MATHEMATICAL METHODS IN THE APPLIED SCIENCES
Math. Meth. Appl. Sci. 2002; 25:955–980 (DOI: 10.1002/mma.323)
MOS subject classication: 35 B 40; 35 L 05; 35 L 70; 73 B 30; 73 K 03
Asymptotic behaviour and exponential stability for a
transmission problem in thermoelasticity
Alfredo Marzocchi1; † , Jaime E. Muñoz Rivera2; ∗; ‡ and Maria Grazia Naso3; §
1 Dipartimento
di Matematica e Fisica; Universita Cattolica del Sacro Cuore; Via Musei 41; I-25121 Brescia;
Italia
2 National Laboratory for Scientic Computation; Rua Getulio Vargas 333; Quitadinha-Petr
opolis 25651-070;
Rio de Janeiro; RJ; Brazil
3 Dipartimento di Matematica; Universit
a degli Studi di Brescia; Via Valotti 9; I-25133 Brescia; Italia
Communicated by Y. Shibata
SUMMARY
We show that the solution of a semilinear transmission problem between an elastic and a thermoelastic
material, decays exponentially to zero. That is, denoting by E(t) the sum of the rst, second and third
order energy associated with the system, we show that there exist positive constants C and satisfying
E(t)6C E(0)e−t
Moreover, the existence of absorbing sets is achieved in the non-homogeneous case. Copyright ? 2002
John Wiley & Sons, Ltd.
KEY WORDS:
transmission problem; thermoelasticity; exponential decay; simultaneous stabilization;
asymptotic behaviour; absorbing set
1. INTRODUCTION
The wave equation without any dissipation is a conservative system, that is, its total energy is
constant for any time. Several authors introduced dierent types of dissipative mechanisms to
stabilize the oscillations. For example, the frictional damping ut (see Reference [1]) where
the dissipation works in the whole domain, or frictional boundary conditions (see References
[2; 3]) where the dissipation is working in a part of the boundary or also localized frictional
damping, that is when the frictional damping is of the form (x)ut where vanishes in
∗ Correspondence
to: J. E. Muñoz Rivera, National Laboratory for Scientic Computation, Rua Getulio Vargas 333,
Quitadinha-Petropolis 25651-070, Rio de Janeiro, RJ, Brazil
† E-mail: [email protected]
‡ E-mail: [email protected]
§ E-mail: [email protected]
Contract=grant=sponsor: CNPq-BRASIL; contract=grant number: 305406=88-4
Contract=grant=sponsor: Italian MURST
Copyright ? 2002 John Wiley & Sons, Ltd.
Received 19 September 2001
956
A. MARZOCCHI, J. E. MUÑOZ RIVERA AND M. G. NASO
Figure 1.
some part of the domain (see References [4–6]). So, the direction of the research seems
to nd the minimal dissipation such that the solution of the corresponding dissipative wave
equation decays uniformly to zero, as time goes to innity. Concerning the localized frictional
damping the main question could be: in which part does have to be positive in order to
assure a uniform rate of decay? This question somehow is vinculated with control theory. In
Lions’ book [7], it is proved that if we can control a system then we can stabilize it. More
precisely, if we can control a system acting over a subset ! of , then it is possible to
stabilize the system introducing a damping mechanism eective only over !. This somehow
leaves the problem of nding the smallest possible part ! for which the wave equation can
be controllable with controls working on !. An answer to this question is given in the work
of Bardos et al. [8] where the authors nd a sucient condition for the controllability and
the stabilization of the solution for second order hyperbolic equations when the controls are
eective in a part of the domain.
In this paper we will use the above theory to show stability results for composed materials.
That is, we will consider a material which has thermoelastic properties over one part while
the other part is indierent to the change of temperature. This situation can be considered
as a system with localized damping. Since the density and the elastic coecients of each
component are not necessarily the same, we have to deal with a system with discontinuous
coecients. The resulting mathematical model is called Transmission problem. We are mainly
interested in the asymptotic properties: that is, whether the weak dissipation given by the
thermal eect is enough to stabilize the system, when this dissipation is eective only over a
part of the material.
Let us consider a one-dimensional body which is congurated in the interval [0; L3 ] ⊂ R.
Given L1 ; L2 ∈]0; L3 [, we denote by the set ]0; L1 [∪]L2 ; L3 [. We assume that, the material is
thermoelastic over and elastic on ]L1 ; L2 [ (see Figure 1).
Let x + u(x; t) be the position of the material particle x at time t in the thermoelastic part,
and let us denote by x + v(x; t), the position on the elastic part. We denote by the dierence
of temperature between the actual state and a reference temperature. Then the system that
models the above situation is given by
ut t − auxx + mx + f(u) = h1 ;
in × ]0; ∞[
(1)
t − xx + muxt = h2 ;
in × ]0; ∞[
(2)
in ]L1 ; L2 [×]0; ∞[
(3)
vt t − bvxx = h3 ;
where a; b; and m are positive constants, hi : → R (i = 1; 2) and h3 : ]L1 ; L2 [ →R are given
functions and f : R → R is a non-linear function whose properties will be specied later. The
Copyright ? 2002 John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci. 2002; 25:955–980
TRANSMISSION PROBLEM IN THERMOELASITICITY
957
system is subjected to the following boundary conditions:
u(0; t) = u(L3 ; t) = (0; t) = (L3 ; t) = 0
u(Li ; t) = v(Li ; t);
x (Li ; t) = 0;
(4)
aux (Li ; t) − m(Li ; t) = bvx (Li ; t);
(i = 1; 2)
(i = 1; 2)
(5)
(6)
and initial conditions
u(x; 0) = u0 (x);
ut (x; 0) = u1 (x);
(x; 0) = 0 (x);
x∈
v(x; 0) = v0 (x);
vt (x; 0) = v1 (x);
We denote
F(s) =
x∈
(7)
(8)
s
x ∈ ]L1 ; L2 [
(9)
f() d
0
and we assume that f satises
sf(s)¿0;
∀s ∈ R
(h1)
Let us mention some other papers related to the problems we address. The asymptotic
behaviour as t → ∞ of solutions to the equations of linear thermoelasticity in a bounded
domain has been studied by many authors. In one dimension, it is well known that solutions
decay (to zero) exponentially for all the classical boundary conditions (see, for example,
References [9–18]), while in two or three dimensions, the situation becomes more delicate.
Dafermos [19] (also cf. References [20; 21]) investigated the linear equations of n-dimensional
thermoelasticity and showed that, e.g. if the displacement u and the temperature dierence satisfy Dirichlet boundary condition, then tends to zero and u tends to a function ũ as time
goes to innity. Whether the function ũ is zero, it depends on the geometry of the domain,
e.g. ũ = 0 for a rectangle but ũ = 0 for the unit ball in R2 . However, no decay rate was given
in References [19; 21]. Henry [22], Henry et al. [11] proved that in more than one dimension,
there is no uniform decay rate of solutions for a spatially periodic boundary condition or for
the domain containing a nite cylinder whose ends are in the boundary. Racke [23] studied
some special boundary conditions and proved the exponential decay of and of the curl-free
part of u. Recently, Jiang et al. [24] showed that solutions with spherical symmetry decay
exponentially in the annular domains with appropriately large diameter. We also mention the
works of Carvalho Pereira and Perla Menzala [25] who showed that if an additional damping
term ut is added to the equations, then solutions converge to zero exponentially. See also
Racke [23].
The main result of this paper is to prove that in the linear homogeneous case (f ≡ 0; hi ≡ 0;
i = 1; 2; 3), the solution of the above system tends to zero with an exponential rate, as time
goes to innity. This is also interesting from a physical point of view, since this stability
is given from the thermoelastic property of the (possibly small) thermoelastic part of the
material and from the boundary conditions (5).
Copyright ? 2002 John Wiley & Sons, Ltd.
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A. MARZOCCHI, J. E. MUÑOZ RIVERA AND M. G. NASO
In the non-linear case, this property is replaced by the existence of an absorbing set in the
space of solutions provided that is suciently small, in comparison to the energy function
chosen.
The remaining part of this paper is organized as follows. In Section 2, we set the problem
and in Section 3 we show the existence of weak and strong solution to system (1)–(9).
In Section 4 we derive the various energy estimates and we state the exponential decay
of the solution. Finally, in Section 5 we prove the existence of absorbing sets in the nonhomogeneous case.
2. FUNCTIONAL SETTING AND NOTATION
Let I ⊂ R be a bounded interval. With usual notation we introduce the spaces L2 (I ); H 1 (I )
and H01 (I ) acting on I . Let ·; ·
and · denote the L2 -inner product and L2 -norm,
respectively.
We will also consider spaces of functions dened on an interval I with values in a Banach
space X such as C(I; X ); C 1 (I; X ), Lp (I; X ) and H p (I; X ), with the usual norms. We further
introduce the following spaces:
HL1 () = {w ∈ H 1 () : w(0) = w(L3 ) = 0}
V = {(u; v) ∈ HL1 () × H 1 (]L1 ; L2 [) : u(Li ) = v(Li );
i = 1; 2}
Note that V is a closed subspace of HL1 () which together with the norm
L2
2
2
(u; v)V :=
|ux | dx +
(|v|2 + |vx |2 ) dx
L1
is a Hilbert space. In order to simplify the notation, we will omit the indication of the space
domain of the variables, when it is understood. For example, v ∈ C([0; T ]; H 1 ) will mean
v ∈ C([0; T ]; H 1 (]L1 ; L2 [)); u ∈ C([0; T ]; L2 ) will mean u ∈ C([0; T ]; L2 ()), and so on.
We conclude this section with the following lemma.
Lemma 2.1
Let us suppose that z ∈ H 1 (0; T; L2 (x1 ; x2 )) and q ∈ C 1 (x1 ; x2 ). Then for any function w ∈
H 2 (I; L2 (x1 ; x2 )) ∩ L2 (I; H 2 (x1 ; x2 )) satisfying
wt t − wxx = z
(10)
where ¿0, we have that
d
dt
x2
x1
qwt wx dx =
x2
x1
qwx z dx +
q(x2 )
(|wt (x2 ; t)|2 + |wx (x2 ; t)|2 )
2
q(x1 )
1
(|wt (x1 ; t)|2 + |wx (x1 ; t)|2 ) −
−
2
2
Copyright ? 2002 John Wiley & Sons, Ltd.
x2
x1
(qx |wt |2 + qx |wx |2 ) dx
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TRANSMISSION PROBLEM IN THERMOELASITICITY
959
Proof
Multiplying the equation by qwx and integrating over [x1 ; x2 ] we arrive at
x2
1 x2 d
1 x2
d
d x2
2
2
qw1 wx dx =
q
|wt | dx +
q
|wx | dx +
qwx z dx
dt x1
2 x1 dx
2 x1
dx
x1
Performing an integration by parts our conclusion follows.
3. EXISTENCE AND UNIQUENESS OF SOLUTIONS
In this section, we establish the existence and uniqueness results for problem (1)–(9) where
the non-linearity is assumed to be a real function with conditions (h1).
First of all, we dene what we will understand for weak solution of problem (1)–(9).
Throughout this section, we set I = [0; T ], with T ¿0.
Denition 3.1
Let h1 ; h2 ; h3 ∈ L2 . We say that (u; ; v) is a weak solution of (1)–(9) when
(u; v) ∈ L∞ (I; V );
(ut ; vt ) ∈ L∞ (I; L2 × L2 )
∈ L∞ (I; L2 ) ∩ L2 (I; HL1 )
satisfying the identities
T
0
u1 (0) dx −
u0 t (0) dx +
0
=
T
L2
L1
v1 w(0) dx −
L2
L1
L2
L1
0
T
{ut t + aux x − mx + [f(u) − h1 ]} dx dt +
(vwt t + bvx wx − h3 w) dx dt
v0 t (0) dx
(−
t + x
x − mux
t
− h2 ) dx dt =
for all (; w) ∈ C 2 (I; V ),
0 (0) dx + m
u0x (0) dx
∈ C 2 (I; HL1 ) and a.e. t ∈ I such that
(T ) = t (T ) = (T ) = w(T ) = wt (T ) = 0
The existence of solutions to system (1)–(9) is given in the following theorem:
Theorem 3.1
Let us suppose that f is a C 1 -function verifying (h1). Let us take initial data satisfying
(u0 ; v0 ) ∈ V;
Copyright ? 2002 John Wiley & Sons, Ltd.
(u1 ; v1 ) ∈ L2 × L2 and 0 ∈ L2
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A. MARZOCCHI, J. E. MUÑOZ RIVERA AND M. G. NASO
Then, there exists a solution (u; v; ) of the system (1.1)–(1.9) satisfying
(u; v) ∈ C(I; V ) ∩ C 1 (I; L2 × L2 )
∈ C(I; L2 ) ∩ L2 (I; HL1 )
In addition, if
(u0 ; v0 ) ∈ (H 2 × H 2 ) ∩ V;
(u1 ; v1 ) ∈ V; 0 ∈ H 2 ∩ HL1
verifying the compatibility conditions
au0x (Li ) − m0 (Li ) = bv0x (Li );
i = 1; 2
then the solution satises
(u; v) ∈ C(I; (H 2 × H 2 ) ∩ V ) ∩ C 1 (I; V ) ∩ C 2 (I; L2 × L2 )
∈ C(I; H 2 ∩ HL1 ) ∩ C 1 (I; L2 )
In this case we will say that (u; v; ) is a strong solution.
Proof
We follow a standard Faedo–Galerkin method and we divide the proof into four steps.
Step 1 (Faedo–Galerkin scheme): Let us denote by {(’i ; wi );i ∈ N} a basis of
V ∩ (H 2 × H 2 ) and by { i ; i ∈ N} a basis of H 2 ∩ HL1 . We denote
V = span{(’1 ; w1 ); : : : ; (’ ; w )};
H = span{ 1 ; : : : ;
}
Let us write
(u ; v ) =
i=1
ai (t)(’i ; !i );
=
i=1
bi (t)
i
where u and v satisfy
[utt ’i + aux’i; x − m ’i; x + f(u )’i − h1 ’i ] dx
L2
+
L1
(t
i
+ x
(vtt !i + bvx !i; x − h3 !i ) dx = 0
i; x
+ muxt
(u (0); v (0)) = (u0 ; v0 );
i
(11)
− h2 i ) dx = 0
(12)
(ut (0); vt (0)) = (u1 ; v1 );
(0) = 0
(13)
for a.e. t 6T and where ’0 ; !0 and 0 are the zero vectors in the respective spaces. Re-casting
exactly the classical Faedo–Galerkin scheme, we obtain a system of ODE in the variables
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TRANSMISSION PROBLEM IN THERMOELASITICITY
ai (t) and bi (t). According to standard existence theory for ODE, there exists a continuous
solution of this system, on some interval (0; Tn ). The a priori estimates that follow imply that
in fact Tn =+∞.
Step 2 (Energy estimates): Multiplying (11) and (12) by ai (t), integrating by parts and
summing over i, we have
d E (t) + m
dt
x ut dx =
h1 ut dx +
L2
L1
h3 vt dx
where
E (t) =
1
2
1
2
[|ut |2 + a|ux |2 + 2F(u )] dx +
L2
L1
(|vt |2 + b|vx |2 ) dx
Multiplying (19) by bi (t), integrating by parts and summing up over i, we have
1 d
2
2
| | dx + |x | dx − m
x ut dx =
h2 dx
2 dt Summing up the above two identities we get
d ˆ
E =−
dt
where Eˆ (t) = E (t) +
1
2
|x |2 dx +
h1 ut dx +
L2
L1
h3 vt dx +
h2 dx
| |2 dx. Using Cauchy inequality we can write
d ˆ
E (t)6E (t) − dt
|x |2 dx + C
(|h1 |2 + |h2 |2 ) dx +
L2
L1
|h3 |2 dx
Setting
ch = C
(|h1 |2 + |h2 |2 ) dx +
L2
L1
|h3 |2 dx
we have, after an integration over (0; t); t ∈ (0; T ), that
Eˆ (t) + et
t
e
−b
0
|x (
)|2 dx d
6Eˆ (0)eT +
ch T
e
Thus, we conclude that
(u ; v ) is bounded in L∞ (I; V )
(ut ; vt ) is bounded in L∞ (I; L2 × L2 )
is bounded in L∞ (I; L2 )
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A. MARZOCCHI, J. E. MUÑOZ RIVERA AND M. G. NASO
which implies that
u * u
weak ∗ in L∞ (I; H 1 )
* weak ∗ in L∞ (I; L2 )
v * v
weak ∗ in L∞ (I; H 1 )
ut * ut
weak ∗ in L∞ (I; L2 )
vt * vt
weak ∗ in L∞ (I; L2 )
u → u
strongly in L2 (I; L2 )
In particular, we have that
whence it follows that
u → u a:e: in and then
f(u ) → f(u) a:e: in Since u is bounded in L∞ (]0; T [× ) we conclude that
f(u ) is bounded in L∞ (I; L2 )
Therefore
f(u ) * f(u)
weak in L2 (I; L2 )
The rest of the proof of the existence of a weak solution is a matter of routine.
Step 3 (Regularity of solution): To get the regularity result, let us dierentiate the approximated equation and, using similar arguments of step 2, we get
d E (t) + dt 2
|xt |2 dx = −
f (u )ut utt dx +
h1 utt dx +
L2
L1
h3 vtt dx +
h2 dx
where
E2 (t) =
1
2
(|utt |2
+
a|uxt |2
1
+ |t | ) dx +
2
2
L2
L1
(|vtt |2 + b|vxt |2 ) dx
Since
f (u )
is bounded in L∞ (]0; T [ × )
our conclusion follows.
Step 4 (Uniqueness and continuous dependence on initial data): Suppose that y1 = (u1 ; ; v1 )
and y2 = (u2 ; 2 ; v2 ) are two solutions of (1)–(9) with initial data (u10 ; 10 ; v10 ) and (u20 ; 20 ;
˜ ṽ) = y1 − y2 and ỹ = y10 − y20 . Taking the dierence
v20 ), respectively, and let ỹ = (ũ; ;
0
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TRANSMISSION PROBLEM IN THERMOELASITICITY
of (1)–(3) with y1 and y2 in place of y = (u; ; v), we get
1 d
2 dt
(|ũt |2 + a|ũx |2 ) dx +
L2
L1
(|ṽt |2 + b|ṽx |2 ) dx +
|˜|2 dx
= −
|˜x |2 dx − [f(u1 ) − f(u2 )]ũ1 dx
6− [f(u1 ) − f(u2 )]ũt dx
(14)
Concerning the uniqueness, we have to use the regularity of the solution. Any weak solution
of system (1)–(3) satises u ∈ L∞ (I; H 1 ), and therefore u ∈ L∞ ( × ]0; T [). Let us denote by
M ¿0 the number such that |u(x; t)|6M a.e. on × ]0; T [. Now, since f is a C 1 -function,
the number
Mf = sup {|f (x)| : |x|6M }
is nite. Under these notations we have that
|f(u1 ) − f(u2 )| |ũt | dx 6 Mf
|u1 − u2 | |ũt | dx
6 Mf
|ũ| |ũt | dx
6 Mf
(|ũ|2 + |ũt |2 ) dx
Finally, since u(0; t) = u(L3 ; t) = 0, Poincare inequality implies
|f(u1 ) − f(u2 )| |ut | dx6Mf C (|ũx |2 + |ũt |2 ) dx
where C = max {CP ; 1}. Hence (14) leads to
d
dt
(|ũt | + a|ũx | ) dx +
2
2
L2
L1
(|ũt |2 + a|ũx |2 ) dx +
6C̃
(|ṽt | + b|ṽx | ) dx +
2
|˜|2 dx
2
L2
L1
(|ṽt |2 + b|ṽx |2 ) dx +
|˜|2 dx
where C̃ is a positive constant. Using Gronwall’s lemma, the uniqueness and the continuous
dependence on initial data follows.
Remark 3.1
With the same above procedure we can prove that when the initial data satises
(u0 ; v0 ) ∈ (H 3 × H 3 ) ∩ V;
(u1 ; v1 ) ∈ (H 2 × H 2 ) ∩ V;
Copyright ? 2002 John Wiley & Sons, Ltd.
(u2 ; v2 ) ∈ (H 1 × H 1 ) ∩ V;
0 ∈ H 3 ∩ HL1
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A. MARZOCCHI, J. E. MUÑOZ RIVERA AND M. G. NASO
verifying the compatibility conditions
au0x (Li ) − m0 (Li ) = bv0x (Li );
i = 1; 2
where
(u2 ; v2 ) = (au0; xx − m0; x − f(u0 ) + h1 (x);
bv0; xx + h3 )
then the solution satises
(u; v) ∈
2
i=0
C i (I; (H 3−i × H 3−i ) ∩V ) ∩ C(I; L2 × L2 )
∈ C(I; H 3 ∩ HL1 ) ∩ C 1 (I; H 1 )
When (u; v; ) satises the above regularity we will say that (u; v; ) is an H 3 -solution.
4. ENERGY ESTIMATES
In the next lemmas we show the dissipative properties of system (1)–(9). We assume that f
is a Lipschitz function and ¿ 0 is its Lipschitz constant. With these notations, we have
Lemma 4.1
Let us suppose that (u; ; v) is a strong solution of system (1)–(9). Then the energy identity
can be written as
L2
d
2
E1 (t) = − |x | dx + (h1 ut + h2 ) dx +
h3 vt dx
(15)
dt
L1
where
E1 (t) =
1
2
(|ut |2 + a|ux |2 + ||2 ) dx +
1
2
L2
L1
In particular, if hi ≡ 0 (i = 1; 2; 3); we have
d
E1 (t) = −
dt
(|vt |2 + b|vx |2 ) dx+
F(u(t)) dt
|x |2 dx
Proof
Multiplying Equation (1) by ut , Equation (2) by and Equation (3) by vt ; we get, integrating
over the respective intervals,
1 d
2 dt
[|ut | + a|ux | + 2F(u)] dx + m
2
x ut dx
2
h1 ut dx − [aux (L1 ; t) − m(L1 ; t)]ut (L1 ; t) + [aux (L2 ; t) − m(L2 ; t)]ut (L2 ; t) (16)
=
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1 d
2 dt
1 d
2 dt
|| dx + L2
L2
=
L1
uxt dx =
2
L1
|x | dx + m
2
h2 dx
(17)
(|vt |2 + b|vx |2 ) dx −bvx (L2 ; t)vt (L2 ; t) + bvx (L1 ; t)vt (L1 ; t)
h3 vt dx
(18)
Summing up identities (16)–(18) our conclusion follows.
Lemma 4.2
Let us suppose that (u; ; v) is an H 3 -solution of system (1)–(9). Then we have that
d
E2 (t) = − |xt |2 dx− f (u)ut utt dx
dt
(19)
where
E2 (t) =
1
2
(|utt |2 + a|uxt |2 + |t |2 ) dx +
1
2
L2
L1
(|vtt |2 + b|vxt |2 ) dx
Proof
Dierentiating Equations (1)–(3) with respect to t and using the same procedure as in
Lemma 4.1, we get (19).
Lemma 4.3
Let us suppose that (u; ; v) is an H 3 -solution of the system (1)–(9). Then we have that
d
E3 (t) = −a
dt
|xx |2 dx + a
f(u)uxxt dx − ma[x (L3 ; t)uxt (L3 ; t) − x (0; t)uxt (0; t)]
+ m[t (L1 ; t)utt (L1 ; t) − t (L2 ; t)utt (L2 ; t)]
L2
− a (h1 uxxt + h2 xx ) dx − b
h3 vxxt dx
(20)
L1
where
E3 (t) =
1
2
a(|uxt |2 + a|uxx |2 + |x |2 ) dx +
1
2
L2
L1
b(|vxt |2 + b|vxx |2 ) dx
In particular, if hi ≡ 0 (i = 1; 2; 3); we obtain
d
E3 (t) = −a
dt
|xx | dx + a
f(u)uxxt dx − ma[x (L3 ; t)uxt (L3 ; t) − x (0; t)uxt (0; t)]
2
+ m[t (L1 ; t)utt (L1 ; t) − t (L2 ; t)utt (L2 ; t)]
Copyright ? 2002 John Wiley & Sons, Ltd.
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A. MARZOCCHI, J. E. MUÑOZ RIVERA AND M. G. NASO
Proof
Multiplying Equation (1) by −auxxt , Equation (2) by −axx and Equation (3) by −bvxxt ,
integrating over the respective intervals and summing up the product results, our conclusion
follows.
Dene the quantity
(N0 uxt − ut uxx + c0 uut ) dx + c0
K (t) =
L2
L1
vvt dx
where N0 and c0 are positive constants, and let us denote
B(t) = |uxt (L1 ; t)|2 + |uxt (L2 ; t)|2 + |utt (L1 ; t)|2 + |utt (L2 ; t)|2
B0 (t) = |uxt (0; t)|2 + |uxt (L3 ; t)|2
Now, we have
Lemma 4.4
If (u; ; v) is an H 3 -solution of the system (1)–(9) then the following inequality holds:
d
a
K (t) 6 −
dt
4
−
−
mN0
|uxx | dx −
4
2 N0
|uxt | dx +
m
2
ac0
8
1
4a
|ux |2 dx + c0
L2
L1
|xx |2 dx
2
|vt |2 dx + B(t)
|utt |2 dx + C
|x |2 dx
(|h1 | + |h2 | ) dx +
+C
2
L2
2
L1
|h3 | dx
2
(21)
for 12 2 ¡ac0 =20; N0 large enough and c0 a positive constant to be xed later.
Proof
Multiplying Equation (2) by uxt we get
d
dt
uxt dx =
t uxt dx +
uxtt dx
xx uxt dx − m
=
|uxt | dx −
x utt dx +
2
h2 uxt dx
− [(L2 ; t)utt (L2 ; t) − (L1 ; t)utt (L1 ; t)]
Copyright ? 2002 John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci. 2002; 25:955–980
967
TRANSMISSION PROBLEM IN THERMOELASITICITY
2
6
m
m
|xx | dx −
2
2
|uxt | dx +
N
0
[|utt (L2 ; t)|2 + |utt (L1 ; t)|2 ] + C
N0
+
|utt |2 dx
2
|h2 |2 dx + C
|x |2 dx
(22)
Multiplying Equation (1) by −uxx we get
−
d
dt
ut uxx dx = −
utt uxx dx −
ut uxxt dx
=−a
|uxx |2 dx + m
h1 uxx dx +
−
x uxx dx +
f(u)uxx dx
|uxt |2 dx
+ ut (L2 ; t)uxt (L2 ; t) − ut (L1 ; t)uxt (L1 ; t)
(23)
Using Equation (1) we have
|utt |2 ¿
a2
|uxx |2 − c|x |2 − 4|f(u)|2 − 4|h1 |2
2
and we nd
d
−
dt
a
ut uxx dx 6 −
4
1
|uxx | dx −
2a
|utt | dx + C
2
|uxt | dx +
2
2
+ [|uxt (L1 ; t)|2 + |uxt (L2 ; t)|2 ] + C
512 2
|x |2 dx + C
|ux |2 dx
|h1 |2 dx
(24)
Multiplying Equation (1) by u and (3) by v; we get
d
dt
uut dx =
|ut | dx − a
|ux |2 dx + a[u(L1 ; t)ux (L1 ; t) − u(L2 ; t)ux (L2 ; t)]
2
−m
x u dx −
f(u)u dx +
h1 u dx
and
d
dt
L2
L1
vvt dx =
L2
L1
|vt |2 dx − b
L2
L1
|vx |2 dx
+ b[v(L2 ; t)vx (L2 ; t) − v(L1 ; t)vx (L1 ; t) +
Copyright ? 2002 John Wiley & Sons, Ltd.
L2
L1
h3 v dx
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A. MARZOCCHI, J. E. MUÑOZ RIVERA AND M. G. NASO
Summing up the above two equations, we get
d
dt
uut dx +
vvt dx
L1
|ut |2 dx +
=
L2
L2
L1
|vt |2 dx − a
|ux |2 dx − b
L2
|vx |2 dx
L1
+ m[u(L1 ; t)(L1 ; t) − u(L2 ; t)(L2 ; t)] − m
x u dx −
f(u)u dx
h1 u dx +
+
|ut |2 dx +
L2
+C
a
2
|vt |2 dx −
L1
h3 v dx
L1
6
L2
|ux |2 dx − b
|x | dx + C
L1
|h1 | dx + C
2
2
L2
L1
L2
|vx |2 dx
|h3 |2 dx
where is such that 12 2 ¡ac0 =20 and 1 is the Poincare constant depending on .
We conclude that
a
d
K(t) 6 −
dt
2
+
−
|uxx | dx −
2
N0 2
m
ac
0
2
N0 m
− C − c 0 1
2
1
4a
|xx |2 dx −
−
512 2
|uxt |2 dx
|utt |2 dx + B(t)
|ux | dx + c0
2
+ C
L2
L1
|vt |2 dx
|x |2 dx + C
(|h1 |2 + |h2 |2 ) dx + C
L2
L1
|h3 |2 dx
+ m[u(L1 ; t)(L1 ; t) − u(L2 ; t)(L2 ; t)]
Since
m[u(L1 ; t)(L1 ; t) − u(L2 ; t)(L2 ; t)]6
|ux | dx + C
|x |2 dx
2
for ¡ac0 =4, our conclusion follows.
Copyright ? 2002 John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci. 2002; 25:955–980
TRANSMISSION PROBLEM IN THERMOELASITICITY
969
Let us now introduce the integrals
J1 (t) = −
L1
w1 utt uxt dx −
0
J2 (t) = −
L2
L1
L3
L2
w2 utt uxt dx
w3 vtt vxt dx
where
w1 (x) = x −
L1
;
2
w2 (x) = x −
L2 + L 3
;
2
w3 (x) =
x ∈ [0; L1 ]
x ∈ [L2 ; L3 ]
L2 − L3 − L1
L1
(x − L1 ) + ;
2(L2 − L1 )
2
x ∈ [L1 ; L2 ]
Lemma 4.5
Let (u; ; v) be an H 3 -solution of problem (1)–(9) and let h1 ∈ L2 . Then the following inequalities hold:
d
J1 (t)6 − l[B(t) + B0 (t)] + D2 [|utt |2 + (a + 2 )|uxt |2 + |xt |2 ] dx
(25)
dt
and
d
J2 (t) 6
dt
2a2
l
L3 + L1 − L2 L2
+1
B(t) −
(|vtt |2 + b|vxt |2 ) dx
b
2
4(L2 − L1 ) L1
2
+ 2m 1 |xt |2 dx
(26)
where l = (a + 1)=2 min{L1 ; L3 − L2 } and D2 = max{2 L=2; (mL + L + 1)=2} depends on m and
the domain.
Proof
Applying Lemma 2.1 with z = −m#xt − f (u)ut ; x1 = 0; x2 = L1 ; = a to the time derivative
of Equation (1) with w1 in place of q; we get
−
d
dt
0
L1
L1
[a|uxt (L1 ; t)|2 + a|uxt (0; t)|2 + |utt (L1 ; t)|2 ]
2
1 L1
2
2
2
2
2
2 L1
|utt | + (a + 1)|uxt | + (m |xt | + |ut | )
dx
+
2
2 0
w1 utt uxt dx 6 −
Copyright ? 2002 John Wiley & Sons, Ltd.
(27)
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A. MARZOCCHI, J. E. MUÑOZ RIVERA AND M. G. NASO
Similarly,
d
−
dt
L3
L2
w2 utt uxt dx 6 −
L3 − L2
[a|uxt (L2 ; t)|2 + a|uxt (L3 ; t)|2 + |utt (L2 ; t)|2 ]
2
1
+
2
L3 L3 − L2
|utt | + a|uxt | + (m |xt | + |ut | )
dx
2
L2
2
2
2
2
2
2
(28)
Summing up inequalities (27) and (28) and denoting
2
L mL + L + 1
D2 = max
;
2
2
we have that
d
J1 (t) 6 −l[|uxt (L3 ; t)|2 + |uxt (L2 ; t)|2 + |uxt (L1 ; t)|2
dt
+ |uxt (0; t)|2 + |utt (L2 ; t)|2 + |utt (L1 ; t)|2 ]
+ D2
(|utt |2 + a|uxt |2 + |xt |2 + 2 l|uxt |2 ) dx
From boundary conditions (5) we observe that
b2 |vxt (Li ; t)|2 6 2a2 |uxt (Li ; t)|2 + 2m2 |t (Li ; t)|2
6 2a B(t) + 2m 1
2
|xt |2 dx;
2
i = 1; 2
and applying Lemma 2.1 to the time derivative of (3), we get
l
d
J2 (t) 6 [|vtt (L1 ; t)|2 + b|vxt (L1 ; t)|2 + |vtt (L2 ; t)|2 + b|vxt (L2 ; t)|2 ]
dt
2
−
6
−
L3 + L1 − L2
4(L2 − L1 )
L2
L1
(|vtt |2 + b|vxt |2 ) dx
2a2
l
+1
B(t) + 2m2 1 |xt |2 dx
b
2
L3 + L1 − L2
4(L2 − L1 )
Copyright ? 2002 John Wiley & Sons, Ltd.
L2
L1
(|vtt |2 + b|vxt |2 ) dx
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971
TRANSMISSION PROBLEM IN THERMOELASITICITY
Lemma 4.6
Let (u; ; v) be an H 3 -solution of problem (1)–(9) and let h1 ; h2 ; h3 ∈ L2 . Then the following
inequality holds:
0 b
d
K (t) + 0 J1 (t) +
J2 (t)
dt
8(2a2 + b)
6−
mN0
10
1
8a
|uxt |2 dx −
0 (L3 + L1 − L2 )
−
32b(L2 − L1 )
L2
L1
|utt |2 dx −
l0
[B(t) + B0 (t)]
8
|uxx |2 dx −
2 N0
(|vxt | + |vtt | ) dx + c (|x | + |xt | ) dx +
m
(|h1 | + |h2 | ) dx +
+c
2
2
a
2
2
2
L2
L1
2
|xx |2 dx
2
|h3 | dx
2
(29)
Proof
Using the rst part of Lemma 4.5 we get
mN0
d
[K (t) + 0 J1 (t)] 6 −
− 0 D2 (a + )2
|uxt |2 dx
dt
4
−
1
− 0 D 2
4a
|utt |2 dx −
a
2
|uxx |2 dx
− (l0 − )[B(t) + B0 (t)] + c0
+
2 N0
m
L1
|vt |2 dx
|xx |2 dx + c
(|x |2 + |xt |2 ) dx
(|h1 | + |h2 | ) dx + c
+c
L2
2
2
L2
L1
|h3 |2 dx
Then we take small such that ¡l0 =2 and we nd
mN0
d
[K (t) + 0 J1 (t)] 6 −
dt
8
a
−
2
Copyright ? 2002 John Wiley & Sons, Ltd.
1
|uxt | dx −
8a
|utt |2 dx
2
l0
[B(t) + B0 (t)] + c0
|uxx | dx −
2
2
L2
L1
|vt |2 dx
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A. MARZOCCHI, J. E. MUÑOZ RIVERA AND M. G. NASO
2 N0
2
+
|xx | dx + c (|x |2 + |xt |2 ) dx
m L2
2
2
+ c (|h1 | + |h2 | ) dx + c
|h3 |2 dx
L1
Since
a2 |ux (Li ; t)|2 ¿b2 |vx (Li ; t)|2 − m2 |(Li ; t)|2 ;
i = 1; 2
whence we have that
mN0
1
d
2
[K (t) + 0 J1 (t)] 6 −
|uxt | dx −
|utt |2 dx
dt
8 8a L2
a
l0
2
[B(t) + B0 (t)] + c0
−
|uxx | dx −
|vt |2 dx
2 4
L1
2 N0
2
|xx | dx + c (|x |2 + |xt |2 ) dx
+
m L2
2
2
|h3 |2 dx
+ c (|h1 | + |h2 | ) dx + c
L1
Using the second part of Lemma 4.5 we get
0 b
d
K (t) + 0 J1 (t) +
J2 (t)
dt
8(2a2 + b)
mN0
1
a
l0
[B(t) + B0 (t)]
6−
|uxt |2 dx −
|utt |2 dx −
|uxx |2 dx −
8 8a 2 8
L2
0 b(L3 + L1 − L2 )
−
(|vxt |2 + |vtt |2 ) dx
32(2a2 + b)(L2 − L1 ) L1
L2
2 N0
+ c0
|vt |2 dx +
|xx |2 dx + c (|x |2 + |xt |2 ) dx
m L1
L2
+ c (|h1 |2 + |h2 |2 ) dx + c
|h3 |2 dx
L1
Since
L2
L1
|vt | dx 6 2(L2 − L1 )|vt (L1 ; t)| +2(L2 − L1 )
2
2
6 2(L2 − L1 )c
0
Copyright ? 2002 John Wiley & Sons, Ltd.
L1
2
L2
L1
|uxt | dx+2(L2 − L1 )
2
|vxt |2 dx
2
L2
L1
|vxt |2 dx
Math. Meth. Appl. Sci. 2002; 25:955–980
TRANSMISSION PROBLEM IN THERMOELASITICITY
973
taking c0 such that
2(L2 − L1 )2 c0 ¡
0 b(L3 + L1 − L2 )
64(2a2 + b)(L2 − L1 )
and N0 large enough, our conclusion follows.
We introduce the following functional:
L(t) = M0 E1 (t) + M0 E2 (t) + N E3 (t) − Na
f(u)uxx dx
+ K (t) + 0 J1 (t) +
0 b
J2 (t)
8(2a2 + b)
where M0 , N , 0 are positive constants. Under these conditions we are able to show the main
result of this section.
Theorem 4.1
Let (u; ; v) be a strong solution of problem (1)–(9) and let h1 ; h3 ∈ H 1 , h2 ∈ L2 . Then
d
L(t)6 − L(t) + (30)
dt
c1
where ; c1 ; are positive constants. In particular, if hi ≡ 0, i = 1; 2; 3. We have that
E1 (t) + E2 (t) + E3 (t)6[E1 (0) + E2 (0) + E3 (0)]e−()=(c1 )t
Proof
We will suppose that the (u; v; ) is an H 3 -solution; our conclusion will follow by standard
density arguments. Note that
d
N E3 (t) − Na f(u)uxx dx
dt
2
6 −Na |xx | dx − Na f (u)uxx ut dx − Nma[x (L3 ; t)uxt (L3 ; t) − x (0; t)uxt (0; t)]
− Nm[t (L2 ; t)utt (L2 ; t) − t (L1 ; t)utt (L1 ; t)] − Na[h1 (L3 )uxt (L3 ; t) − h1 (L2 )uxt (L2 ; t)]
− Na[h1 (L1 )uxt (L1 ; t) − h1 (0)uxt (0; t)] − Nb[h3 (L2 )vxt (L2 ; t) − h3 (L1 )vxt (L1 ; t)]
L2
−N
h1x uxt dx − N
h3x vxt dx
L1
Using Gagliardos–Niremberg and Cauchy inequalities we get
1=2 |x (x; t)| 6 c
|x |2 dx
6 c
|x |2 dx + Copyright ? 2002 John Wiley & Sons, Ltd.
1=2
|xx |2 dx
|xx |2 dx
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974
A. MARZOCCHI, J. E. MUÑOZ RIVERA AND M. G. NASO
Therefore, we obtain
d
N E3 (t) − Na f(u)uxx dx
dt
6[|uxt (L3 ; t)|2 + |uxt (0; t)|2 ]]
+ [|utt (L1 ; t)|2 + |utt (L2 ; t)|2 ] + [|vxt (L1 ; t)|2 + |vxt (L2 ; t)|2 ]
− Na
|xx |2 dx + NaL
+
L2
L1
N2
|uxt |2 dx +
(|xt |2 + |x |2 ) dx
|vxt | dx + C
[|h1x | + |h1 | ] dx +
2
2
2
L2
L1
|h3x | dx
2
(31)
From (31) we get
0 b
d
N E3 (t) − Na
f(u)uxx dx + K(t) + 0 J1 (t) +
J
(t)
2
dt
8(2a2 + b)
mN0
1
a
6−
− NaL
|uxt |2 dx −
|utt |2 dx −
|uxx |2 dx
8
8a 2 −
−
0 (L3 + L1 − L2 )
−
32b(L2 − L1 )
l0
[B(t) + B0 (t)] + C
8
L2
L1
N0
(|vxt |2 + |vtt |2 ) dx − Na − −
|xx |2 dx
m
(|xt |2 + |x |2 ) dx
(|h1 | + |h1x | + |h2 | ) dx +
+ C
2
2
2
L2
L1
(|h3 | + |h3x | ) dx
2
2
Taking ¡mN0 =(20NaL) and N such that
Na − −
N0 Na
¿
2
2
we nd
mN0
d
L(t) 6 −
20
dt
−
|uxt |2 dx −
1
8a
|utt |2 dx −
a
2
|uxx |2 dx
L2
0 (L3 +L1 −L2 )
Na
−
(|vxt |2 +|vtt |2 ) dx−
|xx |2 dx
32b(L2 −L1 )
m
L1
Copyright ? 2002 John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci. 2002; 25:955–980
975
TRANSMISSION PROBLEM IN THERMOELASITICITY
− (M0 − C )
(|h1 | + |h1x | + |h2 | ) dx +
+ C
f (u)ut utt dx
(|xt |2 + |x |2 ) dx − M0
2
2
2
L2
L1
(|h3 | + |h3x | ) dx
2
2
Using
M0
f (u)ut utt dx 6 M0 |ut | |utt | dx
6 8M02 2 a
|ut |2 dx +
6
8M02 2 12
a
a
16
|uxt |2 dx +
|utt |2 dx
1
16a
|utt |2 dx
and taking such that
mN0 8M02 2 12
= k0 ¿0
−
20
a
and
M0 − C = k1 ¿0
we conclude that
1
a
|utt |2 dx −
|uxx |2 dx
8a
2
L2
0 (L3 + L1 − L2 )
−
(|vxt |2 + |vtt |2 ) dx
40b(L2 − L1 )
L1
Na
2
−
|xx | dx − k1 (|x |2 + |xt |2 ) dx
2 L2
2
2
2
+ C
(|h1 | + |h2 | ) dx +
|h3 | dx
d
L(t) 6 −k0
dt
|uxt |2 dx −
L1
Recalling the denition of L and using Cauchy inequality, we see that there exist two positive
constants c0 and c1 such that
c0 [E1 (t) + E2 (t) + E3 (t)]6L(t)6c1 [E1 (t) + E2 (t) + E3 (t)]
(32)
It is not dicult to see that there exists ¿0 such that
d
L(t) 6 −[E1 (t) + E2 (t) + E3 (t)] + C
dt
6 − L(t) + c1
Copyright ? 2002 John Wiley & Sons, Ltd.
(|h1 |2 + |h2 |2 ) dx +
L2
L1
|h3 |2 dx
(33)
Math. Meth. Appl. Sci. 2002; 25:955–980
976
A. MARZOCCHI, J. E. MUÑOZ RIVERA AND M. G. NASO
where
(|h1 |2 + |h2 |2 ) dx +
= C
L2
L1
|h3 |2 dx
whence the exponential decay holds.
Remark 4.1
Note from the proof that if we take L1 = and L2 ≈ − + L3 , for small , the dissipative part
of the material is small. Let us consider smaller than the other constants in (30). Then we
conclude that in (33) is a multiple of , that is
= c
where c is a parameter which depends on the coecients.
Taking hi ≡ 0; i = 1; 2; 3; we conclude that
L(t)6L(0)e−(c)=(c1 )t
(34)
and this means that the decay rate depends on the amplitude of the dissipative of the material.
If → 0 then we lose the uniform decay.
5. ASYMPTOTIC BEHAVIOUR
In this section we state some asymptotic properties of the solutions of our equations.
Denition 5.1
Let B(0; R) be the open ball with centre 0 and radius R¿0 in H. A bounded set B0 ⊂ H is
called an absorbing set for problem (1)–(9) if for any initial value y0 ∈ B(0; R) ⊂ H there
exists tH = tH (R) such that every solution starting from y0 satises
y(t) ∈ B0 ;
∀t ¿tH
Theorem 5.1
Under the hypotheses made on f, there exists an absorbing set in the space V.
Proof
From Theorem 4.1, we have
(1 − e−t )
and using the compatibility condition of the initial data, we get
L(t)6L(0)e−t +
E1 (0)6c[E2 (0) + E3 (0)]
so we have
E2 (t) + E3 (t)6c1 [E2 (0) + E3 (0)]e−t + c2
for appropriate constants c; c1 ; c2 . This implies that every ball in V of radius greater than c2
is absorbing, since E2 + E3 is an equivalent norm in V.
Copyright ? 2002 John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci. 2002; 25:955–980
TRANSMISSION PROBLEM IN THERMOELASITICITY
977
In the linear case, we can also obtain the decay in the rst-order energy.
Theorem 5.2
If f ≡ 0; hi ≡ 0 (i = 1; 2; 3) then there exists a constant c¿0 such that
E(t)6E1 (0)e−t
Proof
We introduce the following functions:
U (x; t) =
t
u(x; s) ds + 1 (x);
(x; t) ∈ ×]0; ∞[
(x; s) ds + 2 (x);
(x; t) ∈ ×]0; ∞[
v(x; s) ds + 3 (x);
(x; t) ∈ ]L1 ; L2 [×]0; ∞[
0
(x; t) =
t
(35)
0
W (x; t) =
t
0
Integrating on [0; t] each Equation (11)–(13) with f ≡ 0; hi ≡ 0 (i = 1; 2; 3), substituting (35),
we observe that (U; ; W ) verify
Utt − aUxx + mx + a1xx − m2x − u1 = 0
t − xx + mUxt + 2xx − mu0x − 0 = 0
Wtt − bWxx + b3xx − v1 = 0
We choose functions i (i = 1; 2; 3) as solutions to the elliptic system
a1xx − m2x − u1 = 0
2xx − mu0x − 0 = 0
b3xx − v1 = 0
where u0 ; u1 ; 0 ; v1 are dened on initial conditions (7)–(9), so that (U; ; W ) satisfy
Utt − aUxx + mx = 0
t − xx + mUxt = 0
Wtt − bWxx = 0
From Theorem 4.1 applied to the above system, we immediately nd
L(U (t); (t); W (t))6L(U (0); (0); W (0))e−t
Copyright ? 2002 John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci. 2002; 25:955–980
978
A. MARZOCCHI, J. E. MUÑOZ RIVERA AND M. G. NASO
Now, notice that in the linear homogeneous case there exists a positive constant c3 such that
E3 (t)6c3 E2 (t)
(36)
since
Uxx =
1
(Utt + mx )
a
xx =
1
(t + mUxt )
k
and
so that
|Uxx |2 6
m2
1
|Utt |2 + 2 |x |2
2
a
a
|xx |2 6
m2
1
2
|
|
+
|Uxt |2
t
k2
k2
Thus, using Poincare inequality, for x and inserting these estimates into the expression of E3 ,
we nd (36). At this point, there exists a positive constant c4 such that
E1 (u(t); (t); v(t)) = E2 (U (t); (t); W (t))6L(U (t); (t); W (t))6L(0)e−t
6 c1 [E2 (0) + E3 (0)]e−t 6c4 E2 (0)e−t = c4 E1 (u(0); (0); v(0))e−t
Theorem 5.3
If f ≡ 0 in (1), then the solution tends exponentially to the solution (u∗ ; v∗ ; ∗ ) of the static
problem
∗
−auxx
+ mx∗ = h1 ;
in ×]0; ∞[
∗
−xx
= h2 ;
in ×]0; ∞[
∗
−bvxx
= h3 ;
in ]L1 ; L2 [×]0; ∞[
Proof
We put
y1 = u − u ∗ ;
y2 = v − v ∗ ;
y3 = − ∗
then (y1 ; y2 ; y3 ) veries system (1)–(3) with hi ≡ 0 (i = 1; 2; 3) and f ≡ 0. Therefore E1; y (t)
→ 0, and, by the expression of E1; y (t), we nd that y → 0.
Copyright ? 2002 John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci. 2002; 25:955–980
TRANSMISSION PROBLEM IN THERMOELASITICITY
979
Figure 2.
Remark 5.1
The same procedure applies also, with slight modications, to the case where the body has
only an elastic part, followed by a thermoelastic one (that would be if L1 = 0) (see Figure 2).
ACKNOWLEDGEMENTS
The authors express their appreciation to the referees for their valuated suggestions which improved
this paper. This work has been partially supported by a grant 305406=88-4 of CNPq-BRASIL and by
Italian MURST through the Research Programme “Modelli Matematici per la Scienza dei Materiali”.
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