For proving the global existence for problem (3), we cite the local well-posedness result presented in .
Lemma 3.1 ()
Let with . Then the Cauchy problem (3) has a unique solution where depends on .
Assume with . Then there exists a unique solution to problem (3) and
with the maximal existence time . First, we study the differential equation
Lemma 3.2 Let , and let be the maximal existence time of the solution to problem (3). Then problem (5) has a unique solution . Moreover, the map is an increasing diffeomorphism of R with for .
Proof From Lemma 3.1, we have and . Thus we conclude that both functions and are bounded, Lipschitz in space and in time. Using the existence and uniqueness theorem of ordinary differential equations derives that problem (5) has a unique solution .
Differentiating Eq. (5) with respect to x yields
which leads to
For every , using the Sobolev imbedding theorem yields
It is inferred that there exists a constant such that for . It completes the proof. □
Lemma 3.3 Let with , and let be the maximal existence time of the problem (3). We have
where and .
Proof Using Eqs. (2) and (6)-(8), we have
Using and solving the above equation, we complete the proof of the lemma. □
Remark 1 From Lemma 3.3, we conclude that, if , then . Since the operator preserves positivity, we get . Similarly, if , we have and .
Lemma 3.4 If , , such that (or ) and , then there exists a constant such that the solution of problem (3) satisfies .
Proof We will prove this lemma to assume which results in from Lemma 3.1. For , from Lemma 3.3, we have . Then does not change sign. From the assumption one derives
where c is a positive constant. Then
On the other hand, we have
which results in
We conclude from Eqs. (11) and (13) that . To complete the proof, we use a simple density argument . Setting , we have and . Applying when , we have . □
Using the first equation of system (3) one derives
from which we have the conservation law
Lemma 3.5 (Kato and Ponce )
If , then is an algebra. Moreover
where c is a constant depending only on r.
Lemma 3.6 (Kato and Ponce )
Let . If and , then
Lemma 3.7 Let and the function is a solution of problem (3) and the initial data . Then the following results hold:
For , there is a constant c only depending on a and b such that
Proof Using , the Gronwall inequality and Eq. (14), one derives Eq. (15).
Using and the Parseval equality gives rise to
For , applying to both sides of the first equation of system (3) and integrating with respect to x by parts, we have the identity
We will estimate the terms on the right-hand side of Eq. (17) separately. For the first term, by using the Cauchy-Schwartz inequality and Lemmas 3.5 and 3.6, we have
Using the above estimate to the second term yields
Using the Cauchy-Schwartz inequality and Lemma 3.5, we obtain
For the last term in Eq. (17), using results in
For , it follows from Eq. (20) that
For , applying Lemma 3.5 derives
It follows from Eqs. (18)-(23) that there exists a constant c such that
Integrating both sides of the above inequality with respect to t results in inequality (16). □
Proof of Theorem 1 Using Eq. (16) with , we obtain
Applying the Gronwall inequality, we get
Using Eq. (15) and Lemma 3.4, we complete the proof of Theorem 1. □
Remark 2 In fact, using with , Eqs. (15) and (26), we derive that the solution of Eq. (2) in space blows up in finite time if and only if .