(2.18)
U
(2.19)
la(u’),
(2.20)
(2.21)
M(a (um) )AUm --’+
It must still be proved that, in fact
(2.22)
M(a(u))Au
(2.22) will be shown to follow from the Lemma below, whose proof, here
reproduced, was given by J.L. Lions [ 3] and [4].
LEMMA.
The mapping v --+
PROOF.
The function
M(a(v))Av from V into H is monotonic.
M(s)ds is non-decreasing (because M’ ()
()
O
M(O)
>
0) and convex (because "()
M’ ()
> 0).
Take
(a(v)),
(v)
It is easy to see that
’
for v in V
has a Gateau derivative,
(v)
2M(a(v))Av, for v in V,
and that # is convex, i.e., for 0 < p < i,
#(pv + (l-o)w) < O#(v) + (l-o)#(w), for v, w in V.
This inequality can be written in the form.
(w + ,(v-w))
(w) -< (v)
(w)
Passing to the limit, as p + 0 it follows that
(’ (w). v-w) -<
(v)
(w)
(w)
(v)
and, interchanging the roles of v and w,
(’ (v). w-v)
<
Adding the two inequalities above, one obtains:
(’(w)
’(v). w-v)
>_
0.
A NON-LINEAR HYPERBOLIC EQUATION
513
This proves the Lemma.
It can now be shon that (2.22) holds.
Indeed, because of the Lemma, for all v in
ITcMcaCum))Au
m)
Because (u
is bounded in
Jection of V in H
strongly in
MCaCv))Av), um
m
o
L=(0,T;V)
m)
is compact, (u
L2(O,T;H).
L2(0,T;V),
and
(u)
v)dt
0
L=(0,T;H)
in
and the in-
can, further, be supposed to converge to u
Hence, as m
/ (R):
M(a(v))Av, u- v)dt
T(
it is true that
> 0
o
Set u
v
pw, p > O, divide the inequality by p and let p
T(@_
M(a(u))Au, w)dt
/
0, to obtain:
> 0
o
This holds for all w in L
(0,T;V), hence
M(a(u))Au.
In the following, let k be fixed, k < m; take v in V and let m
k
(2.19) and (2.20) imply that, in D’(0,T)
dt
(u(t)
V)
-+
d-(u’ (t),v)
/
.
(2.23)
(2.24)
Passing to the limit in
(2.8), then (2.23) and (2.24),
with
(2.17), (2.18),
(2.21) and (2.22) ensure that
_d [(u,(t) v) + la(u’(t) v)] + (Au(t) Av) +
dt
+
holds in
D’ (0,T),
all v in V.
[a
+ M(a(u(t)))] a(u(t),v)
for all v-in V
k.
(2.25)
(f(t),v),
By density, (2.25) holds in
’
(0,T), for
514
E.H. de BRITO
Therefore, u is, indeed, a weak solution of (2.3a).
It must still be shown, in order to complete the proof of existence, that
u satisfies
(iv)
(2.4ab).
Initial Conditions
(2.19) means that, for v in V and 0 in C’ (0,T)
0(T)
such that 0(0)
i and
0, as m
o
v)O(t)dt
(u’(t)
m
(2.26)
(u’(t),v)(t)dt
-+
o
Because of (2.5) and (2.16), integrating (2.26) by parts, it follows that
But u
om
(Uom, V)
-
-+
(u(0),v), for v
(2.27)
in V.
u in V; hence (2.27) yields
o
(u(0),v), for v in V, i.e., u satisfies (2.4a).
(Uo,V)
To show that u satisfies (2.4b), consider equations (2.3b) and (2.8) for
as
It follows, using (2.17)-(2.18), (2.21) and (2.22), that,
w., j --1,2
v
3
m/o
__d
dt
converges to
t
(u(t)
w.) +
(u’ (t),wj) + Xa(u’
3
(t),wj)
(2.28) means that for v in V, 0 in
-
IT t
o
o
(2.28)
Xa(u(t),wj)]
weak star in
CI(0,T)
L=(0,T).
such that 0(0)=i, O(T)
[(u(t)’wj)
+ Xa(u’(t)
wj )] 0(t)dt
m
[(u’(t),wj)
+ Xa(u’(t),Wo)] 0(t)dt.
0,
(2.29)
3
Because of (2.6), (2.19) and (2.20), integrating (2.29) by parts, it follows
that
A NON-LINEAR HYPERBOLIC EQUATION
+
(Ulm,Wj)
But
to (u
u
Ulm--+
I
in
%a(Ulm,Wj) --+
515
(u’ (0),wj) + %a(u’ (0),wj),
(2.30)
W, hence the left-hand side of (2.30) converges also
l,wj + a (ul,wj).
Therefore
+
(u’(O),wj)
+
(Ul,Wj)
%a(u’(O),wj)
(2.31)
%a(ul,wj)
1,2,..., it follows that, in fact, for all v in V:
As (2.31) holds for j
(u’ (0),v) + la(u’ (0),v)
(Ul,V)
+ la(ul,v).
In other words
(I + lA)u’(0)
[6]) u’(0)
But this implies,[see
3.
(I + lA)u
Ul,
I
in
V’
i e. u satisfies
(2 45)
UNIQUENESS
Let u and 5 be two solutions of (2.3a) with the same initial conditions
(2.4ab).
Thus w
(I + kA)w" +
satisfies
u
A2w +
Aw + M(a(u))w +[M(a(u))
0,
w(0)
w’(0)
M(a(5))] A5
(3.z)
0,
(3.2ab)
0.
The standard energy method cannot be used to prove uniqueness, because,
and not to
L(0,T;V).
L2(0,T;V’),
(3.1) belongs to
while the left-hand side of
u’ belongs
to
L(0,T;W)
A modification has to be made; this procedure can be
f.ound in Lions [3].
Consider:
s
-ftw()d
for t < s
0
for t > s
(3.3)
z(t)
and
wl(t)
w()d,
(3.4)
516
E.H. de BRITO
Then
o
z(s)
z’(t)
and
As w
L(R)(0,T;V),
LI(0,T;V).
(3.6)
w(s)
z(O)
(3.7)
(3.8)
w(t)
[see
it is clear
(3.8).]
(3.3) and
that z and
z’
are in
Hence, it follows from (3.1) that
< (I
+ lA)w"(t), z(t)->
dt
(Aw(t), Az(t))dt +
+
O
O
+
I
a
s
(Aw(t),z(t))dt +
o
I
s
M(a(u(t)))(Aw(t),z(t))dt +
o
s
+
[M(a(u(t)))
M(a((t)))](AS(t),z(t))dt
(3.9)
0.
o
Bu,
[ (3.8)]
<
(l+IA)w"(t) ,z (t) >
t((l+IA)w’(t),z(t))- ((I+IA)w’(t),z’(t))
((Z+A)w’ (t),z())
Therefore, using (3 2ab) and (3 7)
+
s
+ )w"(t),z()
o
No, [, ( 3. ) ]
Thus,
[see
(Aw(t)
Az(t))
(3.6) and
(3.7)]
(Az’(t)
Az(=))d=
o
((Z+XA)w(t),w(=))
it follows that (remember
ka (w))
< (
y
> d
-
Az(t))
2
i
l2
__d
1/2lAw(s)[2
dt
’[w[ 2
(3.0)
]Az(t)
(3.n)
A NON-LINEAR HYPERBOLIC EQUATION
As
lwll >-lwl,
I() 12
+
+
517
(3.9), (3.10) and (3.11)yield
IA (s)12 -< 21=I
l(w(t), Az())Idt
O
M(a(u(t)))l(w(t) Az(t))Idt
2
O
IM(a(u(t)))
+ 2
M(a((t)))
l((t), Az(t))Idt
(3.12)
O
I
As u, uEL (0,T;V) and, for s > 0, M > 0 is a C function, with
there is a constant C > 0 such that
M(a(u(t)))](w(t) Az(t))[dt
2
< 2C
O
O
[w(t) IAz(t)[dt
(3.13)
O
And
IM(a(u(t))
2
l(5(t),
M(a((t)))
Az(t))
O
s
]a(u(t))
-< 2C
O
< 2C 2
o
< 2C 3
o
[see
I
I
s
](A(u(t) + (t)), w(t))] ]Az(t) ldt
O
s
[w(t) [Az (t)
(3.14)
O
(3.5)
]
2[w(t) ]Az(t)[
<
Notice that,
a((t))] [(t)[ IAz(t)[dt
O
2[Iw(t)]2 + iAwl(t [2 ] + IAwl(s) 12
(3.15)
Hence, it follows from (3.15) that
2
O
Hence (3.13) and (3.15) give
O
+
iAWl(t)12 ]dr +
a(3.16)
E.H. de BRITO
518
s
M(a(u(t)})l(w(t),
2
Az(t)}
o
<_
E
2co
IAw(t)12]d + Co IAw ()12
lw(t)12 +
o
(3.17)
Now (3.14) and (3.15) give
s
IMCaCuCt)))
2
MCaCCt)))l ICe,t),
AzCt))
o
co
-<
[
o
lw(t)12 + IAWl(t)i 2 ]dt + C3o
It now follows from (3.12), with (3.16)
lw(s) 12 +
where C
Take s
I=l
o
(
IAw (s) 12
s
(3.17) that
E lw(t)12 +
cs)lAw()12 -< 2C
(3.8)
o
IAwl(t)
(3.19)
+ C o + Co3.
such that
for 0 -< s
So’
1 <i- CS < I.
Hence (3.19) yields
for0Sss:
o
lw() 12 +
I
IAw()l 2
I
< 2O
s
IAWl(t) 123dt.
E lw(t)
2
+
2
+
lw(t) 123dt.
A fortiori, for 0
lw()l 2 +
IAl(,)l 2
_<
[lwCt)
4c
o
Applying Gronwall inequality, it then follows that
w(s)
0, for 0 < s < s o
0, for s o < s
0, for 0 < s < T.
Similarly, it is proved that w(s)
then follows that, in fact, w(s)
The proof of uniqueness is complete.
<
s
o
+
,
with
0.
It
A NON-LINEAR HYPERBOLIC EQUATION
4.
519
APPLICATION
n,
For
a bounded open set in R
H
L
with regular boundary, consider
2()
V
H
I()
H
o
2(D)
n
Let A be the Laplace and V the gradient operators in R
A
*
A, hence A
Hypothesis on A are satisfied.
V.
case, the condition (Av,v) >
klvl 2,
respectively.
Take
Notice that, in this
for v in V, is the Friedrichs
Poincar
inequality; the compactness of the injection of V in H is the Rellich theorem.
It is clear that
Now (,) and
are
W
L2(),
if I
W
HI(),
if I > 0
respectively
0
the inner product and the norm in
L2().
Given
u
u
f
H
o
I()
2
H ()
o
I
L
L2(),
if I
2(0,T;
L ()),
O; u I t
HI(),
if I > 0,
2
the theorem proved above ensures existence and uniqueness of weak solution for
the non-linear hyperbolic equation
(I
satisfying
u(0)
ACKNOW__EDGBMENT.
2
l&)u + A u- [e
Uo, u’ (0)
u
M(IVu
2)
Au
f,
I.
This research was supported by FIMEP and CEPG- UFRJ.
Special
acknowledgement is due to Professor L.A. Medeiros, who drew my attention to this
problem, for his generous help and encouragement.
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13
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2]
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