Decay rates of solutions to thermoviscoelastic plates with memory

MA Journal of Applied Mathematics (1998) 60,263-283
Decay rates of solutions to thermoviscoelastic
plates with memory
JAIME E. Mur;joz RIVERAt
National Laboratory for Scientific Computation, Department of Researchand Development,
Rua Lauro Miiller 455, Botafogo Cep. 22290, Rio de Janeiro, RJ, Brasil
AND
RIOCO KAMEI BARRETO
University Federal Fluminense,Rio de Janeiro, RJ, Brasil
[Received 13 March 1997and in revised form 7 August 19971
We considerthe thermoelastic
plateunderthe presence
of a long rangememory.We find
uniform ratesof decay(in time) of the energy,providedthat suitableassumptions
on the
relaxationfunctionsaregiven.Namely,if therelaxationdecaysexponentiallythenthefirst
orderenergyalsodecaysexponentially.Whentherelaxationg satisfies
--c&)r+“P
< g’(t) < -~g(t)‘+“~;
andg, g’-“P E L’(R) with p >
2
thentheenergydecaysasl/( 1 + t)P. A newLiapunovfunctionalis built for thisproblem.
1. Introduction
In this paper we study the thermoelastic plate model endowed with long range memory.
Our attention will be focused on the asymptotic behaviour of the solution as time goes
to infinity. More precisely, let us denote by 52 an open bounded set of Iw* with smooth
boundary lY We assumethat the boundary is divided into two parts, such that
r = ro U rl
with
TO nF1 = 0
and ro # 0.
(l-1)
Let us denote by v = ( ~1, ~2) the external unit normal to r, and by q = (-~2, vi) the
unitary tangent positively oriented on r. The equationswhich describesmall vibration of
a thin homogeneous,isotropic thermoviscoelasticplate of uniform thickness h are given
by:
t
utt - hdu,, + A*u + ad0 s0
et -
KAO
+ y0
u(x9 y, 0) = Uo(& y>;
t Also at: Institute
of Mathematics,
Federal
g(t - r)A*u(t)
- aAut
=
0
dt = 0
in Qx]O,
u&L y9 0) = uI(x, y>
University
in 52x10, oo[,
OO[,
(W
(l-3)
(x9 Y> E 52
(1.4)
of Rio de Janeiro, RJ, Brasil.
@ Oxford University
Press 19%
264
JAIME
with the following
u=-
dU
E. MUfiOZ
RIVERA
AND
RIOCO
KAMEI
BARRETO
boundary conditions:
= 0
on rox]O, oc[,
(13
dV
t
l31u+ae
431
autt
&~-h~+~~-a,
(s
0
g@ - r)u(t)
(s
ae
on rx]O,
= 0
on ri x10, oo[,
(W
t
ae
G + he = 0
dt
o la-
t)u(t>
dt
I
= 0
on ri x10, oo[, (1.7)
oo[,
WV
dAu
&u = - av +(l
&U = Au + (1 - ,u)Blu,
-P>-
aB2u
ar7 ’
and B1 and B2 are given by
0
= 2vp2--
a2u
axay
-
,a224 2 a2u
‘1 ay2 - ‘2 ax2’
B2u = (v; - v,“) &+v1v2{$-3
In (1.2), u = u (x, y, t) denotes the position of the plate. We may interpret equation (1.2)
as saying that the stresses at any instant depend on the complete history of strains which
the material has undergone. By g E C2(Iw, lEX>we denote a positive real function and the
constant ,X is assumed to be in the interval IO, i[. The constants K, y, a, h, and h are
assumed to be positive.
The main result of this paper (Sections 3 and 4) is the uniform rate of decay of the
solution to the system (1.2) to (1 .S). We prove that the rate of decay of the solution depends
on the rate of decay of the relaxation kernel g. That is, when g decays to zero exponentially
then the vertical deflection and the thermal difference also decay to zero exponentially.
Moreover if the relaxation decays polynomially then the vertical deflection and the thermal
difference also decay polynomically. This shows that the memory effect is strong enough
to produce the dissipation which enables us to show that the energy decays uniformly
(exponentially or algebraically).
Let us mention some other papers related to the problems we address. Dafermos (1970)
proved that the solution to viscoelastic systems goes to zero as time goes to infinity, but
without giving explicit rate of decay. Lagnese (1989) considers the linear viscoelastic equation, obtaining uniform rates of decay but introducing additional damping terms acting on
the boundary. Uniform rates of decay for the solutions of linear viscoelastic equations with
memory were obtained recently by Mufioz Rivera (1994).
Concerning the asymptotic stability of the solution for inhomogeneous anistropic
thermoelasticity we have the pioneering work of Dafermos (1968). In that work it is proved
that the solution is asymptotically stable as time goes to infinity. In particular that the thermal difference as well as the rotational part of the displacement goes to zero; but no rate of
decay is shown. For 3-dimensional isotropic homogenous materials the work of Dassios &
DECAY
RATES
265
OF SOLUTIONS
Grillakis (1984) shows that a displacement vector field can be decomposed into two parts,
the rotational and selenoidal components. The rotational component decays uniformly to
zero like t-5 while the selenoidal component conserves its L*-energy. This result was
improved in (Mufios Rivera, 1993). The above results illustrate that the displacement, in
general, does not go to zero. This because for n-dimensional thermoelastic bodies, the displacement is a vector-valued field and the dissipation given by the thermal difference has
only one degree of freedom. By contrast, in the l-dimensional case, both the thermal difference and the displacement have the same degree of freedom. So, uniform rate of decay
is expected as was shown in (Hansen, 1992; Henry et al., 1993; Kim, 1992; Mufioz Rivera,
1992; Renardy, 1993). For plates the situation is similar to the case of l-dimensional
thermoelasticity, because the vertical deflection as well as the temperature are scalar functions. So, uniform rate of decay is also expected and was proved in (Avalos & Lasiecka, to
appear; Kim, 1992; Liu dz Renardy, 1995; Mufioz Rivera & Racke, 1995; Mufioz Rivera &
Shibata, 1997).
Adding the viscoelastic damping to the thermoelastic system, we should expect that
the elastic and thermal parts of the energy continue to decay to zero exponentially even
if the relaxation kernel decays polynomially, but this is not true as we can see in Remark 1.1 below. So, the memory effect prevails over the thermal difference and introduces
a new difficulty to the problem. Unfortunately the method used to obtain uniform rates of
decay in the above works are based on second-order estimates, which make the problem
of securing estimates more complicated. Thus, the methods that have been used for establishing uniform rates of decay fail in the case of a thermo-viscoelastic system, and so new
asymptotic techniques have to be devised.
The method we use here is based on the construction of a functional C for which an
inequality of the form
d
zL(t)
< --cc(t)‘+;
holds, with c, p > 0.
Let us describe briefly all the sections of this paper. In Section 2, we prove the existence
of weak solutions. Furthermore, we show some regularity results. To do this we assume
that g satisfies
00
g,
g’, g” E L’(0, OQ),
1-
s0
g(t) dt > 0.
g(t) 2 0, g’(t) < 0.
(1.10)
Since the methods of Section 2 are quite standard we will only give a brief summary of
the procedure. In Section 3 we show the exponential decay of the energy assuming that g
satisfies additionally
-cog(t) < g’(t) < -c1g(t),
0 < g”(t) < c*g(t)*
(1.11)
Finally, in Section 4, we show that the polynomial rate of decay of the relaxation implies
266
JAIME
E. MUNOZ
RIVERA
AND
RIOCO
KAMEI
BARRETO
the polynomial rate of decay of the solution. To do this, instead of assumption (1.11) we
consider the weaker hypotheses
-cog
l+l'p(t)
<
g'(t)
<
-C*g'+'lP(t),
00
g
‘-*‘P(t)
0 < g”(t)
< c2g1+1’p(t),
dt < 00,
p > 2, (1.12)
p > 2.
(1.13)
Assumptions (1.12) and (1.13) mean that g m (1 + t)-P for p > 2. All the above constants
i = 0, 1,2, are positive.
Ci,
REMARK
1.1 It is well known that the solution of the thermoelastic plate system decays
exponentially as time goes to infinity. But when we introduce the viscoelastic dissipation
with relaxation kernel g satisfying assumptions (I. 12) and (1.13), the solution does not
decay exponentially any more. That is, the memory effect given by the convolution term
prevails over the thermal dissipation. In fact, to facilitate calculations let us assume that the
system (1.2) to (1.4) satisfies the following boundary conditions:
U =
= 8= 0
Au
inr.
It was proved in (Mufioz Rivera & Racke, 1995) that the solution of the thermoelastic plate
system (without memory) decays exponentially as time goes to infinity. Let us supposethat
the same is true for the thermoviscoelastic system with the kernel satisfying assumptions
(1.12), (1.13). Take uo = WI, ut = wi and& = ~1, where wi is the first eigenfunction of
the A operator with Dirichlet boundary condition. Denoting by hl the first eigenvalue we
conclude that the solution can be written as
u(x,
0
=
and
f(Wl
m,
0
=
w)~l,
where f and h satisfy
t
(1 + h&f”
+ h;f - ahlh = A;
g(t - t) f (t) &,
s0
h’+Khlh+Alyh+ahlf’=O.
Let us rewrite equation (1.2) as
t
utt - hAutt + A2u + aAt3 =
s0
g(t - z)A2u(t)
dt
in 52x10, oo[.
(1.14)
Since the elastic and thermal parts of the energy decay exponentially, it is easy to see that
the left-hand side of the above equation also decays exponentially. While the right-hand
side satisfies
t
s0
t
g(t - t)A2u(t>
dt =
s0
g(t - z)f(z)
l
.-
I
dt A2w1.
DECAY
ts
Note that f(z> = o(e-yr),
I 5%
RATES
267
OF SOLUTIONS
2
s
so we have
g(t)e-yct-‘)
g(t - t)emyr dx =
dx = emyf
s0
0
0
‘g(t)eYr
dx.
It is not difficult to see that when t + oo
t
e-Yt
dx = o((1 + t>-P>.
g(t)eyr
s 0
But this is a contradiction because the left-hand side of (1.14) decays exponentially. This
proves that the exponential decay is not expected.
2. Existence and regularity
of weak solutions of
In this section we study the existence as well as the regularity
system (1.2). To do this we introduce the following spaces:
V := { 2) E H’(D);
W :=
2) = 0 on ro) ,
(
3W
wEH*(Q);w=;=Oon~0
.
I
define the bilinear form a (9, l) as follows:
LetOct+;we
a(u,v)=
a*ua*v
--+--+p
ay* ay*
2
a*u a*v
ax* ay*
--+--
a224a*v
ay* aX*
2
+ 2(1 -/.&au
axay axay
Thanks to Korn’s inequality the bilinear form a(*, l) defines a norm in W equivalent to the
usual norm of H*(Q). Note that the space V together with the norm
is a Hilbert space.
Frequently, we shall use the Green’s formula for A* which is proved in (Duvaut & Lions,
1976). For proof we reproduce the proof here.
LEMMA 2.1
Let u and v be functions in H4(i2)
(A 2 u)v dA = a@, 21)+
s L?
Pro@
From Green’s formula we get
n W. Then, we have
(B2u)v - (&u)-
av
drl
au I
cw
268
JAIME
s
(A 2 u)v dA=
52
s
f,
-s l-1
(-
E. MUROZ
ilAU
RIVERA
)vdrr--
av
+(1-p)
RIOCO
KAMEI
BARRETO
Au Av dA
a,;drl+/-
s l-1
drr -
( *)v
av
AND
$2
Au;
s l-1
a2u a2v
a a3 a3
dI-i +a@, v)
a2u a2v
a2u a2v
dA.
sQ axay axay
--+$-ay2dA-2(1-p)
s
--
Recalling the definition of Br and B2 and using
s
a2u a2v
--a2u a2v + --dA-2
52ax2 a9
ay2 ap
a2u a22,
s szaxay axay
--dA=
our result follows
cl
To showthe existenceof regular solutionswe first prove, using the Galerkin method, the
existence of weak solutions (seeDefinition 2.1 below), then using elliptic and parabolic
regularity we will obtain the regularity of solutions to the thermoviscoelasticplate equation. To simplify our analysis, we introduce the following binary operators:
g 0 a224=
ts
g(t - ee40
go”u=~tg(t-r)~
- U(t), u(t) - u(z)) dL
lVu(x,
t)
- Vu(x, t)l
2
dAdt,
With this notation we have the following.
LEMMA
2.2
For any v E C*(O, T; H2(L?)) we get
tsg(t
a(
t
t
t)
- t)v(t)
dt, v,) = -ig(t)a(v,
V) +
igf
0
a22,
0
s(t - t)Vv(t)
ssIn 0
go - t)v(x,
dt
l
t) dtv,(x,
Vv, dA =
dA =
‘“( goav2 -(~+w],
-Tdt
DECAYRATESOFSOLUTIONS
269
Pro@ From the symmetry of a (0, 0)we get
d
dt {g cl a2v} = g’ cl a2v - 2
t
s0
t
go - @Wt>,
v(t)) dt + 2
go - t>aW),
vt(t>>
f
=g’oa2v-2
s0
-g(OO, v>.
g(t - t) dt a(v, v,)
s0
d
t
dt + dt
g(t) dt a(v, v>
(s 0
I
cl
Which showsthe first identity. The proofs of the others identities are similar.
The definition of weak solution we use in this work is given asfollows.
DEFINITION
2.1
We say that the couple (u, v) is a weak solution of equations (1.2) to
(1.8) if
8 E CO([O,T]; H’(52))
u E C’([O, T]; V) n CO([O,T]; W),
and the following identities are satisfied:
T
T
-utqt - hVu VP, dA +
ss0
n
s0
T
ss0
T
T
a@,
l
p)dt
+a
ss 0
BAcpdxdti-2
-sL?
u&,
0)
dA
+hsA-2
VulVspL
0)
dA
T
T
6+ktdA dt +
ss0
52
a(g*u, p)dt
s0
KV~V+ + ye@ dAdt + a
ss0
L?
In
Au@, dAdt
T
-
Oo+(*, 0) dA - a
Auolc/(*, 0) dA - h
s 52
s 52
ss 0
9@drdt,
WI
l-1
for any function cp E C’([O, T]; W) such that (p(*, T) = 0, p,(*, T) = 0 and @ E
C’([O, T]; V) suchthat @(-, T) = 0.
Note that since & # 0, Korn’s lemmaimplies that ,/m
is a norm equivalent to the
usualSobolev norm I] 1(2 on W. Let us introduce the energy function
l
2 + h]Vvt12dA + (1 -
s0
‘(gdt)a(v,
v)
+g
0 a2v +ll+12dAJ
;
let us denote by (wi E W; i E Af) an orthonormal basisof W. In theseconditions we
able to show the existenceof weak solution to the thermo-viscoelasticplate equation.
2.1 Let us supposethat g satisfies(1.9) and (1.10). Then for any initial data
(UO,u 1, 60) E W x V x L2(s2), h > 0 and T > 0, there exists a weak solution to equation
(2.2).
THEOREM
270
Pro@
JAIME
E. MUROZ
RIVERA
AND
RIOCO
KAMEI
BARRETO
Our starting point is to construct the Galerkin approximation urn of the solution,
m
urn(-,
t)
=
e”(*,
hi,m(t)wi(‘),
c
i=l
t)
=
~k,.m(t)s(*),
i=l
which is given by the solution of the approximated equation
s
s
VuzVWj
s 52
uzwjdA+h
52
dA + a(um, Wj) - a(g * Urn, Wj) + a
O) =
8”Awj
dA = 0,
(23
VBmVvj dA + Y
BmAvj dx +a
Auyvj
8,FVj dA + K
s sz
s 52
s In
62
um(*9
s i2
Uy(‘,
UO,m,
0)
=
Ul,m
UO,m =
0)
0”(*,
Ul,m,
dA = -h
emvj dr.
(24
=
00,m
ulwi dA
=
sr
I
Wi 9
Let us denote by A = (aij) the matrix given by
Uij
=
(s
Lz
wiwj
dA+h
s sz
Vwi
l
Vwj dA
.
>
We can easily verify that A is a positive definite matrix, so the existence of the approximated solution urn is guaranteed. Let us multiply equation (2.3) by h) m, equation (2.4) by
kj 9m then summing up the product result in j and using Lemma 2.2 we conclude that:
]u~12+hlVu~~2dA+
= ig’ cl a2um - ig(t)a(u”,
( 1 -~gd~)a(um,um)+g~~2um+~~~~2dA]
urn) -
K
s
In
IVBm12 dA - y
10” I2 dA - h
10” I2 dr.
s L?
sr
In view of the hypotheses on g we get
E(t, Urn, em) < E(0, Urn, em);
and from our choice of ~0,~~and u1 ,nzit follows that
DECAY
RATES
271
OF SOLUTIONS
Urn
is bounded in
C([O, T]; W) n C’([O, T]; V),
em
is bounded in
C([O, T]; L2(Q))
n L2([0, T]; H’(Q)).
Multiplying equation (2.3) by /3 E C2([0, T]), such that p(T)
[0, T] we have
= 0 and integrating over
T
-
Uywj/&
i-2
ss 0
+ hVuy
l
VWjbt - aedwjp
dA dt
T
+
--
s0
T
a(um, wj)B dt -
ul,mwjp(O)
s 52
s0
a(g *
urn,
wj)p
dA + h In Vul,mVwj~(O)
s
dt
dA.
Letting m + oo and using the density of the terms { WjB; j E N, /I E C([O, T])}, we get
that u satisfies (2.2). Similarly we get that 8 also satisfies the weak formulation. The proof
cl
is now complete.
To prove the regularity of the solution we introduce the following definition.
2.2
DEFINITION
We say that (UO, ~1, eo) is k-regular if
u j E H2+k-j(S2)
j = 0,. . . , k,
n W,
where uj and 8j are obtained by the following
uk+l E V,
Bj E Hk-j(52),
recursive formula:
Uj+2 - hAuj+2 = A2uj - ad6j
- f,,(O),
8.J+l - KAej + y8j = aAuj+l,
h-
au J+2
*
ae
=
B2Uj
+
3V
a-onrt,
tlV
Mj
x
+
Aej
= 0 onr,
and also
aUj
ui
=
av
= 0
on
ro,
44
=0
on
rr,
Vj = l,-,k,
where fj is given by
j-0 := 0,
.f j+l = fi(t> - g(t)A2uj,
fj E LYO, T; L2(Q)),
To show the regularity result we will usethe following lemma.
j = 1,
l
l
* , k.
272
JAIME
2.3
LEMMA
E. MUROZ
Suppose that
RIVERA
AND
RIOCO
KAMEI
BARRETO
f E L2(i2), g E Hi (Ii) and h E Hs (l-i); then any solution
of
d-b w> =~fwdA+~l~wd~i+~lh~drr
VWEW
satisfies
v E H4(J2)
and also
f
A2v=
VC--
3V
au
&v = h,
Moreover, if
Pro05
(f, g, h) E Hk(i2)
ina,
= 0
on PO,
&v = g
x Hk+1/2(I’l)
on&
x Hk+3/2(l-‘l),
then v E Hk+4(i2).
cl
See (Lions & Magenes, 1972).
The regularity of the solution is established in the next theorem.
2.2 Let us suppose that the initial data (~0, u I,&) is k-regular (k > 2) and
hypothesis (1.1) holds. Then there exists only one solution of equation (1.2) satisfying
THEOREM
U E c’([o,
T]; v n Hk+‘(i2))
0 E c’([O,
n C’([O, T]; w n Hk+2(Q)),
T]; Hk(52)),
and also
u E Cj([O, T]; V n Hk+2-j(i2)),
8 E Cj([O,T];
J c0,o.o~.
Vn Hk-j(Q)),
Proo$ Let us take
VO =
uk+l,
W I=
uk+2,
00
:=
f
ek+l,
:=
fk+l*
From Theorem 2.1 there exists only one weak solution (v, #) of equation (2.2) satisfying
v E c’([O,
T]; V) n C’([O, T]; w),
0 E c’([O,
Tl; L2(a))
n L2([0, T]; v).
Let us write
s
11
v(t) dt dtl 0
0
It is easy to see that
l
l
DECAY
RATES
273
OF SOLUTIONS
* E ck+‘uo, n V) f-l Ck([O, T]; W); 8 E P([O, T]; L2(i2))
and satisfies equation (2.2) for u(O) = ~0, ~~(0) = ur and f = 0. Taking I.J = ~0 with
8 E Cr(O, T), w E W in equation 2.2 and q = it with z E C,“(O, T), w E V,
integrating by parts with respect to the time variable and using the du Bois-Reymond
lemma, we conclude that
a(u,
w>
- a(g * u, w) = -
s
sz
(utt - yAutt + aA0)w
dx -
si
au,,
ae
- czz
w dlj.
I
l-1 y av
To prove the regularity result we reason by induction. Let us suppose that k = 2. From the
parabolic regularity we conclude that
8 E C(0, T; H2(1n)).
So, Lemma 2.3 implies that u - g * u E H4(Q),
conditions:
and also satisfies the following
&u--y~+ch5-+2g*u=0
boundary
on&
Using the resolvent equation, it follows that u E H4(i2). So, the result is valid for k = 2,
which means that u satisfies (1.2) to (1.7) in the strong sense. Differentiating (k - 2)-times
equation (1.2) to (1.7) we get:
$’ -hA
(f +A2 (ki2) +aA
(k-2)
0
’
-
go
s0
(k-1)
(k-2)
u
a (ki2)
=-
(k-2)
8 +6
t
W-2)
u 4%
(s
0
a?
=+a--
av
(k-2)
a
8
av+A
(s 0
k-2
dt=
(k-I)=0
u
= 0
I
t
ae
x32
(t)
W-2)
u (0, t)dz
go -t)
(k-2)
fJ2(k;2)-y
U
so
e=O
fk
in 52 x10, oo[,
inQx]O,
oo[,
on ro x10, m,
=0
ih
a
(k-2)
(k-2)
(k-2)
8 +y
0 -aA
-/cd
8
- t)A2
-t)
(k-2)
u (9, t)dt
on rx]O,
oo[.
on G xl09 m,
on fi xl& cd,
274
JAIME
E. MUROZ
RIVERA
AND
RIOCO
Using the resolvent Volterra equation we have that
KAMEI
BARRETO
(k-2)
u satisfies
(k-2)
A* u = F,
(k-2)
u
x31 (ku2)=
a
d (kii2)
=-
(k-l)
8
=0
av
onro,
*
a
r(t
t)
-
(k-1)
8
dt
onri,
s 0
a?
yav
x3* (ku2)=
(k-11
-
a
a0
-+
aP
(k-1)
a %’
a 0
- t)x
- adt
a,cc
t
s
r(t
0
l
on
r1,
-
.-
vG
where Y is the Volterra kernel associated to g and
dt
- (ii’ (*, t) + A ‘II’ (0, t) +
(k-1)
aA 0 +fk.
From hypothesis (1.1) and the elliptic regularity given in Lemma 2.3 we onclude that
% C(0, T; H*(C?))
(k-2)
+
u E C(0, T;
'ki2'~ C(0, T; H4(i2))
+
‘kU4& C(0,
'ki4'E C(0, T; H6(i2))
+
‘kU6’~ C(0, T;
H4(52)) and
T; H6(s2)) and
H8(Q))
and
(k-2)
8 E C(0, T; H*(Q)),
(k-4)
8 E C(0, T; H6(D)),
W-6)
8 E C(0, T; H’(C)).
Reasoning by induction we get
(k-2 j)
u E C(0, T; H*'+*(n))
e
u E Ck-*'(O 9T; H*'+*(Q)).
Using the intermediate derivative theorem (see (Lions & Magenes, 1972, Theorem 2.3,
p. 15 and Theorem 9.6, p. 43)) our conclusion follows. Uniqueness is immediate for kcl
regular solutions (k > 2). The proof is now complete.
3. Asymptotic
behaviour:
Exponential
decay
In this section we study the asymptotic behaviour of the solution of (1.2) to (1.7). Note that
as deduced in Section 2, the energy function satisfies
d
zE(t,
U,
e) =
ig’
I
a*u -
ig(t)a(u,
U) -K
s
IVBl’dA-y
52
s sz
lel*dA -h
s
18l*dr.
r
(3.1)
DECAY
RATES
275
OF SOLUTIONS
To prove exponential decay, first write w = u - g * u. It follows that w satisfies
wit - hdw,,
+ A2w + g’(O) {u - hdu}
= 0
+aA0
6 -
Kde
-
+ g(0) (ut - hdu,)
(W
in Dx]O, oo[,
= 0
dUt
w9 y, 0) = Uo(& y>,
+ g” * {u - hdu}
wtk
in Inx]O, oo[,
y9 0) =
Ul(%
Y)
-
(33
gWuo(L
yL
(3.4
with the following boundary conditions:
dW
(33
on GGw, a,
au
B*w +ae = 0 on ri x10, oo[,
WC---=(-)
W)
au
af
awt
au,
W-J - hx
+ g(O)h~ + g’(o)h,v + h7
ae
z +
he
= 0
on rx]O,
*u
ae
= 0 on rl x]O, oo[, (3.7)
+ “ay
oo[.
68)
Next we introduce the new energy function associatedto system(3.2) to (3.8),
6
Iw,12
+
hlVw,12
+
lAv12
dA
+
x
W),
i=l
where Si (t) is given by
&(t)
= - ~h(~[~Vu]2+(~gdr)]Vu]2-2(g*Vu)Vu]
Sdt)=T
(so [g~O)iul2+ ([or)
dA-hgDVu],
lul2] dA 3’
s,o=~h(~g~O,[~V,lz+(~g’d~)
s,(t)=-;
s,(t)=-;h
ivui2]
(8” clu+l
(p”
q
vU+s,
[/g’*ul2[Ig’*vu12-
(pr)
0 U})
dA-g’nvuJ.
lulj d+
(I’fdr)
/vui2]
,,)*
The point of this study is to establishan energy inequality given in the next lemma.
3.1 Under the above conditions, the solution of system (3.2) to (3.8) satisfies
the following inequality:
LEMMA
276
JAIME
d
,w
E. MUtiOZ
<cc1s s2g(t)IVu12
+C (g(t>a(u,
RIVERA
AND
RIOCO
KAMEI
dA + g •I Vu + s, WI”
U> + g I a2u} + E
BARRETO
,A)
IVu12 dA - g(0)
lut12 + hlVu,12 dA.
s 52
s 52
Pro~$ Let us multiply equation (3.2) by wt, applying Green’s formula and using the
boundary conditions we get
Izo,12dA+h
s 52
Vwt12 dA + a(w, w)
P
- g’(O)
I 62uwt
L
f
+ hVuVw,
dA -g(O)
/ \
Y
:= I1 + 12
I s2
utwt + hVu,Vw, dA
2
Y
:=
-s
13 +
14
(,,, * u + hg” * Vu) wt dA
52
awtt
-h
-
aw
-
av
d(o)av
u-4 -
g(0)
2)
wtdr
l
-11
[(&w)wI
- (&w)%]
-
.-
yO
Now we will considereach term Zi. Using Lemma 2.2 we get
g’(O)
h(t)
=
d
-2dt
s
52
M2 CM- g’(o); / (g * u)u dA + g’(0) s (g * u)ut dA
lu12+
g
* uu + Q1
5 (I’pdr)
lu12] d,“-
;g au]
:= sl(t)
I
+g (0)
Tg’ou-
duagw
2
lu12dA.
s r(2
Similarly we get that
12(t)
=
d
$(t)
g’(o)
+ hTgt
•I Vu -
g’(o)g(t)
2
h
s
Now we will consider 13:
IVu12 dA.
L2
dr.
DECAY
= -g(O)
13(t)
--
RATES
277
OF SOLUTIONS
g(O)uu, + (g’ * u)u, dA
s sz
S
g(0)
L?
:=
+-
g(O)
2 g” I
u -
i3(t)
S
g mm
2
In
lu12dA.
By symmetry we get
14(t)
=
d
lVul12dA + Thg”
s 62
- gum
--&94(t)
•I Vu - g(“)Zg’(t)h
/
lOuI dA.
52
Finally we consider 25:
I5(t) = -
S
g” * uut -
(g’ * u) + g(O)u(g’
g(O)u(g"
* u) -
* u) dA
l2
-s s
u
‘{g(0)gN(t
sz
- t) - g’(O)g’(t
- z)} (u(t)
- u(t)}
0
-W)g’(t)
dt dA
S52Id2 dk
- s’mw
analogously
16(t)
=
d
Slvu12dA
- h(g(O)g”(t)- g’@)g’(t))
-&s6(t)
-hlVu*(l
{g(O)g’(t
Q
- t) - g’(O)g(t
- t)} {Vu(t)
- Vu(t)}
dt
dA.
From the above inequalities, Poincare’sinequality and the hypotheseson g we obtain
d
Ii(t)
<
zSi(t)
+
CE
g(t)(Vu12dA+g
•I Vu
+c
I
s i2
PI dA,
for i # 3,4, and
d
13(t)
=
$53(t)
14(t)
=
$940)
d
- g(O) ~lutl~+C(~OU+~~f~~lul~dA],
- g(W
put I2+ c g El vu + g(t) s, IWdA}
s 52
l
278
JAIME
E. MUROZ
RIVERA
AND
RIOCO
KAMEI
BARRETO
Finally, we will estimate the term Zo:
Z()(t)
=-as
Q
=a
s sz
u,AO - g’* uA8 - g(O)Ak
dA
vevut + veg’* vu + g(o)vevu dA
<G(ssz IVei*dA+g
q au +S
I
s In
lVutl*dA.
cl
So, from the last inequality our conclusion follows.
REMARK
3.1
It is not difficult to see that there exists a positive constant C such that
E(t) < CE(t).
To prove it we will show that there exists a positive constant c such that
s
(wt12 + hlVw,l*
r(2
dA + a(w, w) < cE(t).
We only prove the inequality
s sz
Iwt12 dA < cE(t);
the others are similar. Note that
t
wt
=
ut
-
mu
g’(t - z){u(*, t) - ~(0, t)} dt.
s 0
From this it follows that
s
sz
Iw,(*dA
< c
(s 52
I4 I2dA + g(t)
ssz
lul*dA+g
•I u .
J
Using Kern’s inequality our conclusion follows.
Let us introduce the functional
J(t, u) :=
utu + hVu,Vu
s 52
dA.
Now we are in conditions to prove the exponential decay of the solutions.
3.1 Suppose that the initial data (~0, u 1) is 2-regular and the kernel g satisfies
conditions (1.11) and (1.9) then there exist positive constants ~0 and ~1 such that
THEOREM
E(t, u, v) < KoE(O, u, v)e-?
279
DECAYRATESOFSOLUTIONS
Proofi Multiplying equation (3.2) by w, integrating over Q and using the boundary conditions we get
d
J(t) =Q
dt
s
lut12+ wut
-a
<
sQ
+&
t
2dA - a(u,u)+a
go - t u(t) dt, u
>
A8udA
WV
(3.10)
sr(2
l&l2 + Wk
2dA - (l-
r,
(I’Sdr)g
cl Pu -J,
c)a(w,w)+C,
P
WI2 dA
s52
0
IAvl”dA.
(3.11)
Using Lemma 3.1 we find that
d
dt
g(O)
NE(t) + E(t) + 2 J(t)
I
I
l .-
at>
<--Ko
(sszlut12+ hph,12 + lOI2dA + a(u, U) + g I
a2u .
:= N(t)
It is not difficult to seethat
@N(t)
< C(t) < NN(t),
for N large enough. From the last two inequalitiesour conclusion follows.
cl
3.2 Note that h > 0 hasno role in the above estimates,so a similar result holds
for plates with thicknessh = 0.
REMARK
4. Polynomial decay
In this section we will showthat when the kernel g decayspolynomially then the first-order
energy also decaysat the samerate. To do so, we considerhypotheses(1.9), (1.12), (I. 13).
To prove the main result of this section we will usethe following lemma.
4.1 Supposethat g and h are continuous functions satisfying the conditions
l+l’q(O
00) n L1 (0,oo) for some4 > 1 and gr E L1 (0,oo) for some0 < r < 1;
CL
then we have ;hat
LEMMA
tsI&t
-wwl
dt<ist
0
0
X
IgO - t>l l+cl-r)‘qlh(~)I dt
IgO - ~>l’lfwl dt
l/(q+l)
q/(4+1)
I
l
280
JAIME
E. MUROZ
RIVERA
AND
RIOCO
KAMEI
BARRETO
Pro05 For any fixed t we have
t
t
Jg(t - t) Irl(q+l) Ih@) 1lbl+l) Jg(t _ t) 1l-r/(q+l) lh@) Id(q+l) dt.
Ig(t-t)h(t)ldt
=
/L
/
Y
Sk0
v
s0
l .-
l .-
u
V
Note that u E LP(O, 00), v E LP’(0, 00), where p = 4 + 1 and p’ = (4 + 1)/q. Using
Holder’s inequality, we get
WI+*)
t
lg(t - wml
/0
dt <
v
ot lg@- wfwl~~
I
t
X
(s 0
I& - Gl l+(l-r)‘qlh(t)ldz
I
q/(4+*)
.
cl
This completes the proof.
4.2 Let us suppose that z E C(0, T; E?*(Q)) and g is a continuous function
satisfying hypotheses (1.12), (1.13); then for 0 < r < 1 we have
LEMMA
V(1+(1-r)p)
1+1/p
1g
t
(s
I
a*2}
(1-r)p/(l +u -4p)
9
while for r = 0 we get
g cl a22 < 2
ll(p+l)
llmll~ dt + IIW 11;
I g 1+1/p
I
0
c3
a2dp’(1+p)
l
Pro05 From the hypotheses on z and Lemma 4.1 we get
t
g c3 a22 =
s0
g(t - t) La(z(t)
- z(z),
) llw-r)p+l)
s’(t - t)h(z)
(s 0
G { gr
z(t) - z(t)> / dt
:= h(t)
t
<
v
I
a*z}
( Jd’
gl+l,p(t
_
t)h(t)
dr}(l-r,rlirl.-nuii,
dt
l/((l-r)P+l)
{ gl+l/p
I
a*~}(l-‘)Pl((l-r)P+l)
.
(4-l)
grI a22= stg'(t - M(z(t)
For 0 < Y < 1 we have
- z(t), z(t) - z(Wr
0
t
a
g’@)
, 7-+2)*
7
dtIkli&
s0
From here the first inequality of Lemma 4.2 follows. To prove the last part, let us take
r = 0 to obtain
I I
a22
a(z(t) - z(t), z(t) - z(t))dt
=
I 0
t
< wml~
+2
s 0
IIW 11;dt*
DECAY
RATES
281
OF SOLUTIONS
Substitution of the above inequality into (4.1) yields the second inequality. The proof is
now complete.
cl
From the above lemma we get
g q
forO<r
azu
<
Co (gl+l/p
(
a2U)(l-r)pl(l+(l-r)p)
(4-2)
c 1.
LEMMA 4.3 Let us supposethat f is a non-negative C1 function satisfying
f’(t) < -kdf wll+l’P + (1 +k;)p+l
for somep > 1 and positive constantsko and kl. In theseconditions, there exists a positive
constant cl such that
f (0 G (f (0) +
/(
2h
P
1 + c’{ f (0)p
P
>
.
Pro05 Let us denote by h(t) and F(t) the functions defined by
h(t) :=
2Kl
F(t) := f(t) + h(t).
p(1 + t)P’
So we have
F’(t) =
2Kl
f’(t) -
< -Ko[f
(1 +
(t)]l+l’P
t)p+l
-
(1 +
K1
t)p+’
1+1/p
[f
(t)]l+‘lP
+
+[h(t)]l+l’p
2KOK’
From here it follows that there exist a positive constant c for which we have
F’(t) < -c {[f
(t)]‘+“P + [h(t)]*+‘lp} < -c[F(t)]‘+‘lp,
which gives the required inequality.
cl
LEMMA 4.4 Under the above conditions, the solution of system (3.2) to (3.8) satisfies
the following inequalilt7t’:
d
ZW)
G c,
s(t) I’vu12 + JV0i2 dA + gl+‘lp •I Vu
is 52
+c {g(t>a(u u) + g’+‘lp ~1 a2u} + E IVul dA - g(0) )u,12+ hlVu,12 dA.
sQ
s 52
282
JAIME
E. MUROZ
RIVERA
AND
RIOCO
KAMEI
BARRETO
ProoJ: The only difference with respect to the proof of Lemma 3.1 is to estimate the following term:
-s s
In
u
‘(g(o)g”(t
0
- t) - g’(O)g’(t
- z)} {u(t)
- u(t)}
dt dA.
All other estimates follows using the same argument and the hypotheses on g. Writing
h(t - t) = g(O)g”(t
- t) - g’(O)g’(t
- t)}
we have that
Ih(
< cg’+“P
from which our conclusion follows.
cl
We are now in condition to prove the main result of this section.
4.1 Let us suppose that the initial data (UO, ur ) is 2-regular, and that (1.9),
(1.12), (1.13) hold; then any solution of system (1.2) to (1.7) satisfies
THEOREM
W, 0 < CE(O, @(I + t)-P
for p > 2.
Pro05 As in Theorem 3.1 we arrive at the following
d
dt
inequality:
g(O)
NE(t) + E(t) + 2 J(t)
:=&I)
< -Ko
lu,12 + hlVu,12 + 1612dA + a(u, u) +g’+“P
/
7
u 52
•I a2u .
:=N(t)
Since the energy is bounded, Lemma 4.2 implies that
w>
2 CJW (l+U-~)PMl--~)p
9
q
8 1+1/p
azu
2
c Ig
I
a2u](1+(1-r)P)/(1-r)P
.
It is not difficult to seethat we can take N large enoughsuchthat C satisfies
c (E(t, u)} < L(t, u) < Cl {N(t) + gl+‘lp cl
a2U}(1-r)p”1+(1-~)p).
From here it follows that
d
--L(t,
u) < --qL(t,
U)o+(l-r)p)l(l-r)p,
which implies that
at, u> G cao,
1
4
(1
+
t)(‘-‘)P
l
(43
DECAY
RATES
OF SOLUTIONS
283
From this it follows that the energies decay to zero uniformly. Using Lemma 4.2 with r = 0
we get that
N(t) > cJv(t)(‘+P)‘~,
g •I a2u 2 c {g 0 a2u}~1+p)~p.
Repeating the same reasoning as above we get
L(t, u) < CC(0, u)-
1
(1 + ty’
From this our result follows. The proof is now complete.
cl
Acknowledgments
The authors express their appreciation to the referee for a valuable suggestion which improved this paper. The research was supported by a grant from CNP, (Brazil).
REFERENCES
E. C. & LASIECKA,
I. SIAMJ. Math. Anal. to appear.
C. M. 1970 An abstract Volterra equation with application to linear viscoelasticity. J.
Dtfferential Equations 7,554-589.
DAFERMOS,
C. M. 1968 On the existence and the asymptotic stability of solutions to the equation
of linear thermoelasticity. Arch. Rat. A4ech. Anal. 29,241-271.
DASSIOS,
G. & GRILLAKIS,
M. 1984 Dissipation rates and partition of energy in thermoelasticity.
Arch. Rat. h4ech. Anal. 87,49-91.
DUVAUT,
G. & LIONS, J. L. 1976 Inequalities in Mechanics and Physics. New York: Springer.
HANSEN,
S. W. 1992 Exponential energy decay on a thermoelastic rod. J. A4ath. Anal. Appl. 167,
429-442.
HENRY, D. B., PERISSINITTO,
A., JR. & LOPES, 0. 1993 Asymptotic behaviour of a semigroup
of the thermoelasticity. J. Nonlinear Anal. 7’MA 21, 65-75.
KIM,
J. U. 1992 On the energy decay of a linear thermoelastic bar and plate. SIAM J. Math. Anal.
23,889-899
LAGNESE,
J. E. 1989 Asymptotic Energy Estimatesfor KirchhoffPlates Subject to Weak Viscoleastic
damping. International Series of Numerical Mathematics 9 1. Basle: Birkhauser.
LIONS,
J. L. & MAGENES,
E. 1972 Non-Homogeneous Boundary Value Problems and Applications,
Vol. 1. New York: Springer.
LIU, 2. Y. & RENARDI,
M. 1995 A note on the equation of a thermoelastic plate. Appl. Math. Lett.
8, l-6.
MUROZ
RIVERA
J. E. 1992 Energy decay rates in linear thermoelasticity. Funkcialaj Etvacioj 35,
19-30.
MU~~OZ
RIVERA,
J. E. 1993 Decomposition of the displacement vector field and decay rates in
linear thermoelasticity. SIAM J. Math. Anal. 24, 390406.
MU~~OZ
RIVERA,
J. E. 1994 Asymptotic behaviour in linear viscoelasticity. Quart. Appl. Math. 3,
257-273.
MuRoz RIVERA,
J. E. & RACKE, R. 1995 Smoothing properties, decay and global existence of
solutions to nonlinear coupled systems of thermoelastic type. SIAM J. Math. Anal. 26, 15471563.
Mur;joz RIVERA,
J. E. & SHIBATA, Y. 1997 A linear thermoelastic plate equation with dirichlet
boundary condition. J. Math. Meth. Appl. Sci. 20,915-932.
RENARDY,
M. 1993 On the type of certain Co-semigroups. Comm. Part. Difl Equns. 18,1299-l 307.
AVALOS,
DAFERMOS,