In this section, we introduce some lemmas which play a crucial role in proof of our main result in next section.

**Lemma 2.1**. *E*(*t*) is a non-increasing function.

**Proof**. By differentiating (1.9) and using (1.2) and (A1), we get

Thus, Lemma 2.1 follows at once. At the same time, we have the following inequality:

**Lemma 2.2**. Assume that

*g*(

*t*) satisfies assumptions (A1) and (A2),

*H*(

*t*) is a twice continuously differentiable function and satisfies

for every *t* ∈ [0, *T*_{0}), and (*u*(*x*, *t*), *v*(*x*, *t*)) is the solution of the problem (1.2).

Then the function *H*(*t*) is strictly increasing on [0, *T*_{0}).

**Proof**. Consider the following auxiliary ODE

for every *t* ∈ [0, *T*_{0}).

It is easy to see that the solution of (2.4) is written as follows

for every *t* ∈ [0, *T*_{0}).

By a direct computation, we obtain

for every *t* ∈ [0, *T*_{0}).

Because *g*(*t*) satisfies (A2), then *h*'(*t*) ≥ 0, which implies that *h*(*t*) ≥ *h*(0) = *H*(0). Moreover, we see that *H*'(0) > *h*'(0).

Assume that (2.6) is not true, let us take

By the continuity of the solutions for the ODES (2.3) and (2.4), we see that

*t*_{0} > 0 and

*H*' (

*t*_{0}) =

*h*' (

*t*_{0}), and have

This contradicts *H*'(*t*_{0}) = *h*'(*t*_{0}). Thus, we have *H*'(*t*) > *h*' (*t*) *≥* 0, which implies our desired result. The proof of Lemma 2.2 is complete.

**Lemma 2.3**. Suppose that

, (

*u*_{1},

*v*_{1}) ∈

*L*^{2}(Ω) ×

*L*^{2}(Ω) satisfies

If the local solution (

*u*(

*t*),

*v*(

*t*)) of the problem (1.2) exists on [0,

*T*) and satisfies

then
is strictly increasing on [0, *T* ).

**Proof**. Since

, and (

*u*(

*t*),

*v*(

*t*)) is the local solution of problem (1.2), by a simple computation, we have

Therefore, by Lemma 2.2, the proof of Lemma 2.3 is complete.

**Lemma 2.4**. If

, (

*u*_{1},

*v*_{1}) ∈

*L*^{2}(Ω) ×

*L*^{2}(Ω) satisfy the assumptions in Theorem 1.2, then the solution (

*u*(

*x*,

*t*),

*v*(

*x*,

*t*)) of problem (1.2) satisfies

for every *t* ∈ [0, *T*).

**Proof**. We will prove the lemma by a contradiction argument. First we assume that (2.9) is not true over [0,

*T*), it means that there exists a time

*t*_{1} such that

Since

*I* (

*u*(

*t*,

*x*),

*v*(

*t*,

*x*)) < 0 on [0,

*t*_{1}), by Lemma 2.3 we see that

is strictly increasing over [0,

*t*_{1}), which implies

By the continuity of

on

*t*, we have

On the other hand, by (2.2) we get

It follows from (1.9) and (2.11) that

Thus, by the Poincaré's inequality and

, we see that

Obviously, (2.15) contradicts to (2.12). Thus, (2.9) holds for every *t* ∈ [0, *T*).

By Lemma 2.3, it follows that

is strictly increasing on [0,

*T*), which implies

for every *t* ∈ [0, *T*). The proof of Lemma 2.4 is complete.