Abstract
A nonlinear partial differential equation, which includes the Novikov equation as
a special case, is investigated. The wellposedness of local strong solutions for
the equation in the Sobolev space
MSC: 35Q35, 35Q51.
Keywords:
local strong solution; local weak solution; generalized Novikov equation1 Introduction
Novikov [1] derived the integrable equation with cubic nonlinearities
which has been investigated by many scholars. Grayshan [2] studied both the periodic and the nonperiodic Cauchy problem for Eq. (1) and discussed
continuity results for the datatosolution map in the Sobolev spaces. A Galerkintype
approximation method was used in Himonas and Holliman’s paper [3] to establish the wellposedness of Eq. (1) in the Sobolev space
We note that the coefficients of the terms
which takes a key role in obtaining various dynamic properties in the previous works.
Motivated by the desire to extend parts of local wellposedness results in [3,11,12], we study the following model:
where m, a and
The objective of this paper is to investigate Eq. (2). Since m, a and
The rest of this paper is organized as follows. The main results are given in Section 2. The proof of a local wellposedness result is established in Section 3, while the existence of local weak solutions is proved in Section 4.
2 Main results
Firstly, we state some notations.
The space of all infinitely differentiable functions
where
For
Defining
and setting
We consider the Cauchy problem for Eq. (2)
which is equivalent to
Now, we give our main results for problem (3).
Theorem 1Let
It follows from Theorem 1 that for each ε satisfying
has a unique solution
Theorem 2If
Theorem 3Suppose that
3 Proof of Theorem 1
Consider the abstract quasilinear evolution equation
Let X and Y be Hilbert spaces such that Y is continuously and densely embedded in X, and let
(I)
and
(II)
(III)
Kato theorem (see [25])
Assume that (I), (II) and (III) hold. If
Moreover, the map
We set
Lemma 3.1The operator
Lemma 3.2Let
Lemma 3.3For
The above three lemmas can be found in Ni and Zhou [12].
Lemma 3.4Letrandqbe real numbers such that
This lemma can be found in [25,26].
Lemma 3.5Let
Proof Using the algebra property of the space
which completes the proof of (9). Using
and
Using (12) and (13) yields
which completes the proof of inequality (10). □
Proof of Theorem 1 Using the Kato theorem, Lemmas 3.1, 3.2, 3.3 and Lemma 3.5, we know that system (3) or problem (4) has a unique solution
□
4 Proofs of Theorems 2 and 3
Using the first equation of system (3) derives
from which we have the conservation law
Lemma 4.1 (Kato and Ponce [26])
If
wherecis a constant depending only onr.
Lemma 4.2 (Kato and Ponce [26])
Let
Lemma 4.3Let
For
For
Proof Using
Using
For
We will estimate the terms on the righthand side of (19) separately. For the first term, by using the CauchySchwarz inequality and Lemmas 4.1 and 4.2, we have
Using the above estimate to the second term yields
Using the CauchySchwarz inequality and Lemma 4.1, we obtain
For the last term in (19), using
For
For
It follows from (20)(25) that there exists a constant c such that
Integrating both sides of the above inequality with respect to t results in inequality (17).
To estimate the norm of
Applying
For the righthand of Eq. (28), we have
Since
using Lemma 4.1,
and
Using the CauchySchwarz inequality and Lemma 4.1 yields
Applying (29)(33) into (28) yields the inequality
for a constant
Lemma 4.4 ([17])
For
wherecis a constant independent ofε.
Proof of Theorem 2 Using notation
Letting
Applying the Hölder’s inequality yields
or
where
Since
where c only depends on m, a, b.
Using the algebraic property of
and
where c is a constant independent of ε. From (45), we have
It follows from (43) and (46) that
It follows from the contraction mapping principle that there is a
has a unique solution
Using Theorem 2, (17), (18) and (44), notation
and
where
Proof of Theorem 3 From Theorem 2, we know that
with
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The article is a joint work of two authors who contributed equally to the final version of the paper. All authors read and approved the final manuscript.
Acknowledgements
Thanks are given to referees whose comments and suggestions are very helpful to revise our paper. This work is supported by both the Fundamental Research Funds for the Central Universities (JBK120504) and the Applied and Basic Project of Sichuan Province (2012JY0020).
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