Proof Multiplying the first equation of (1.1) by u and integrating with respect to x on , using integration by parts, we obtain
(4.1)
Similarly, we get
(4.2)
and
(4.3)
Summing up (4.1)-(4.3), we deduce that
By integration by parts and the Cauchy inequality, we obtain
(4.5)
Using integration by parts, we obtain
(4.6)
Combining (4.4)-(4.6) yields
Integrating with respect to t, we have
Differentiating (1.1) with respect to , we obtain
(4.8)
Taking the inner product of with the first equation of (4.8) and using integration by parts yield
Similarly, we get
and
Combining (4.9)-(4.11) yields
Using integration by parts and the Cauchy inequality, we obtain
(4.13)
Using integration by parts, we have
(4.14)
Combining (4.12)-(4.14) yields
In what follows, we estimate (). By integration by parts and the Hölder inequality, we obtain
where
It follows from the interpolating inequality that
From (2.2), we get
where
When , we have , and the application of the Young inequality yields
(4.16)
where
From integration by parts and the Hölder inequality, we obtain
(4.17)
where
Similarly,
(4.18)
and
(4.19)
where
By integration by parts and the inequality, we have
where
When , we have , and the application of the Young inequality yields
(4.20)
where
Combining (4.15)-(4.20) yields
From the Gronwall inequality, we get
Multiplying the first equation of (1.1) by and integrating with respect to x on , and then using integration by parts, we obtain
Similarly, we get
and
Collecting (4.22)-(4.24) yields
Thanks to integration by parts and the Cauchy inequality, we get
(4.26)
It follows from (4.25)-(4.26) and integration by parts that
In what follows, we estimate ().
By (2.3) and the Young inequality, we deduce that
(4.28)
By (2.3) and the Young inequality, we have
(4.29)
Similarly, we obtain
and
(4.32)
Combining (4.27)-(4.32) yields
From (4.33), the Gronwall inequality, (4.7) and (4.21), we know that . Thus, can be extended smoothly beyond . We have completed the proof of Theorem 3.1. □