We shall divide the proof into two steps: in Step 1, we shall use the semigroup approach to prove the existence of global solutions and the Remark 2.1; Step 2 is devoted to proving the uniform decay of the energy by the boundary control and the multiplier method.
Step 1. Existence of global solutions.
The proof is based on the semigroup approach (see [4, 12]) that can be used to reduce problem (1.1)-(1.7) to an abstract initial value problem for a first-order evolution equation. In order to choose proper space for (1.1)-(1.7), we shall consider the static system associated with them (see ). Considering the energy and the property of operator A, we can choose the following state space and the domain of operator A for problem (1.1)-(1.7):
Using the same method as in [4, 12], we can prove that the operator A generates a -semigroup of contractions on the Hilbert space H. Define
Then by Theorem 2.3.1 of  about the existence and regularities of solutions, we can complete the proof.
Step 2. Uniform decay of the energy.
In this section, we shall assume the existence of solutions in the Sobolev spaces that we need for our computations. The proof of uniform decay is difficult. It is necessary to construct a suitable Lyapunov function and to combine various techniques from energy method, multiplier approaches and boundary control (see [10, 11]). We mainly refer to Racke  for the approaches of thermo-elastic models with second sound.
Multiplying (1.1) by , (1.2) by , (1.3) by , (1.4) by , and (1.5) by in , respectively, and summing up the results, yields
Similarly, we can get
Multiplying (1.1) by in , we get
Multiplying (1.2) by , (1.3) by in and summing them up, yields
Combining (3.5) with (3.6), we get
Now, we conclude from (1.1), (1.4), (1.5) and Poincaré inequality
Multiplying (1.1) by u in , we obtain
From (1.2) and (1.3), we get
Multiplying (3.10) by , (3.11) by in , and summing up the results, we get
The boundary terms are estimated as follows.
for some ,
Combining (3.13)-(3.16), we get
Differentiating (1.1) with respect to t and multiplying by , where
Multiplications of (1.2) by and (1.3) by yield
which implies, using (1.4) and (1.5),
Combining (3.19)-(3.21), we conclude
Using (1.2) and (1.3), we get
With (3.17) and (3.23), we can estimate
Define . Multiplying both sides of (3.12) by ξ and combining the result with (3.7) and (3.24), we obtain for sufficiently small the estimate
where , and . Now, we can define the desired Lyapunov functional . For , to be determined later on, let
Then we conclude from (3.3), (3.4), and (3.15)
By using (3.8), we arrive at
while (3.9) yields
Combining (3.27)-(3.29), we conclude
We choose such that all terms on the right-hand side of (3.30) become negative,
Choosing ε as in (3.31), we obtain from (3.30)
There exist positive constants , and such that for any and , it holds
where , are determined later on. In fact,
At this point, we choose , . Finally, we choose . Thus, we have the validity of (3.34). Combining (3.33) with (3.34), we get
Hence, it follows from (3.36), . Applying (3.34) again, we can conclude (2.5) with . The proof is complete.