by Veron [13]. In particular we obtain the analogue of the Pazy theorem (see [7], theorem 2.2) for a second order evolution equation of the form where A is a ...
Nonlinear Analysis, Theory, Printed in Great Britain.
Methoa!s & Applications,
ASYMPTOTIC
Vol. 6, No.
11, pp. 12451252,
BEHAVIOUR EVOLUTION ENZO
Istituto
di Matematica
Key words andphrases: be haviour.
OF SOME SECOND EQUATIONS-f
order equations,
ORDER
MITIDIERI
dell’Universit8 (Received
0362-546X/82/1 11245-08 $03.00/O @ 1982 Pergamon Press Ltd.
1982.
di Trieste,
10 February monotone
34100-Trieste,
Italy
1982)
operators,
convergence
condition,
asymptotic
1. INTRODUCTION
IN THIS paper we study some sufficient conditions that ensure the strong convergence, as t-+ 00, of the bounded solution of some second order evolution equations. The results that we obtain are motivated by the theory of contraction semigroups in Hilbert spaces, and by an interesting counter-example constructed by Veron [13]. In particular we obtain the analogue of the Pazy theorem (see [7], theorem 2.2) for a second order evolution equation of the form
where A is a maximal monotone operator on a real Hilbert space H. The first results concerning the strong convergence of the bounded solution of (1.1) were obtained by Barbu [l] and Brezis [6]. We obtain such results as a consequence of our main theorem (see Section 3).
2. SECOND
ORDER
EVOLUTION
EQUATION:
NOTATIONS
AND
KNOWN
RESULTS
Let A be a (possibly multivalued) maximal monotone operator on a. real Hilbert We denote by ( , ) the inner product of H, and )I (1the norm. We call D(A)
= {X E H : Ax + #I} andR(A)
=
U
space H.
{AX}
xED(A)
the domain and the range of A. We put [x, y] EA @y E Ax. It is well known that for every x E D(A), Ax is a closed convex set. We denote by Aox its element of minimum norm. If 0 E R(A), we make F={xEH:OEAx}. Since A is maximal -F Work
supported
monotone,
by C.N.R.
A -’ is also, and therefore
(Italy). 1245
F is a closed convex
set.
E. MITIDIERI
We shall denote by P the projection onto F. Consider problem in the half axis R+: find u in a suitable function
now the following space such that
boundary
value
d2u sEAu
(2.1) u(0) = uo, Barbu
[1] and Brezis
THEOREM problem
[4] prove
sup llW)II < + O” t>O I
for (2.1) the following
existence
theorem:
2.1. If A is maximal monotone on H and 0 E R(A) then for every uo E D(A) the (2.1) has a unique solution u E W$$(O, + 00; D(A)) satisfying the following properties
u E C(0, +y du % Moreover
du t-p
D(A)),
du
t3’2
E L2(0, +yH),
L”(0,
;
E
+QyH)
L’(O, +a; H).
(2.2)
if uo E D(A) then du ; E L’(O, +a; H)
u E C1(O, + 00; D(A)),
(2*3)
and if we put u(t) = SIR(t)uO, we have 0i ( ii ) .. . ( 111)
&/2(O)Uo
S/2@
+
=
s>uo
IIWt)uo -
uo
=
t,s ER+
&/2(t>&/2(+40,
&/2(t>~oll
s
Iluo
-
uoll uo,
U&D(A)
and so there exists a maximal monotone operator A lU2that generates Q(t) on D(A). For other interesting details the reader is referred to Barbu [2]. A recent result on the asymptotic convergence of u, as t leads to infinity, is the following: THEOREM 2.2. ([ll], [13]). Th e solution of problem (2.1) converges weakly as t+ 00 to a point of F. Remark 2.1. It is possible 3. THE
CONVERGENCE
to deduce CONDITION
Now
we introduce the convergence Pazy theorem for (2.1). Definition 3.1. A maximal
this theorem
monotone
OF PAZY
condition
operator
as a special AND
case of ([8], corollary
OTHER
SUFFICIENT
2.5).
CONDITIONS
of Pazy [7] on A and we prove the analogue
on H satisfies
(i) F is not empty, (ii) If [&, Yn] E A, II&II s c, JJy,ll s c, and lim (yn, xn - Px,) = 0, n-,m
the convergence
condition
of
if:
1247
Asymptotic behaviour of some second order evolution equations
then lim inf d(x,, F) = 0 n where d(x, . ) is the distance function. The main result of this paper is the following:
THEOREM 3.1. If A satisfies
the convergence condition then converges strongly as t--, + 00 to a point of F. Moreover if p = lim u(t), we have t+ +a lip - puoll s II& - P&II. The proof
of theorem
3.1 is based
LEMMA 3.1. If u is the solution bounded
on the following
of problem
and therefore
(3-I)
observation.
(2.1) with u. E D(A) then the function (obviously
of equation
- Pu(t)/*
(3.2)
it is non-increasing.
Proof. By well known have for every X, y E H
properties
of the projection
onto convex
sets of Hilbert
(IPX - Pyl12a (Px - Pv, x - Y) s llx - YII20 Choosing
(2.1)
on [0, +m [)
h(t) = +(t) is convex,
simple
the solution
x = u(t + h), y = u(t) in (3.3)
IIW + wIIs h2
spaces
we
(3.3)
with h > 0, we obtain
Pu(t + h) - Pu(t) u(t + h) - u(t) ? h h
~ Ilu(t + h) - u(t) Ii2 . h2
(34
Since uo E D(A), by equation (2.3) of theorem 2.1, u E C’ (0, + 00; D(A)) and so dPu/dt exists a.e. because P is nonexpansive. Passing to the limit as h + 0 in (3.4), we obtain
Now we have $ and by lemma
(t) = ($
(t), u(t) - Pu(t))
- (%(t),
u(t) - Pu(t))
2.4. ([7]),
dPu -&- (t), u(t) - Pu(t)
= 0.
Finally $
(t) = ($
and by the monotonicity
(t), u(t) - Wt))
of A, together
+ ll$(t)i[
with equation
- (F(t)$(t)) (3.9,
1248
E. MITIDIERI
In view of (3.6) increasing. n
h(t) is convex
and,
being
bounded
on [0, + ~0 [, it follows
that
it is non-
Proof of theorem 3.1. We prove the assertion for uo E D(A), since by virtue of the nonexpansive property of the solution [see (iii) theorem 2.11, the case u. E D(A) follows by a simple approximation procedure. If u. E D(A), then by equation (2.3) d2u s~L2(0, By this fact and the monotonicity
+yH).
of A, the function du ;,
g(t) =
P-7)
u(t) - Pu(t) u(t) - Pu(t) is bounded.
is nonnegative and belongs to L2(R+), because a sequence t+,-+ + 00 such that
Therefore
there exists,
lim g(t,) = 0. t-++w Using
the convergence
condition
for xn = u(tJ, =-
h
d2u dt2 @n)
we have:
0 = lim inf d(u(t,), n Since by the preceding
lemma
F) = lim inf /u(tJ n
- Pu(t,>(I
h(t) is non-increasing,
the function
it follows
(3.8) that
lim /u(t) - Pu(t)ll = 0. We conclude
the proof
by observing
that,
if h > 0
II40 - UP+ h)II6 II@) - Pu(0II+ II@ + h) - w> II and
(3.9)
lb4 + v - Pu(t> IIs Ilw - P”wIIa
(3.10) (3.11)
Therefore II@ + h) and so by (3.9)
there
@NIs 2 lb49- p”wII
exists a p E F, such that lim t-+x
Letting
h+
+OQ in (3.11)
we obtain,
u(t) = p.
for t = 0, (3.1)
W
There are many known maximal monotone operators that satisfy the convergence We refer the interested reader to [7], for some significant examples.
condition.
Asymptotic
behaviour
of some second
order
COROLLARY 3.1 ([ 11, [4]). If A is maximal monotone continuous function on [0, + m[ such that
the solution
Proof. so-called
of (2.1) converges
1249
equations
and there exists a positive
non-decreasing
for every [x, y] E A.
01743 !fall)~ Then
evolution
strongly
as t+
+ CCto zero. In fact, in this case A satisfies
It is sufficient to apply the preceding theorem. n ‘uniform convergence condition’ (see [7]).
the
Now we study the convergence of the solution, under different hypotheses on A. The first one is the compactness of the resolvent [i.e. (I + A)-’ compact]. It is well known that in general, an operator with compact resolvent does not satisfy the convergence condition (this is true for subdifferentials). However for our problem [equation (1.1) is elliptical], the compactness of (I + A)-’ is sufficient for the convergence of the solution. THEOREM
3.2. If A is maximal
monotone
such that:
(i) F is not empty (ii) (I + A)-’ = J1 is compact, then the solution Proof.
of (2.1) converges
It is known
strongly
[7] that the condition ER =
for every R > 0. Consider the solution that
{xE
as t+
00 to a point
(ii) ensures
F.
the precompactness
of
H : llxll 6 R, llyll s R, y E Ax}
of (2.1) with ~0 E D(A).
Choosing
a suitable
sequence
tn -+ + CCsuch
,.j+E$ (tJ = 0 lim
[this sequence
exists by (1.3)],
we have U(tn) E
for some R > 0. By the precompactness
ER
we can assume
t, in such a way that
lim u(tn) = p t,--++m for some p E H. Since by theorem 2.2 u(t) converges that p E F. On the other hand by the semigroup u(t) = Sl,,(t)uo], we have for t 2 tn
IIwbQI - pll= IIq2(t s
We conclude
Ilsl/2(t,)uo
-
trz +
-
weakly as t --) + 00, we have necessarily property of the solution, [by putting
tn)uo
-
S1/2@
PII
l
that lim S&t)uo = lim u(t) = p. t-++m t* + x
t It is clear that for every p E F, we have &,&)p
= p for every t a 0.
n
-
trz)p II ‘r
E. MITIDIERI
1250 COROLLARY
2.2. Let v: ff+]-CQ, + 001be a proper lower semi-continuous convex function, and a~ the subdifferential of q. Assume that the following conditions are satisfied:
(i) F = {X E H : 0 E &p(x)} is not empty (ii) For every R > 0, the set ER = {x E H : q(x) s R, llxll s R} is precompact. Then the solution of (2.1) converges strongly as t+ + 00 to a point F. The proof of this fact follows immediately, by observing that 8~ is maximal monotone and (ii) is equivalent to (I + d&-l compact (see [19]). The following is a trivial consequence of theorem 2.2 and [18]. PROPOSITION
3.1. Let A be a maximal
monotone A(x)
such that, for every x E D(A),
operator
= -A(
-x)
then the solution of (2.1) converges strongly as t+ + 00 to a point In the case where A = &p, we have a more general result.
THEOREM
2.3. Let q be a proper
(i) D(q) (ii) there
1.S.C. convex
F.
on H in ] - 00, + m 1. If
function
= {X E H : q(x) < + m} is symmetric, esists a continuous function a : R+ + ] O,l] such that for every x E D(q),
v(x)2 43(-a(lW)* Then
the solution
Proof.
Suppose
of (2.1) converges that uo E D(q),
strongly
+ 00, to a point
F.
L*(O, +yH).
3.1 cap. IV [2], the function o-0,
g(t) = 44t)) is absolutely
(3.12)
then d*u d12~
By lemma
as t*
we have
continuous
on the compact
sets of IO, + 00[, and with A(t) E @(u(t))
Multiplying
equation
(2.1) by duldt and using the above
equality
we see that
t>O. Now, since by [ 161 F ( see theorem
is nonincreasing,
and so it is g(t).
3.2) is not empty
we conclude
easily that the function
Asymptotic
By
definition
behaviour
of the subdifferential, &m
2 Q)Ms))
of some second
using condition
evolution
equations
1251
(ii) we find for 0 < t s S:
2 (1-4l49M~))
i.e.
+ ($
3 &4))
0 s (2 Now consider
order
(07
(07 -4ll49ll)~(~)
- 49)
-4l4Glld9 - 49))*
the function H(t)
-
1+
4lM9> {/u(t)112 2
IIU(S)~~2} - use)
/U(t) - U(S)l12
We have
t
$)=($()7
Therefore
H(t) is convex 1+
a (IIu oil s 240s + u(t)) + a(llu(s) II>11% (f) Ii22 0.
and since it is bounded,
it is nonincreasing.
Since H(s) = 0, we find
44~>ll> #40112 - 114412~ 3 ; 4ll~(~Ml40 - dQl12 2
Since a( .) takes values different from zero we conclude The limit is obviously a point of F.
easily that {u(t)} is a Cauchy
sequence.
REFERENCES 1. BARBU V., A class of boundary value problems for second order abstract differential equations, J. Fat. Sci., Tokyo, Sect. 1, 19, 295-319 (1972). 2. BARBU V., Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoff, Amsterdam (1976). 3. BREZIS H., Operateurs Maximaux Monotones et Semigroupes de Contractions dans les Espaces de Hilbert. North-Holland, Amsterdam (1973). d’evolution du second ordre associees a des operateurs monotones, Israel J. Math. 12, 4. BREZIS H., Equations 51-60 (1972). convergence of nonlinear contraction semigroups in Hilbert spaces, J. funct. Analysis 5. BRUCK R E. Jr., Asymptotic 18, 15-26 (1972). behavior of nonlinear contraction semigroups, J. funct. Analysis 6. DAFERMOS C. M. & SLEMROD M., Asymptotic 13, 97-106 (1973). of semigroups of nonlinear contractions in Hilbert space, J. Analyse math. 36, 7. PAZY A., Strong convergence l-35 (1978). of nonlinear contractions in Hilbert space, J. funct. Analysis 8. PAZY A., On the asymptotic behavior of semigroups 27, 292-307 (1978). 9. BAILLON J.-B., Un exemple concernant le comportement asymptotique de la solution du probleme (duldt) + aq(u) 3 0, J. funct. Analysis 28, 369-376 (1978). 10. BAILLON J.-B. Quelques proprietes de convergence asymptotique pour les contractions impaires, C.r. hebd. Seanc. Acad. Sci. Paris 283, 587-590 (1976). 11. MOROSANU G., Asymptotic behavior of solutions of differential equations associated to monotone operators, Nonlinear Analysis 3, 873-883 (1979). du second ordre associes a des operateurs monotones, C.r. hebd. Seanc. Acad. 12. VERON L., Problemes d’evolution Sci. Paris 278, 1099-1101 (1974).
1252
E . MITIDIERI
13. VERON L., Un example concernant le comportement asymptotique de la solution bornee de l’equation (d2u/dt2 E &p(u), Mh. Math. 89, 57-67 (1980). 14. HARAUX‘4.) Nonlinear evolution equations-global behavior of solutions, Lecture Notes in Mathematics, p. 841,
Springer, Berlin (1981). 15. HARAUX A., Thesis, Paris VI (1978). 16. MITIDIERI E., On the strong convergence of an iterative scheme related to subdifferentials (preprint). 17. PAZY A., Private communication (1981). 18. BAILLONJ.-B., Quelques proprietes de convergence asymptotique pour les semi-groupes de contractions impaires, C.r. hebd. Skanc. Acad. Sci. Paris 283, 75-78 (1976). 19. PASCALI D. & SBURLAN S., Nonlinear Mappings of Monotone
Type. Sijthoff & Noordhoff,
Amsterdam
(1978).
