In this section we prove the exponential stability. The great difficulty here is to deal with the boundary terms in the interface of the material. This difficulty is solved using an observability result of the elastic wave equations together with the fact that the solution is radially symmetric.

Lemma 4.1.

Let one suppose that the initial data

is 3-regular; then the corresponding solution of the system (1.1)–(1.13) satisfies

where

with

and
.

Proof.

Multiplying (1.1) by

, integrating in

, and using (2.16) we have that

Multiplying (1.2) by

, integrating in

, and using (2.17) we have that

Multiplying (1.3) by

, integrating in

, and using (2.11) we have that

Multiplying (1.4) by

, integrating in

, using (2.11), and performing similar calculations as above we have that

Adding up (4.4), (4.5), (4.6), and (4.7) and using (1.12) and (1.13) we obtain

In a similar way we obtain (4.2).

Lemma 4.2.

Under the same hypotheses as in Lemma 4.1 one has that the corresponding solution of the system (1.1)–(1.11) satisfies

where

,

are positive constants and

Proof.

Multiplying (1.1) by

, integrating in

, and using (1.10) we have that

Similarly, multiplying (1.2) by

, integrating in

, and performing similar calculations as above we obtain

Multiplying (1.3) by

and integrating in

we have that

Using (1.10) and (2.9) and performing similar calculations as above we obtain

Replacing (1.1) in the above equation we obtain

Multiplying (1.4) by

, integrating in

, and performing similar calculations as above we obtain

Adding (4.19) with (4.27) we have that

Adding (4.20) with (4.28) we have that

Moreover, by Lemma 2.2, there exist positive constants

,

such that

The result follows.

Lemma 4.3.

Under the same hypotheses of Lemma 4.1 one has that the corresponding solution of the system (1.1)–(1.13) satisfies

where
,
and
are positive constants.

Proof.

Multiplying (1.1) by

, integrating in

, using (2.9), and performing straightforward calculations we have that

Multiplying (1.2) by

, integrating in

, and performing similar calculations as above we obtain

Multiplying (1.3) by

and integrating in

, we have that

Performing similar calculations as above we obtain

Multiplying (1.4) by

, integrating in

, and performing similar calculation as above we obtain

Adding (4.37), (4.38), (4.40), and (4.41), using (1.13), and performing straightforward calculations we obtain

Using the Cauchy inequality we have that

and, from trace and interpolation inequalities, we obtain

Replacing in the above equation we obtain

The result follows.

We introduce the following integrals:

with
, where
is a ball with center
and radius
.

Lemma 4.4.

Under the same hypotheses as in Lemma 4.1 one has that the corresponding solution of the system (1.1)–(1.13) satisfies

where
,
, and
are positive constants and
,
.

Proof.

Using Lemma A.1, taking

as above,

,

, and

, we obtain

Applying the hypothesis on

and since

Using (2.8) and the Cauchy-Schwartz inequality in the last term and performing straightforward calculations we obtain

Finally, considering (1.1) and applying the trace theorem we obtain

with
; there exists a positive constant
which proves (4.51).

We now introduce the integrals

Lemma 4.5.

With the same hypotheses as in Lemma 4.1, the following equality holds:

Proof.

Differentiating (1.2) in the

-variable we have that

Multiplying the above equation by

and integrating in

we obtain

On the other hand, using Lemma A.1 for

,

,

, and

we obtain

Multiplying (4.61) by

and adding with (4.62) we obtain

The result follows.

We introduce the integral

where
and
are positive constants.

Lemma 4.6.

Under the same hypotheses as in Lemma 4.1 one has that the corresponding solution of the system (1.1)–(1.13) satisfies

Proof.

From (4.11), (4.12), and (4.34), using the Cauchy-Schwartz inequality and performing straightforward calculations we have that

where

,

, and

are positive constants. By Lemma 2.2, there exist positive constants

and

such that

Hence, taking

,

, and

we obtain

Using (1.10), we have that

where
.

where
and
are positive constants.

Theorem 4.7.

Let us suppose that

is a strong solution of the system (1.1)–(1.13). Then there exist positive constants

and

such that

Proof.

We will assume that the initial data is 3-regular. The conclusion will follow by standard density arguments. Using Lemmas 4.3 and 4.5 and considering boundary conditions, we find that

From (4.1), (4.2), and (4.75) we have that

Using the Cauchy inequality, we see that there exist positive constants

,

such that

Then

. Note that for

large enough we have that

From the above two inequalities our conclusion follows.