145
Internat. J. Math. & Math. Sci.
(1990) 145-150
VOL. 13 NO.
PERIODIC WEAK SOLUTIONS
TIME
ELIANA HENRIQUES DE BRITO
lnstituto de Matemtlca
Unlversldade Federal do Rio de Janeiro
C.P. 68530, Rio de Janelro, Brasil
(Received December 30, 1987)
ABSTRACT.
uniqueness
In continuing from previous papers, where we studied the existence and
the global solution and its asymptotic behavior as time t goes to
of
infinity, we now search for a tlme-perlodlc weak solution u(t) for the equation whose
weak formulation in a Hilbert space H is
d
d--{ (u’
+ 6(u’ v) +
v
the
Gateaux
derivative
a
of
)
convex
0, hence B
when
>
function of t with values in H.
W when
0 and V
>
b(u,v), a(u,v) are given forms on
0, 8 ) 0 are constants and s + 8 > 0; G
U,
[0,) for V
functional J: VcH
in H;
d/dr;(,) is the inner product
subspaces UcW, respectively, of H; 6 > 0,
where:
is
(h,v)
b(u v) + 8a(u,v) + (G(u),v)
e
0; v is a test function in V; h is a given
Application is given to nonlinear inltial-boundary value problems in a bounded
domain of R
n.
KEYWORDS AND PHRASES. Periodic weak solution, Gateaux derivative.
1980 AMS SUBJECT CLASSIFICATION CODE. 35L20, 35B40.
I.
INTRODUCTION.
In continuation of Brito [I], [2], where we studied existence and uniqueness of
the global solution and its asymptotic behavior as time t goes to infinity, Zwe now
search for a time-periodic weak solution u(t), i.e., such that
u’(T)
u(T); u’(0)
u(0)
for the equation whose weak formulation in a Hilbert space H is
td-(u’,v)
+ (u’,v) + sb(u,v) + 8a(u,v) + (C(u),v)
(1.1)
(h,v)
where
--d/dr; (,)
are
given
forms
constants and
J: V cH
=+
on
subspaces
8>0;
[0,), for V
H; b(u,v), a(u,v)
respectively, of H;
> 0,
is the inner product in
U
W,
0, B
0 are
G is the Gateaux derivative of a convex functional
U, when
>
0, and V
W, when
test function in V; h is a given function of t with values in H.
O, hence 8
>
O; v
is a
E.H. DE BRITO
146
Application is given to initial-boundary value problems in a bounded domain
n
O:
O, k
2 depends on n, a
O, B
R for the followlng equations, in which p
u" +
u
lulP-2u
Au +
u" + u’
h
aA2u-
BAu + u
A2u
{ + k
+
u" + u’ +
>
>
>
of
+’’lul p-2 u=h
f (Vu)2dR}gu
(1.4)
h
and the generalization of (1.4) in a Hilbert space H
u" +
u’+ A2u
+ BAu +
M(IAI/212)Au
(1.5)
h
for A a linear operator in H, M a real function.
For related problems, we refer to Biroll [3], Lovlcar [4]; see also references in
Brlto [I].
2.
PRELIMINARIES.
Uc Wc H each continuously embedded and dense in
We consider three Hilbert spaces
the following.
We assume the injection W cH compact.
Let (,) denote the inner product in H and
its norm.
Let a(u,v) and b(u,v) be two continuous, symmetric, billnear forms in W and U,
We shall assume
We shall write a(v) for a(v,v), b(v) for b(v,v).
respectively.
I/2" defines in W a norm equivalent to the norm of W and, similarly, that
I/2"
defines in U a norm equivalent to the norm of U.
(b(v))
that (a(v))
Let c
>
0 be such that
clvl 2
(2.1)
a(v) for v in W.
Let A be a linear operator in H, with domain D(A) such that UcD(A)c W and
a(u,v)
(Au,v) for u in U, v in W
(Au, Av) for u,v in U
b(u,v)
>
0, a
0, B
V
U if a
>
V
W if a
0 are constants and a +
B
>
0.
As s ume
0, B
0, B
0;
>
0.
Consider a convex functional
J: V
Let G: V
H be
[0,(R)) such that J(0)
the
Gateaux
With
these
hypothesis,
we
of
derivative
differentiable, locally Lipschitz and G(O)
respectively
0.
have,
J.
We
assume
G
is
Gateaux
O.
from
[2]
Theorem
3.1
and
[I]
Lemma 2.1,
THEOREM 2.1.
Given u 0 in
V, u
147
PERIODIC WEAK SOLUTIONS
TIME
in
H, h in
L2(O,’r;H),
there is a unique function u
such that
(0,T;V); u’
n (O,T;H); G(u)
g (0,T;II)
a)
u e U
b)
for all v in V, u satisfies,
d
(u’,v) + 6(u’,v) + ab(u v) + Ba(u,v) +
dt
e
c)
u satlsfies the initial conditions
d)
u satisfies the energy equation
no;
u(0)
u’(0)
t
0
(G(u),v)
(h v)
(2.2)
u
t
lu (s)12d s
E(t) +
g
I
E(0) +
r2.43
(h(s) u (s))ds
0
where
lu’(t)l 2
2E(t)
+ b(u(t)) + 8a(u(t)) + 2J(u(t)).
(u(L),u’(t)
S(uo,U 1)
is, for fixed t, (sequentially) weakly continuous (i.., if
then
S(#n)
VH given by
In the conditions of Theorem 2.1, the map S: VH
THEOREM 2.2.
> s(#)
weakly in V
n>
we.;ly in V x H,
H).
We shall, further, assume that
2J(v)
(G(v),v) < 0 for v in V.
3.
(2.5)
EXISTENCE OF TIME-PERIODIC WEAK SOLUTIONS.
We shall refer to u(t) in the coudltlons of Theorem ’.. a the solution of (2.2)
H, given by (2.3).
m
with initial conditions
(Uo,U I)
C([O,T];H)
If h
THEOREM 3.1.
there
is
at
least
,ne
solution of
(2.2) with
H such that
initial condition in V
(3.1)
,’(T).
it
add
to the
and
0
2y
>
constant
by
(2.2
u(t)
mul.[plied
in
PROOF.
an,
with
(2.5),
,nultiplied by 2, to ,,btain,
energy equation (2.4) differentiated
u(T); u’(0)
u(O)
Take v
d---
+ b(u) + ga(.) + 2J(u) + 2T(u,u )}
u
+ 27{ u’
+ 2(8-2y)
For 0
<
7
<
+ eb(u) + Sa(u) + 2J(u) + 2y(u,u )}
u’
2(h,u’ + yu).
+ ’u’,u]
6/2, let
w(t)
lu’ + "ul 2
+ ab(u) + a(u) + 2J(u).
Then we have
w’(t) + 2yw(t)
2(h,,a’ + 7u)
2
d
2
2(6-2)(u’,u’ + 7.) +
2
(3.2)
E.H. DE BRITO
148
The right-hand side of the above Inequality is equal to
2(h,u’ + yu) +
2(y2u-
(6- 2y)u’,u’ + yu)
2(6- 2y)(u’ + u,u’ + yu) + 2(6- )(yu,u’ + yu).
2(h,u’ + yu)
Therefore
$)2X2]u]
(
2(h,u’ + yu) +
w’(t) + 2yw(t)
(3.3)
Observing (2.!) and (3.2), we obtain
w(t)
)
u’ +
"(u
>
0.
By assumption, B +
<
We choose 0
y
<
+
u
with e
ac
+ Bc
>
O.
(3.4)
/2 so that
(e-x)
P
2
2
2
x
2
(i-2x)
<
(3.5)
This is possible, because it amounts to choosing y so that
and lira B(’f)
B(X)
(6
-4 e
( 0.
V)2X
2X)
4 e (8
( 0
It follows from (3.3), with (3.4), (3.5), that
,/w(t) + pw(t).
w’(t) + 2wt)
2]h(t)[
Hence for 0 g t
g
T,
w(t) g F(t)
(3.6)
where
F(t)
e -2Tt {w(0)
t
+
f
e2S[21hl4w +
pw]ds}.
(3.7)
0
Therefore, because of (3.6), we have
F’ (t) g (O
Let F(t)
2x)F(t) +
2]h()[/F(t).
r, with
r
)
max
0 t T
O. It follows, using (3.6), (3.7), that if w(O)
Then F’(t)
for 0
t
r then w(t)
r,
T.
Consider
[u
+
K
{(u0,u l) V x H;
We proved that the map S: V x H
S(uo,U x)
Tu012 +
V
ab(u
0)
Ba(u 0) + 2J(u 0)
r}.
H given by
(u(T), u’(T))
takes K into K.
It
is
easy
to
check that K is
a
nonempty,
closed,
bounded,
convex subset
of V x H.
The fact that S has a fixed point, i.e., that (3.1) holds for some
(Uo,U I)
c K,
now follows from Theorem 2.2 as a consequence of the well-known fixed point Theorem:
Let B be
,
149
PERIODIC WEAK SOLUTIONS
TIME
:-;elarable, reflexive Banach space, K a nonempty closed, bounded convex
subset of B, and S a (sequentially) weakly continuous operator of K into K.
Then S
has at least one fixed point In K.
4.
APPLICATIONS
We devote thls Section to applications of Theorem 3.1 involving Inltial-boundary
n
value problems in a bounded domain R with regular boundary in R for equations (1.2),
(1.3), (1.4).
In what follows
H
L2(fl),
A
-A,
W
H0(fl),
and
Le t
H
D(A)
l()
0
(Au,Av),
b(u,v)
0
U
H2(),
H()
0
H2(R).
2
Ho(fl).
Note that similar results are obtained if we suppose U
>
0, h e C([0,T]; H), we have
I, V W
EXAMPLE 4.1. Let a 0, 8
For
[ulPLP()
J(u)
where 2
<
p
2(n-l)/(n-2) if n
>
H0(R)
and
for u in V
2; p
>
2 if n
Then J(u) is well-deflned in V
2.
and
lul
U
H.
We efe .o 2], example 5.1, for the proof.
Theorem 3.
It Is clear that (2.5) holds.
Therefore
ensures existence of a T-perlodlc weak solution of
u" + u’
EXAMPLE 4.2.
For a
>
Au +
IulP-2u
0, 8
0, V
h.
H0(fl)f] H
U
p
- lul LP(R)
2
(fl)
(or H (fl)), let
for u in V
where
2
<
p
2(n- 2)/n-4) if n
>
4; p
>
2 if n
4.
Then J(u) is well-deflned in V and
We refer to [2], example 5.2, for the proof.
Theorem 3.
It is clear that (2.5) holds. Then
ensures existence of a T-perlodlc weak solution of
u" + u’ +
aA2u
Au +
u +
IulP-2u
h.
E.H. DE BRITO
150
EHPLE. 4.3.
J(u)
Then
-
>
For a
k
(a(u)) 2
>
O, k
0, B
or
u in
O, V
U
H ()0H
2(’2.)
(or H 0()),
v.
O(u) --ka(u)Au e H.
We refer to [2], Example 5.3, for the proof.
It is clear that (2.5) holds.
Thus Theorem 3.1. ensures existence of a T-perlodlc weak solution of
aA2u
u" + u’ +
[B + k
Generalizing, let M be a C
M(s)
Take V
)k
J(u)
Then
h.
function suclt that, for s
and M’(s)
U and A as in Section [.
--(Vu)2df/}Au
>
> O,
O.
Let
a(u)
’
M(s)ds
or
u tn V.
0
G(u) -M(a(u)),
Aug H.
We refer to [2], Example 5.3, for the proof. It ts clear that (2.5) holds.
Theorem 3.1 ensures existence of a T-perlodlc weak solution of
At,
+ [B +
u" + u’ +
h.
A2u
Therefore
M(IAI/2UI2)]
REFERENCES
I.
2.
3.
BRITO, E.H., Nonllenar Equations, A_p_Etlcable An_a_ly_i_s., Vol. 24, 13-26, (1987).
BRITO, K.I{., lonlinear Initial-Boundary Value Problems, Nonlinear Analysis, Vol.
II, No. I, (1981), ]25-137.
BIROLI, M., Sur l’equatlou, de andes avec un Terme Non llnatre, Monotone dens
VIII, Vol. LIII (1972), 359-361., 508-515.
LOVICAR, V., Periodic Solutions of Nonlinear Abstract, Second Order Equations
with Dissipative Terms, Casopls_ rrstova______n__i_i.matematik], roc. 102, Praha,
(1977), 364-369.
Serle
4.