In order to prove Theorem 1.4, we consider the regularized problem for equation (1.1) in the following form:
where , , and a, k are constants. One can easily check that when , equation (3.1) is equivalent to the IVP (1.1).
Before giving the proof of Theorem 1.4, we give several lemmas.
Lemma 3.1 (See )
Let p and q be real numbers such that . Then
Lemma 3.2 Let with . Then the Cauchy problem (3.1) has a unique solution where depends on . If , the solution exists for all time. In particular, when , the corresponding solution is a classical globally defined solution of problem (3.1).
Proof First, we note that, for any and any s, the integral operator
defines a bounded linear operator on the indicated Sobolev spaces.
To prove the existence of a solution to the problem (3.1), we apply the operator to both sides of equation (3.1) and then integrate the resulting equations with regard to t. This leads to the following equations:
Suppose that is the operator in the right-hand side of equation (3.2). For fixed , we get
Since is an algebra for , we have the inequalities
Since , by Lemma 3.1, we get
where , only depend on n. Suppose that both u and v are in the closed ball of radius R about the zero function in ; by the above inequalities, we obtain
where and C only depend on a, k, m, n. Choosing T sufficiently small such that , we know that is a contraction. Applying the above inequality yields
Taking T sufficiently small so that , we deduce that maps to itself. It follows from the contraction-mapping principle that the mapping has a unique fixed point u in .
For , multiplying the first equation of the system (3.1) by 2u, integrating with respect to x, one derives
from which we have the conservation law
The global existence result follows from the integral from equation (3.2) and equation (3.3). □
Now we study the norms of solutions of equation (3.1) using energy estimates. First, recall the following two lemmas.
Lemma 3.3 (See )
If , then is an algebra, and
here c is a constant depending only on r.
Lemma 3.4 (See )
If , then
where denotes the commutator of the linear operators A and B, and c is a constant depending only on r.
Theorem 3.1 Suppose that, for some , the functions are a solution of equation (3.1) corresponding to the initial data . Then the following inequality holds:
For any real number , there exists a constant c depending only on q such that
For , there is a constant c independent of ϵ such that
Proof Using and (3.3) derives (3.4).
Since and the Parseval equality gives rise to
For any , applying to both sides of the first equation of (3.1), respectively, and integrating with regard to x again,using integration by parts, one obtains
We will estimate the terms on the right-hand side of (3.7) separately. For the first term, by using the Cauchy-Schwartz inequality and Lemmas 3.3 and 3.4, we have
Using the above estimate to the second term on the right-hand side of equation (3.7) yields
For the fourth term on the right-hand side of equation (3.7), using the Cauchy-Schwartz inequality and Lemma 3.3, we obtain
For the last term on the right-hand side of equation (3.7), using Lemma 3.3 repeatedly results in
It follows from equations (3.7)-(3.11) that there exists a constant c depending only on a, m, n, s such that
Integrating both sides of the above inequality with respect to t results in inequality (3.5).
To estimate the norm of , we apply the operator to both sides of the first equation of the system (3.1) to obtain the equation
Applying to both sides of equation (3.12) for gives rise to
For the right-hand of equation (3.13), we have
Using Lemma 3.3, and , we have
By the Cauchy-Schwartz inequality and Lemma 3.3, we get
Substituting equations (3.14)-(3.19) into equation (3.13) yields the inequality
with a constant . This completes the proof of Theorem 3.1. □
For a real number s with , suppose that the function is in , and let be the convolution of the function and be such that the Fourier transform of ϕ satisfies , and for any . Thus we have . It follows from Theorem 3.1 that for each ϵ satisfying , the Cauchy problem
has a unique solution , in which may depend on ϵ.
For an arbitrary positive Sobolev exponent , we give the following lemma.
Lemma 3.5 For with and , the following estimates hold for any ϵ with :
where c is a constant independent of ϵ.
Proof This proof is similar to that of Lemma 5 in  and Lemma 4.5 in , we omit it here. □
Remark 3.1 For , using , , equations (3.4), (3.22), and (3.23), we obtain
where is independent of ϵ.
Theorem 3.2 If with such that . Let be defined as in the system (3.21). Then there exist two constants, c and , which are independent of ϵ, such that of problem (3.21) satisfies for any .
Proof Using the notation and differentiating equation (3.21) or equation (3.12) with respect to x give rise to
Letting be an integer and multiplying the above equality by , then integrating the resulting equation with respect to x, and using
we find the equality
Applying Hölder’s inequality, we get
where . Furthermore
Since as for any , integrating the above inequality with respect to t and taking the limit as result in the estimate
Using the algebraic property of with and the inequality (3.26) leads to
where c is a constant independent of ϵ. Using (3.6), (3.29), and the above inequality, we get
where c is independent of ϵ. Furthermore, for any fixed , there exists a constant such that . By (3.6) and (3.26), one has
Making use of the Gronwall inequality with equation (3.5), with , , and equation (3.26), yields
From equations (3.22)-(3.23) and (3.31)-(3.32), we have
For , applying equations (3.28), (3.30), and (3.33), we obtain
It follows from the contraction-mapping principle that there is a such that the equation
has a unique solution . From the above inequality, we know that the variable T only depends on c and . Using the theorem present on p.51 in  or Theorem II in Section 1.1 in  one derives that there are constants and independent of ϵ such that for arbitrary , which leads to the conclusion of Theorem 3.2. □
Using equations (3.5)-(3.6) in Theorem 3.1 and Theorem 3.2, with the notation and with Gronwall’s inequality, results in the inequalities
where , and . It follows from Aubin’s compactness theorem that there is a subsequence of , denoted by , such that and their temporal derivatives are weakly convergent to a function and its derivative in and , respectively. Moreover, for any real number , is convergent to the function u strongly in the space for and converges to strongly in the space for . Thus, we can prove the existence of a weak solution to equation (1.1).
Proof of Theorem 1.4 From Theorem 3.2, we know that is bounded in the space . Thus, the sequences , are weakly convergent to u, in the space for any , separately. Hence, u satisfies the equation
with and . Since is a separable Banach space and is a bounded sequence in the dual space of X, there exists a subsequence of , still denoted by , weakly star convergent to a function v in . As weakly converges to in , as a result almost everywhere. Thus, we obtain . □