I ntnat. J. Math. & Math. Si.
(190) 79-102
NO.
Vol.
79
EXISTENCE AND DECAY OF SOLUTIONS OF SOME NONLINEAR
PARABOLIC VARIATIONAL INEQUALITIES
MITSUHIRO NAKAO
Department of Mathematics
College of General Education
Kyushu University
Fukuoka, Japan
TAKASHI NARAZAKI
Department of Mathematical Sciences
Tokai University
Kanagawa, Japan
(Received January 23, 1979)
ABSTRACT.
This paper
discusses the existence and decay of solutions u(t) of
the variational inequality of parabolic type:
<u’(t) + Au(t) + Bu(t)
for
v e L
P([O,=);V
(p>2)
f(t), v(t)
with v(t) e K a.e.
u(t)> >_ 0
in
[0,=), where K
is a closed
convex set of a separable uniformly convex Banach space V, A is a nonlinear
monotone operator from V to V* and B is a nonlinear operator from Banach space W to
W*.
V and W are related as V
W
c
H for a Hilbert space H.
No monotonicity
assumption is made on B.
KEY WORDS AND PHRASES.
Existence, Decay, Nonlinear pabolic variational
nequaliti
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
35K55.
80
M. NAKAO AND T. NARAZAKI
Introduction
V
be a
real separable uniformly convex Banach space with norm
I Iv
be a real Hilbert space with norm
H
Let
H
densely imbedded in
of
V
containing
II I W
norm
V
V
<W <H.
into
W
H*
V
W
<v*,v>
will be denoted by
W
are compact
H
into
We identify
VW<HW* V*).
(i.e.,
be a Banach space with
We suppose that the natural
and from
and continuous, respectively.
space
be a closed convex subset
K
let
Moreover, let
0.
such that
injections from
and
H
with its dual
V*
Pairing between
v* 6 V*
for
and
and
v e V.
Consider the following variational inequality of parabolic
type
< u’ (t) + Au(t) + Bu(t)
(i)
for
v(t)
LP
A solution
u(t)
u(t) 6 K
(p>2)
[0 ,=) ;V)
u(t)
a.e.
t
6
u
06
is a monotone operator from
bounded operator from
K
a.e.
in
(0,=)
W
following assumptions on
to
6 L
2
loc([0, ) ;H)
and the initial condition
u(0)
A
u’ (t)
C([0,=);H)
[0,=)
(2)
Here
v(t)
with
of (i) should satisfy the conditions
Loc([0,=);V)
for
u(t)> > 0
f(t), v(t)
W*.
them.
K.
V
to
V*
and
B
is a
More precisely we make the
81
NONLINEAR PARABOLIC VARIATIONAL INEOUAIITIES
A
I.
on
V, hemicontinuous on
with some
A 2.
V
C
B
k0 > 0
0(-)
and
and satisfies the inequalities
u
011 u II vp =<
(3)
where
p=> 2,
<__ <u,u>
and
is a monotone increasing function on
is %ke Frchet derivative of a
W
continuous on
function’al
F B (u)
on
W,
+i
with some
k I,>0.
Regarding the forcing term
(t)
[0,).
and satisfies
(5)
A 3.
FA(U)
is the Frchet derivative of a convex functional
A
f L oc ([0,)
max {
t+l
|
< const. <
f (t)
2
;V*)Lloc([0,) ;H)
I
f(s)
a
II,
as
we assume
with
’t+l
I
i/q
If(s)
12
and
as)/}
.
Note that no monotonicity condition on
B
is assumed.
The problem (I) is said ’unperturbed’ if
’perturbed’ if
q=p/(p-l)
B(t) 0.
B(t)=0, and said
The unperturbed problem (i) with the
initial condition (2) is familiar, and the existence and unique-
82
M. NAKAO AND T. NARAZAKI
ness theorems are known in more general situations than ours
(see Lions [5], Brezis [2], Biroli [i], Kenmochi [4], Yamada [13],
etc. ).
However the asymptotic behaviors of solutions as
seem to be known little.
In this note we first prove a decay
property of solutions of the unperturbed problem (1)-(2)
B(t)--0).
t--->
(with
This result is derivea by combining the penalty method
with the argument in our previous paper [i0], where the nonlinear
evolution equations
(not inequalities) were treated.
Next we consider the perturbed problem (1)-(2)
0).
u’ (t) +Au (t) +Bu (t) =f (t)
For the equation
the existence of bounded solutions on
was proved in [8]
(see also [7]).
variational inequality (i)- (2)
[0,)
(i.e., B(t)
(not inequality)
"in the norm
II I V
We extend this result to the
Recently, similar problems were
treated by Otani [12] and Ishii [3] in the framework of the theory
of subdifferential operators.
f(t)-0
or
0
If(s)
the smallness of
IH2
ds
In their works it is assumed that
is small, while here we require only
M_=sup 6(t).
Ishii [3] discussed the decay or
t
blowing up properties of solutions.
We also prove a decay pro-
perty of solutions of the perturbed problem.
Our result is much
better than the corresponding result of [3].
We employ the so-called penalty method introduced by Lions
[5], and the argument is related to the one used in our previous
paper [iI ], where the nonlinear wave equations in noncylindrical
domains were considered.
NONLINEAR PARABOLIC VARIATIONAL INEOUALITIES
i.
83
Preliminaries
We prepare some lemmas concerning a penalty functioanl
8(u).
Let
K
be a closed convex set in
V
J:V --gV*
and let
be the duality mapping such that
Then the penalty functional
(7)
PK
& K)
for
is defined by
pK u)
J(u
is the projection of
K
to
V
K. Recall that
pK u
is determined by
.I
(8)
pK u
8 (u)
8(u)
where
II UIlv2
llUllv,
ll (u) llv,
(6)
u
..IIV
pKu
min
w&K
I
u
w
II V
is also characterized as the unique element of
K
satis-
fying
(9)
<J(u-
pKu)
w-
For a proof see Lions [5].
PKU>
< 0
for
w & K.
The following two lemmas are well
known.
Lemma i.
8(u)
Lemma 2.
Th.__e
(Lions [5]
is a monotone hemicontinuous mapping from
(see, e.g., [6])
projection
PK
i__s
continuous.
V
t_o
V*.
84
M. NAKAO AND T. NARAZAKI
The next lemma plays an essential role in our arguments.
Lemma 3.
u(t)
Le___t
6_
ferentiable on
d
1
(i0)
d--{
I
CI([0,) ;V).
I u(t)-PKU(t) II V2
Then
[0,)
and it holds that
u(t)
PKU(t)II v2
is dif-
u’ (t)>
<8 (u(t))
Proof.
The proof can be given by a
[i, lemma 6].
variant of the way in Biroli
By a standard argument (see Liohs [5, Chap II,
Prof 8.1]) we know
I
(ii)
w
pKw
w,v V.
for
1
I
IIV2
1
v
Then, if t,t+h>0
u(t+h)
pK u(t+h)
_i
2h
It2t2
+h
II
u(s)
pK u(s)
t
(13)
> <8(v), w- v>
we have
llV2
h>0, we have from (12)
If
IIV2
pKv
1
I
u(t)
pK u(t)
I V2
> < 8(u(t)), u(t+h) -u(t)>
(12)
< 8(u(s))
>
tl
for
I
I v2
ds
1
u(s+h)-u(s)
h
t 2>tl__>0, and hence, letting
jftl+h
t
1
> ds
h%0,
I
u(s)(s)
u
-PK
2
llv
ds
NONLINEAR PARABOLIC VARIATIONAL INEOUALITIES
1
I
Y
u(t
2
i
V- g II
PK u (t)II
2
2
1
2h
It2t2+h
[I u(s)-PK u(s)
t
t
for
llV2
i
ds
u(s+h)-u(s)
h
< 8(u(s)),
<
-
h<0, we have
Similarly, if
1
(15)
g
tl+h__> 0
with
II
u(t
2
-PKu (t)
2
pK u(t I)
ilV2
Itltl+h
2
ds
> ds
1
t2>t I
I)
u’ (s) > ds
< 8(u(s))
(14)
u(t
85
and
I12V
1
g
I
u(t l)-pK u (t I)
I12V
< 8(u(s)),u’ (s) > ds
for
t>0.
where we have used the continuity of
t2>tl>0,
pK u(t)
at
The inequalities (14) and (15) are equiavlent to (i0).
We conclude this section by stating a lemma concerning a
difference inequality, which will be used for the proof of decay
of solutions.
Lemma 4.
Let
([9])
(t)
be a nonnegative function on
sup
t<s<t+l
(t) l+r < C 0((t)
[0,)
(t+l)) + g(t)
such that
86
M. NAKAO AND T. NARAZAKI
C0 > 0
with some
(i)
if
(t)<c
r=0
r> 0.
and
Then
g(t)<C 1 exp(-It)
and
exp(-’t)
with some
I>0,
Cl>O
then
CI, I ’>0
for some
and
/
(ii)
and
r>0
if
(t)
2.
<=
lim g (t)
tl+llr=0,
c (l+t)
1/r
then
for some
C
1
> 0
Unperturbed problem
As is mentioned in the introduction we prove here a decay
property of solutions of the unperturbed problem (i)- (2)
Theorem i.
Le__t
u
0
an___d
K
6(t)<C exp(-It)
B(t)-0
let
(I>0)
lim (t) t (p-l) / (p-2) =0 i__f p>2 and
t+
i__f p=2. Then the problem (1)-(2) with
admits a unique solution
(16)
I
u(t)
II v <__ C(II
u0
u(t), satisfying
II v) (l+t) -1/(p-2)
if
p>2
and
(16)’
I
with some
I’>0.
u(t)
II V <__ C(II
u0
llV)
exp(-l’t)
i__f
p=2
NONLINEAR PARABOLIC VARIATIONAL INEQUALITIES
Proof.
-
Recall that the solution
{u (t)}
of
as
e
modified equation
u
is given by a limit function
u (t)
0, where
87
is the solution of the
u’ (t) + Au(t) + 1 8(u)
f(t)
(e>0)
(17)
u(0)
u
0
are monotone hemicontinuous operators from
Since
A
V
V*, the problem (16) has a unique solution
to
and
8
u (t)
such
that
u (t) G
L
;V)
oc ([0,)
and
u (t)
e
e L2loc([0, ) ;H).
(Cf. Lions [5, Chap. 2, Th. 1.2., see also Biroli [i]
where
more general result is given.)
Let
{wj}j= 1
be a basis of
Then, it is known that
V.
u (t)
is given by the limit function of
where
u
m
(18)
m,e
(t)=
=i
.
3, m
(t)w.
3
(t) }
is the solution of
<u’
(t) ,w > + <Au
(t),Wo>
m,e
m,e
j
3
(j=l, 2,
{um
<f(t) ,w >
J
,m)
with the initial condition
(19)
u
(0)
m ,e
u0
m,e
9
u0
in
V.
as
m--9
88
M. NAKAO AND T. NARAZAKI
The problem (17)-(18)
is a system of ordinary differential equa-
tions with respect to
3,m
(t)
nicity and hemicontinuity of
j=l,2,...,m, and by the monotoA
8
and
it is easy to see that
this problem admits unique solution such that
Um, e (t)6 CI([0, ) ;Vm)
V
m
where
CI([0, ) ;V)
is the m-dimensional subspace of
...,}.
V
spanned by
{w I,
For the proof of Theorem I, it suffices to show that
the estimate (16) or (16)’ with
independent of
m
u=u
holds with the constants
m,e
e.
and
By Lemma 3 we have
(20)
Ee
(um ,e (t2))-
Ee
(um ,e (t))
1
+
tl
2
lu’m,e (s)
ds
t
<f (s)
tl
for
t
m,e (s) >ds
2 >tl>0, where
E (u(t))
e
FA(U(t))
+
Also we have easily by (18)
t
AUm, e(s) ,um
{<
t
(21)
(s) > +
1
_I
I
PK u (t)
u(t)
<8(um
(s),um
112V
(s) >}ds
I
I
t
t
2 {<f(s)
,um ,e (s) >
<u’m,e (s)
,um
(s) >}ds.
1
Using the similar argument as in [10], the equalities (20)-(21)
NONLINEAR PARABOLIC VARIATIONAL INEQUALITIES
imply the estimate (16) or (16)’ with
u=u
89
For completeness,
m,e
however, we sketch the proof briefly.
By (20) we have
2 ds < 2{E
m,e
t
e
(Um, e (t)
Ee
(Um, e (t+l))}
+ C6 (t)
(22)
2
--D (t)
constant)
(C>0
On the other hand, using the ineqaulity
+ 1 <8(um ,e (t))
<AUm, e(t),um, e(t)>
Um, e (t) >
> E
e
(um ,e (t))
(see (3) and (9)),
we have from (21)
It+l llf(s) IlV,
t+l
Jt
E (um
e
e(s))ds
ds) 1/2
2
<
t
t+l
+
t
lu’m,e (s)
2
sup
se[t,t+l]
ds) 1/2
II Um, e(s) ii v
sup
t%s=t+l
m,e
(23)
< C(D (t) + 6(t)
e
where hearafter
and
C
E
(um
"
t* 6[t,t+l]
(t*)) < C{D (t) + 6(t)) }
and hence by (20)
I/p
denotes various constants independent of
From (23) there exists
e.
E( e (s))
t.s %t+l ,L,
sup
sup
t<s<t+
such that
e
(s))/P
m
90
M. NAKAO AND T. NAATAKI
E
sup
t< s< t+l e
(um ,e (s))
< C{ (D (t) + 6 (t))
e
<s
t< s< t+l
+ D (t) 2 + D (t)6 (t)}
and by Young’ s inequality,
E (um e (s)) < C{ (D e (t) + 6
sup
t< s< t+ 1 e
(24)
(t))P/(P-1)
+ D (t) 2 + 6 (t)
From (24) we can easily see that
[0,)
E
by a constant depending on
(um
E (um
e
assume, without loss of generality, that
(t))
C(-)
where
quantity.
(26)
m,e (0)
u
u
0
eK
and
IIV
By (20) and (25) we have
sup
C(II
where we set
result.
Since we may
denotes various constants depending on the indicated
E
t<s<t+l
<
is bounded on
(0)).
Ee(Um,e(t)) =< C(Ee(Um,e(O))) =< C(li
(25)
2}
u
0
e
(um ,e (s))
Iv
M) {E
2 (p-l)/p
(Um,
2
M--sup 6(t)
t
(t+l))
E
(Um,
(t)) + 6 (t)
2}
Applying Lemma 4 we obtain the desired
NONLINEAR PARABOLIC VARIATIONAL INEQUALITIES
3.
91
Perturbed problem
In this section we investigate the existence and decay of
solutions of the problem (1)-(2) with
tion A
2.
For this consider the approximate equations
<Um, e (t)
(27)
satisfying the assump-
B
+ Aum,e (t) + Bum,e (t) + 1 8(um ,e (t))
j=l,2,...,m, where we set again
.
f(t), w. >
3
m
Um, e (t)
e
j=l
m,j
(t)w.
3
and we impose
u
for
We also give a rather brief discussion.
(eK)
(0) G K and u
u
in V
m,e
m,e (0) --9 0
Using a similar argument as in [7] we derive a priori estimates
u
mtE
we assume
(t).
p>e+2.
By (27) we have
Se,0(U’m,e(tl))
Ge,0(Um, e(t2))
(28)
First
+
t1
lU’m’e(s)
2
as
t
<f (s) u’
>ds
m, e (s)
where
Ge,0(u(t))
FA(U(t))
+
FB(U(t))
+
and hence, in particular,
(29)
G
e,0
(um ,e (t))
< G
e,0
(um ,e (0))
1
flu(t)
+ 1 8(0)
pK u(t)
if
112v
0<t<l
0,
92
M. NAKAO AND T. NARAZAKI
p>+2
which together with the assumption
I Um,e (t) I V=< c<ll
(30)
if
0<t<l.
u
m
Thus
(t)
If we assume
<
[0,tm]
Ge,0(Um,e (t))<G e,0(um,e (t+l))
tm>l.
some
t>0, we have from (28)
t+l
lu’m,e (s)
t
I v’a 0
exists on an interval, say
with
(31)
u0
implies
for
2 ds < 6 (t) 2 < 2
M
Using (27) and (31)we have
it+l
G
t
e,l
(urn,
(t))ds < M 2 + C
it+l II
(s)II v2
um
t
ds
where we set
(32)
G
e,l
(u)
<Au + Bu + 1 8(u), u>
Since
G,<u>
and since
__>
a+2
koll u IIv kll u IIw
p>e+2, theexists a point
I1 Um,t*
IIv + 1 II Um,
From this and (28)
(t*)
+ i
11
u
pKu
t* [t,t+l]
PKUm
e
(t*)
II v2
IIv2
such that
< C(M)
INEOUALITIES
NONLINEAR PARABOLIC VARIATIONAL
Ge,0(Um,e(t+l))
<
Ge,0(um
93
(t*)) + C6(t) 2 < C(M)
Thus we conclude that
G
e,0
Um, e (t))
and therefore
(32)’
II Um
e
u
(t)
m
(t)
Iv
+ 1
< max(C(M)
max G
0<s<l e,0
exists on
I Us,
(t)
ds
__< C(M,
[0,)
(um ,e (s)))
satisfying
i[ V2
(t)
PKUm
< C(M
I
u
0
II V)
Of course we know
.t+l
I
()
t
lu’m,e (s)
II
u0
II v)
for
t>0.
We have now derived a priori estimate for
u
m,e (t). Using
standard compactness and monotonicity arguments (see Lions [5]
Biroli [i] etc.) we can suppose without loss of generality that
as
(34)
m
,
L ([0, ) V)
u
m, (t)
u (t)
weakly* in
u’m,e (t)
u’ (t)
weakly in
L 2loc
-
weakly** in
AUm, e(t)
BUm, e(t)
+
i 8(Um, e(t))
Bue(t)
Xe(t)
strongly in
0, ) ;v).
Lr ([0,=) ;W*)
L ([0,=);V*)
(r>l)
94
M. NAKAO AND T. NARAZAKI
and
(35)
Au (t) + I
xe(t)
? S(u(t)).
Moreover, with the aid of the inequality
< s(u)
s(v)
u,v
V, we know
for
(36)
lim
m-oo
u- v >
II Um, e (t)
Z (llu- pullv- llv-
PkUm(t) II v
I u e(t) "- PKUe(t)
in
The limit function
u (t)
E
vll v) 2
IV
L 2loc([0’) ).
satisfies
U’ (t) + Au (t) + Bu (t) + 1 8(u(t))
e
e
e
f(t)
a e
(37)
u (0)
u
0
Furthermore, it holds from (32) and (33) that
(t)
(t)
;I
< c(,
and
’
t+l
t
lu’s
s
<
c
u0
llv
I
uo
;Iv)
on [0, =)
--
NONLINEAR PARABOLIC VARIATIONAL INEOUALITIES
for
t>0.
u’e(t)
u e(t)
(38)
suppose as
Then we may
--
2
e’
weakly in
Lloc([0,=) ;V),
u(t)
weakly* in
L ([0, ) ;V),
Au e(t)
Bue(t)
0
u (t)
Clo c([0,) ;H),
and in
and
95
(t)
Bu(t)
----)
L([0, ) ;V*)
weakly** in
Lr([0,);W *) (r>l)
strongly in
Moreover from (32)
u e(t)
PKUe(t)
in
0
L=([0, =) ;V),
which implies easily
u(t) e K
(39)
a.e.
on
[0,)
By a standard monotonicity argument (see Biroli [I]) we see
x(t)=Au(t)
a.e.
on
[0,), and by (37) we have
< u’ (t) + Au(t) + Bu(t)
for
v(t)
eLP([0, ) ;V)
with
f(t)
v(t)
eK
u(t) > > 0
v(t)
a.e.
on
[0, )
We summarize above result in the following
Theorem 2.
Let
p>e+2.
Then under the assumptions
AI,A 2
and A 3, th__e
M. NAKA0 AND T. NARAZAKI
96
problem (i)- (2) admits a solution
t+l
I
u(t
llv
t>0, wher e
four
+
t
we
u(t)
-
such that
lu’ (sl s <=c(,llu 011 v
.se___t M-- sup
(t)
t
Next, we assume
2<p<e+2.
As is already seen, for the
existence of solution it suffices to show the boundedness of
Um, e (t)
by a constant independent of m and
e.
For this we
set further
x Se+2 I
i, ou olI u llv
and
,x (u)
where
,o (u) +
is a constant such tat
S
Note that
(40)
and
x
0>0
(41)
GE, 0(u)
G
e,l
and
(u)>G
max
x>O
e ,0
D0>0
(k0xP
>
e,0(u)
(u)-2k
u
1
+
1
I
for
u II2
u
llu- p<ull v2
I u llw<=sll u llv
Ge, l(u)
II[ u I we+2
llv+2
>__
e,l(U)
u 6V.
__>
for
u@V.
GE, 0(u)
Let us determine
as follows.
k3Se+2x e+2)
Then ’the stable set’
k0x
is defined by
k3Se+2x
e+2
D
O
NONLINEAR PARABOLIC VARIATIONAL INEQUALITIES
{uV
(42)
Ge,l(U)
<
u0e K,
Let us assume the initial value
2/D0-Ge,0(u 0)
M0>0 such tht
m
(>0).
if
G
if
u
IIV <
x0
and let
M < M
6
We shall show that there exists a constant
M<M0’ Um,e (t)
is sufficiently large.
(43)
I
and
D0
97
e,0 (um ,e
6
for
t<__tm
provided that
First, by (29),
(t)) < G
e,0
+
(u)
0
41-
0<t<min(l,tm), for sufficiently small
< D
0
M +
>0
and large
m.
t >i
Thus, if ohr assertion were
m
false, there would exist a time >i such that
The inequality (43) implies
G
(44)
< D
e,0(Um, e(t))
O
if
(um ,e ())
D0
0<t<t
and
G
(45)
e,0
t2= tl=t---i
By (28) with
we have easily
t
(46)
-l
lUm
e
(s)
12
2
ds < M
and hence
(47)
G
-I
,i
(um , (s)) ds<
-i
< 2MS
l-U’m,e (s)+f(s) flUm,e (S)
ix0
98
M. NAKA0 AND T. NARAZAKI
where
S
1
is a constant such that
u
for
Therefore, if we assume
6 [t-l,t]
(48)
where
M<M--D0/2SIX 0,
e V.
there exists a time
t*
such that
Ge,l (Um,e (t*))
x(M)
(<x0)
2MSlX 0
II Um, e (t*) II v
and
< x (M)
is the smaller root of the numerical equation
k0xP
(49)
_<_
Se+2 xe+2
kI
2MSlX 0 (<Do)
We use again (28) to obtain
GE, 0(um, e())
<
(50)
< G
<
Now we determine
(51)
and set
(52)
GE, 0(um, e(t*))
e,l
M’">0
"’0
+ 1 M2 +
2klS
e+2
klSe+2 I
(t*)
u
m
;I ve+2
x (M) +2
as the largest number such that
2klSe+2 x (S"’)
’
1_ M 2 + 2
(um (t*)) +
,e
2MSIX 0
M0--min(M6,M
M2
+
).
+ 2M"’
SIX 0
+ 1 M"’ 2
Then, assuming
Ge,0 (Um, e ())
<
M<M0,
DO
DO
(M’
<
’
we have by (51)
NONLINEAR PARABOLIC VARIATIONAL INEQUALITIES
which contradicts to (45).
[0,)
exists on
m
for large
llUm ,e (t) I V <
53
Consequently, if
M<M0, Um, e
(t)
and it holds that
t+l
x0
99
t
u’m
e
(s)l
2
<
ds < const
and
Ge,0(Um,e (t))
<
t &[0, )
for
D0
Thus, applying the monotonicity and compactness arguments, we
obtain the following
Theorem 3.
Le___t
2<p<u+2
u_
a solution
u(t)
and
M<M
.Then th__e problem
0.
(1)-(2) admits
satisfying
IV<
x0
ad
I
t+l
t
lug-m
(s) 2 ds < const. <
Moreover, we note that the approximate solutions
u
mI
(t)
(m:
large) satisfy
(54)
G
,0
(um , (t))
> G
,0
Um, (t))
klSe+2Xo(e+2)-P) II um,e(t)
- >- (ko
1
with
(k0_klSe+2 x +2)-p )>0.
the section
u-
I vP
exUllv2
Therefore the same argument as in
yields the following
M. NAKAO AND T. NARAZAKI
i00
Theorem 4.
Le___t 2<__p<+2 an___d
M <M 0.
Then the solution
in
Theorem 3
satisfies the decay property
If
(i)
and
p>-2
(t) t (p-l) / (p-2) =0, then
lim
t/
or
I__f p=2
(ii)
6 (t)<C exp{-%t}
and
I
for some
Remark.
and
p>2
IIV <-- C’
u<t)
(C,%>0)
.then
exp{-%’t}
C’ %’>0
lu(t) l<C(l+t) -I/(p-2)
[3], Ishii proved that
In
lu(t)
I<C
exp{-%t}
(C,%>0)
if
for the case
p=2
if
f-0.
It is clear that our result is much better, because the norm
II"
4.
is essentially stronger than the norm
v
An example
in
R
V
with
w’P(),
0<e < pn/(n-l) +2
A; V ---) V*
by
if
H
L
2()
n>p+l
W
L
e+2()
0<e<
if
n<p.
and
and
be a bounded domain
Let
Here we give an typical example.
n and set
We define
NONLINEAR PARABOLIC VARIATIONAL INEQUALITIES
n
u p- 2
x%-x-Tl
< u,v >=
u,v 6W 0l,p(), and
for
d(x)
Bu
d (x)
where
W*
B:W
luleu
I01
u u ax
by
ue L e+2()
for
.
is a bounded measurable function on
Moreover
we set
{ueW
K
where
a, b
01’p()
b(x)
=<
u(x)
=<
a(x)
are measurable function on
Then all the assumptions
AI-A 2
a.e.
-(2) is equivalent in this case to the
n}
a (x) >0>b (x)
with
are satisfied.
on
The problem (i)
problem
f(x,t)
a.e.
on
[0,)
where
b (x) <u (x t) <a (x)
Lu(x,t) < f(x,t)
a.e.
on
[0,)
where
u (x t) =a (x)
Lu(x,t) > f(x,t)
a.e.
on
x [0,)
where
u (x t) =b (x)
Lu(x,t)
with the conditions
u
=0
a.e.
[0,)
on
and
u(x,0)=u 0(x)
(eK)
where
Lu
B__u
t
n
Z
j=l
B
8x.
m
U) + d(x)
uIu
a.e. on
,
M. NAKAO AND T. NARAZAKI
i02
REFERENCES
i.
Sur les inquations paraboliques avec convexe d@pendan du temps:
solution forte et solution faible, Riv. Mat. Univ. Parma (3)3(1974),
Biroli, M.
33-72.
Problemes unilateraux, J. Math. Pures appl.,
5I
(1972), 1-168.
2.
Brezis, H.
3.
Ishii, H.
4.
Kenmochl, N. Some nonlinear parabolic variational inequatities, Israel J Math.
22 (1975), 304-331.
5.
Lions, J. L. Quelques Mthodes de R@solution des Problemes aux Limies Non
Linearies, Dunod, Paris, 1969.
6.
Martin, Jr., R. H. Nonlinear Operators and Differential Equations in Banach
Spaces, J. Wiley & sons, Inc. New York, 1976.
7.
Nakao, M.
8.
Nakao, M.
Asymptotic stability and blowing up of solutions of some nonlinear
equation, J. Differential Euations, Vol. 26, No. 2 (1977), 291-319.
On boundedness, periodicity and almost periodicity of solutions
of some nonlinear parabolic equations, J. Differential Equation_s,
19 (1975), 371-385.
On the existence of bounded solution for nonlinear evolution
equation of parabolic type, Math. Rep. College Gen. Educ., Kyushu Univ.,
XI (1977), 3-14.
9.
Convergence of solutions of the wave equation with a nonlinear
dissipative term to the steady state, Mem. Fac. Sci. Kyushu Univ., 30
Nakao, M.
(1976), 257-265.
i0.
Nakao, M. Decay of solutions of some nonlinear evolution equations,
J. Math. Anal. Appl. 60 (1977), 542-549.
ii.
Nakao, M. & T. Narazaki. Existence and decay of solutions of some nonlinear
wave equations in noncylindrical domains, Math. Rep. Co.lege Gen.
Educ. Kyushu Univ. X_I (1978), 117-125.
12.
tani,
M.
2(u(t))
13.
f(t), J. Fac.
Sci. Univ.
du
I
(u(t))
+
Tokyo, 2_4 (1977), 575-605.
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(t)
Yamada, Y. On evolution equations generated by subdifferential operators,
J. Fac. Sci. Univ. Tokyo, 23 (1976), 491-515.