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We deduce the result from (i). □.

Combining this with (2.5), we deduce the result.

Thus we also deduce the result of Theorem 1.4.

Hence, we can deduce the result from Lemma 2.

Using Theorem 7 and Lemma 1 iii), we can deduce the result.

Proof of Theorem 2.3 From Lemmas 3.1-3.3 3.1-3.3 deduce the result easily.

In Section 4, we will deduce the results of global existence.

As a consequence, we can deduce the results represent a good approximation in a regime of weak nonlinearities U.

From (5.2) and φ ( t ) is nonnegative and nonincreasing, we have deduced the result.

As a consequence of Theorem 2.1, we deduce the following result for the case of a single equation.

As an easy consequence of both Proposition 2.1 and Theorem 2.2, we deduce the following result.