In this section, we deal with the uniform exponential decay of the energy for system (1.1) by using the perturbed energy method. Before we state and prove our main result, we need the following lemmas.
Lemma 3.1. Assume (2.1) and (G 1) hold. Let (u, v) be the solution of the system (1.1), then the energy functional is a decreasing function, that is
(3.1)
Moreover, the following energy inequality holds:
(3.2)
Lemma 3.2. Let (2.1) hold. Then, there exists η > 0 such that for any, we have
(3.3)
Proof. The proof is almost the same that of Said-Houari [26], so we omit it here. □
To prove our result and for the sake of simplicity, we take a = b = 1 and introduce the following:
(3.4)
where η is the optimal constant in (3.3). The following lemma will play an essential role in the proof of our main result, and it is similar to a lemma used first by Vitillaro [27], to study a class of a single wave equation, which introduces a potential well.
Lemma 3.3. Let (2.1) and (G 1) hold. Let (u, v) be the solution of the system (1.1). Assume further that E(0) < E1and
(3.5)
Then
(3.6)
Proof. We first note that, by (2.5), (3.3) and the definition of B, we have
(3.7)
where . It is not hard to verify that g is increasing for 0 < α < α*, decreasing for α > α*, g(α) → - ∞ as α → +∞, and
where α* is given in (3.4). Now we establish (3.6) by contradiction. Suppose (3.6) does not hold, then it follows from the continuity of (u(t), v(t)) that there exists t0 ∈ (0, T) such that
By (3.7), we observe that
This is impossible since E(t) ≤ E(0) < E1 for all t ∈ [0, T). Hence (3.6) is established. □
The following integral inequality plays an important role in our proof of the energy decay of the solutions to problem (1.1).
Lemma 3.4. [28]Assume that the function φ : ℝ+ ∪ {0} → ℝ+ ∪ {0} is a non-increasing function and that there exists a constant c > 0 such that
for every t ∈ [0, ∞). Then
for every t ≥ c.
Theorem 3.5. Let (2.1) and (G 1) hold. If the initial data, satisfy E(0) < E1and
(3.8)
where the constants α*, E1are defined in (3.4), then the corresponding solution to (1.1) globally exists, i.e. T = ∞. Moreover, if the initial energy E(0) and k such that
where k = min{k1, k2}, then the energy decay is
for every t ≥ aC-1, where C is some positive constant.
Proof. In order to get T = ∞, by (2.2), it suffices to show that
is bounded independently of t. Since E(0) < E1 and
it follows from Lemma 3.3 that
which implies that
where we have used (3.3). Furthermore, by (2.3) and (2.4), we get
from which, the definition of E(t) and E(t) ≤ E(0), we deduce that
(3.9)
for t ∈ [0, T). So it follows from (16) and Lemma 3.1 that
which implies
where C is a positive constant depending only on p.
Next we want to derive the decay rate of energy function for problem (1.1). By multiplying the first equation of system (1.1) by u and the second equation of system (1.1) by v, integrating over Ω × [t1, t2] (0 ≤ t1 ≤ t2), using integration by parts and summing up, we have
which implies
(3.10)
For the 11th term on the right-hand side of (3.10), one has
(3.11)
Similarly,
(3.12)
Combining (3.10), (3.11) with (3.12), we have
(3.13)
Now we estimate every term of the right-hand side of the (3.13). First, by Hölder's inequality and Poincaré's inequality
where λ being the first eigenvalue of the operator - Δ under homogeneous Dirichlet boundary conditions. Then, by (3.9), we see that
where c1 is a constant independent on u and v, from which follows that
(3.14)
Since 0 ≤ J (t) ≤ E (t), from (3.2) we deduce that
Hence, by Poincaré inequality we get
(3.15)
where c2 is a constant independent on u and v. In addition, using Young's inequality for convolution ||f * g ||
q
≤ || f ||
r
||g||
s
with 1/q = 1/r + 1/s - 1 and 1 ≤ q, r, s ≤ ∞, noting that if q = 1, then r = 1 and s = 1, we have
(3.16)
and
(3.17)
Hence, combining (3.9), (3.16) with (3.17) we then have
(3.18)
From (3.9), we also have
(3.19)
Combining (3.18) with (3.19), we deduce that
(3.20)
Finally, we also have the following estimate
(3.21)
where c3 is a constant independent on u and v. Combining (3.13)-(3.21), we obtain
(3.22)
where C is a constant independent on u.
On the other hand, from (3.3) and (3.9), we have
which implies
(3.23)
Note that E(0) < E1, we see that
Thus, combining (3.22) with (3.23), we have
that is
(3.24)
Denote
We rewrite (3.24)
for every t ∈ [0, ∞).
Since a > 0 from the assumption conditions, by Lemma 3.4, we obtain the following energy decay for problem (1.1) as
for every t ≥ Ca-1. □