QUARTERLY OF APPLIED MATHEMATICS
VOLUME LXII, NUMBER 2
JUNE 2004, PAGES 273-293
A TRANSMISSION PROBLEM FOR THERMOELASTIC PLATES
By
JAIME E. MUNOZ RIVERA (National Laboratory for Scientific Computation, Department of
Applied Mathematic, Av. Getulio Vargas 333, Quitandinha, 25651-070, Petrdpolis, RJ, Brazil, and
IM, Federal University of Rio de Janeiro)
HIGIDIO PORTILLO OQUENDO (Federal University of Parana, Department of Mathematics,
Politechnic Center, P. O. Box 19081, Jardim das Americas, CEP 81531-990 Curitiva, Parana,
Brazil)
Abstract. In this paper we study a transmission problem for thermoelastic plates.
We prove that the problem is well-posed in the sense that there exists only one solution
which is as regular as the initial data. Moreover, we prove that the local thermal effect
is strong enough to produce uniform rate of decay of the solution. More precisely, there
exist positive constants C and 7 such that the total energy E(t) satisfies
E(t) < CE{0)e"7t.
1. Introduction. From the point of view of applications, the suppression of vibration
of elastic structures is one of the important topics in material science. For example,
engineers at the Ford Motor Company designed a constrained-layer damping patch which
was attached to an elastic plate. They compared the natural frequencies and mode shape
of the plate with and without the patch to ascertain the effect of the patch. Due to the
presence of the patches, the material properties of the structure, such as the elasticity
moduli, damping coefficient, and Poisson ratio, are changed (see [14]). In particular,
jump discontinuity at the location of the edges of the patches is usually introduced to
these properties. In this direction we will consider the model which defines the oscillation
of a plate which is composed of a thermoelastic part and an elastic part. This means
that the thermal constant is discontinuous on the plate, positive over the thermoelastic
part, and vanishing on the elastic part.
We will consider that the plate, in equilibrium, occupies a region Q which is a bounded
open set in Mn with boundary dfl — Ti U T2 where Ti, T2 are two smooth surfaces such
that Fi fl f2 =0. We assume that the plate's particles in are sensitive to change of
temperature and in its complementary part, \ they are not. Let us denote
Received March 19, 2002.
2000 Mathematics Subject Classification. Primary 74F05, 74K20, 35B40.
Supported by a grant of CNPq (Brazil).
©2004 Brown University
273
274 JAIME E. MUNOZ RIVERA and HIGIDIO PORTILLO OQUENDO
by To the common smooth surface between and ^2; a region SI of this type is given
by Fig. 1.
Fig. 1. The set f2
Denoting by u(x,t) and v(x,t) the vertical displacements of the plate and by 9(x,t)
the difference of temperature, the corresponding model can be written as follows:
PiU-tt ~ 7iAtitt + /?i A2u + fiA9 — 0 in Cli x R+, (1-1)
podt — /3qA9 + 7o$ — pAut = 0 in Six x M+, (1.2)
p2Vtt - l2&vtt + /?2A2i> = 0 in Sl2 x R+. (1.3)
We assume that the plate is clamped on the surfaces Ti, T2, i.e.,
f)u c)v
u = — =0 on Ti x ]R+, v = — =0 on x M+. (1-4)
av av
The transmission condition on the interface To is given by
c)i 1 Bv
u = v, — = —, /3iAu + fid — f32Av on T0 x K+, (1.5)
av av
dutt „ dAu d8 dvtt „ dAv , .
-71-5— +Pi~w— = -72-^ on r0xl+ (1.6)
av av av av av
We consider the following condition for the temperature:
9 = 0 on Fo x 1+, ——I- \9 = 0 on Fj x R+, (1-7)
dv
(1.8)
and the initial data
w(0) = uo, ut(0) = u1, 9(0) = 0O in fix,
v(0) = w0, vt(0)=vi in 0,2-
Here, the coefficients pi, 7i, fa, and A are positive, // is different to zero, and uq, u\,9q, fo,
v\ are prescribed functions. To fix ideas we consider /i positive.
Controllability for transmission problems were studied by several authors; for example,
the transmission problem for the wave equation was studied by Lions [6]. He applied
the Hilbert Uniqueness Method (HUM) to show the exact controllability. Later, Lagnese
[5], also applying HUM, extended this result; he showed the exact controllability for
a class of hyperbolic systems which include the transmission problem for homogeneous
anisotropic materials. The exact controllability for the plate equation was proved by Liu
and Williams [9] and Aassila [1].
A TRANSMISSION PROBLEM FOR THERMOELASTIC PLATES 275
Concerning asymptotic stability, second order transmission problems were studied by
Rivera and Oquendo [11], Liu and Williams [8], and Rivera and Ma To Fu [10], while
for beams we have the works of Rivera and Oquendo [12, 13]. Thermoelastic plates were
studied by Lagnese, Avalos, and Lasiecka. In [4], Lagnese obtained the exponential decay
of solutions with the aid of a further mechanical dissipation on the boundary and in [2],
Avalos and Lasiecka obtained the same result removing the boundary dissipation. It
seems to us that there is no result concerning the asymptotic stability of solution for
plates made of different types of materials. So to fill this gap we study this topic here.
The main result of this paper is to show that the dissipation given by the thermal
part of the plate is strong enough to produce uniform stability of the solution, no matter
how thin it is. To attain this goal we will assume that the material type in is more
stiff than that in fi2, that is
Pi > P2, 7i > 72 and ft < ft.
Additionally, some geometric assumptions on fi will be taken into account, as for example
(x — xq) ■ v(x) > So on r0,
(x — x0) ■ v(x) < 0 on r2,
for some xq S R™ and 5o > 0 small. In these conditions we will show that the total
energy associated to the model decays exponentially as time goes to infinity. The idea
we use to achieve our result is based on the energy method; to do so, we need that
the solution enjoys the regularity property. Therefore, in the next section of the article,
we will show that the solution of the above system has the m-regularity result. One of
the main difficulties we have in showing the exponential decay is due to the boundary
conditions. We avoid them using some localized multipliers and some technical ideas
involving the compact embedding of the spaces Hm~l C Hm.
The remaining part of this article is organized as follows. In the next section we will
show that the problem is well-posed in the sense of existence, uniqueness, and regularity
of the solution. To do this we will use the semigroup approach. Finally in Sec. 3 we will
prove that the solution of the system decays exponentially to zero.
2. Existence of solutions. To find a solution for the problem (1.1)—(1.8), we shall
use the semigroup approach. Let us start analyzing the associated stationary problem.
First we shall introduce some notation. Let us consider the following Hilbert spaces
H\ := {(0i,</>2) G H1(fii) x i/1(n2) : 0* = 0 on T,, 0i = 02 on r0},
H% := |(01,02) G [H2(ni) x H2(n2)} n4 : ^ = 0 on ri; ^ on r0J ,
H*:={4>eH\Sh) : 0 = 0 on Tq}, H° := L2(Sh),
276 JAIME E. MUNOZ RIVERA AND HIGIDIO PORTILLO OQUENDO
with the following inner products:
({wi,w2},{<t>i:= / {P\Wi<j>i +7iVwi • V(j)i)dx
Jo. i
+ / {p2W24>2 + 72Vtf2 * V02) dx,
Jq2
[ PiAwi Acfii dx + / 02Aw2A(j)2dx,
I
Jsh
I PoViv • V<£ 4- 70w4> dx + I /3q\w(/) dT,
(w,<j>)fj 0 / poW(j)dx.
jQi
Let us denote the dual space of H£ by f°r s = 1, 2 and the dual space of by
H^1. The following Lemma shows that the norm given by the inner product in H2 is
equivalent to the usual norm of x H2(fl2).
Lemma 2.1. Let us take (f,g) in L2(flx) x L2(tt2). Then there exists a unique couple
(u,v) G H2(fl 1) x H2(fl2)
solution of
A u — f on fii, At; = g on £l2,
satisfying the boundary conditions
du dv
u = 0 on I i. v = 0 on F2, u = v and —— = —— on 1 0 •
av dv
Moreover, there exists C > 0 such that
+ IMI/f2(r22) < C(ll/l|L2(ni) + ll<?IU2(n2))-
Proof. See [3]. d
The corresponding stationary transmission problem for the plate equation is given by
the equations
A 2u = f in fii, A v2 = g in Q2, (2-1)
satisfying the boundary condition
0 on r1; v = ~ = 0 on T2, (2.2)
dv av
du dv . . n . . dAu dAv ^ .
u = v, — = —, f31Au = (32Av, = P2~K~ on F0. (2.3)
av av av av
To find the variational formulation associated to this problem we multiply the first equa-
tion of (2.1) by and the second by f32<f>2 with {<j>i^<j>2) £ H2; next we integrate by
parts to obtain
/ PiAwiAcfri dx + / 02Aw2A<f>2dx = / (3if4>idx+ / f32g4>2dx.
JJQ2 " ^2
The existence and uniqueness of weak and strong solution for this problem is given by
LEMMA 2.2. The following items hold:
A TRANSMISSION PROBLEM FOR THERMOELASTIC PLATES 277
(1) If (/,<?) € H?2, then there exists a unique solution (u,v) £ H£ of (2.1)-(2.3).
Moreover, there exists C > 0 such that
||(w,v)||f/2 < C\\(f,g)\\H-2.
(2) Let m € Zq. If (f,g) € x Hm(p,2), then there exists a unique solution
(u,v) £ Hm+4(rii) x Hm+4(n2) of (2.1)-(2.3). Moreover, there exists C > 0
such that
IMlH™+4(ni) + llullff'"+4(n2) < C(||/Ili?m(fii) + Il5lltfm(n2))-
(3) If (f,g) £ , then there exists a unique solution (u,v) € Hs(ili) x H3(Vl2) of
(2.1)-(2.3). Moreover, there exists C > 0 such that
ll(u,^)lltf3(ni)xw3(o2) ^ C\\(f,g)\\H-i.
Proof. Item 1 is a consequence of Lax-Milgram Theorem; item 2, for the case m = 0,
can be found in [9] and item 3 is a consequence of the Interpolation Theory. Item 2 is a
well known result of elliptic regularity (see [15] for the transmission problem for general
elliptic equations). In Appendix A of this paper we give a simple proof of this item when
/2i//?2 is small or large. □
Now, we shall write system (1.1)—(1.7) in the abstract form of semigroups following
the ideas of Lagnese [4]. Let us consider the operators
Ao : H2 —> H~2, Ai : H\ X H° -> H~l x H° and B0 : H2 x —> H~2 x H'1
given by
(A0{wi,w2},{4>i,(t>2}) ■= {{wi,w2},{<fo.,<fo})H* .
{A1{wi,w2,w3},{<j)i,(l)2,<t>3}) ■= ({wi,W2},{<Al,02})ffi, + {wz,(j>z)H°,
{B0{wi,w2,w3},{<j>i,<j>2,<t>3}) ■= / — Awi<j>3)dx + (w3,<f>3)Hi,
Ja 1
and let us denote by
•Si{<^11^2} := {^o{</>ij <^2}, 0}, <t>2i <^3} := ^4o{<^i; <^2}-
Multiplying equations (1.1), (1-3), and (1.2) by 0i, 4>2 and (p3 respectively with
{<j>i,<j>2, <^3} € £T2 x -^d and performing an integration by parts yields
(Ai-^{ut,vtje},{(f)1,(f>2,<f>3}) = -{Ao{u,v},{4>i,(j)2}) - {Bo{ut,Vt,6},{<j)l,4>2,4>3})-
The above identity can be written as
Aift{ut,vt,9} =-Bi{u,v} - Bo{ut,vt,0} in H^2 x HD\ (2.4)
- dt
Taking into account that
A0jr{u,v} = A0{ut,vt},
'dt1
and denoting by w — (u,v,ut,vt,0), Eq. (2.4) can be written in the following matrix
form
A0 0 \ dw / 0 —B2
0 Ai ) dt \ B\ Bq
w = 0 in Ht 2 x H~ 2 x 1.