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Existence and stability of solitary waves for the generalized Korteweg-de Vries equations
Boundary Value Problems volume 2013, Article number: 121 (2013)
Abstract
In this paper, we consider the fractional Korteweg-de Vries equations with general nonlinearities. By studying constrained minimization problems and applying the method of concentration-compactness, we obtain the existence of solitary waves for the generalized Korteweg-de Vries equations under some assumptions of the nonlinear term. Moreover, we prove that the set of minimizers is a stable set for the initial value problem of the equations, in the sense that a solution which starts near the set will remain near it for all time.
1 Introduction
This paper is devoted to studying the existence and stability of solitary wave solutions of the generalized Korteweg-de Vries equation
where satisfies the following assumption:
-
(A)
, and for some ,
, the Fourier transform .
When and , equation (1.1) is the well-known Korteweg-de Vries equation, introduced by Korteweg and de Vries in 1895 (cf. [1]). The existence and stability of solitary waves of the Korteweg-de Vries equation is considered by Benjamin in [2]. Recently, in [3], Pelinovsky obtained a Korteweg-de Vries equation with a forcing term, which is a simple analytical model of tsunami generation by submarine landslides.
Here, we consider the generalized Korteweg-de Vries equation (1.1). Let . Since the functionals
and
are two conserved quantities with (1.1), for studying the existence of solitary wave solutions to (1.1), by the variational methods, the solitary wave solutions to equation (1.1) will be founded as minimizers of
where . Denote the set of minimizers of the problem () by
Inspired by the methods used in [4, 5], by studying the problem (), we obtain the existence of solitary waves for equation (1.1) with some special nonlinearities , where , and general nonlinearities satisfying the assumption (A). Moreover, we prove that the set of minimizers is a stable set for the initial value problem of equation (1.1) in the sense that a solution which starts near the set will remain near it for all time. In order to obtain those results, we have to overcome one main difficulty: the minimization problem () is given in the unbounded domain ℝ which results in the loss of compactness. As is done in [4, 6], we overcome the difficulty of loss of compactness by the method of concentration-compactness introduced by Lions in [7, 8] for solving some minimization problems in unbounded domains.
Now we give our main results.
Theorem 1.1 Suppose that and satisfies condition (A) and for some . Then there exists such that is not empty. Moreover, if is a minimizing sequence for the problem (), then there exist a sequence and such that contains a subsequence converging strongly in to g, and
where is the norm of .
Theorem 1.2 Under the assumptions of Theorem 1.1, the set is -stable with respect to equation (1.1), i.e., for any , there exists such that if
then the solution to equation (1.1) with initial data satisfies
for any .
Theorem 1.3 Suppose that and satisfies condition (A) and for some . Then there exists such that is not empty. Moreover, if is a minimizing sequence for the problem (), then there exists a sequence and such that contains a subsequence converging strongly in to g, and
where is the norm of given in Section 5.
Theorem 1.4 Under the assumptions of Theorem 1.1, the set is -stable with respect to equation (1.1), i.e., for any , there exists such that if
then the solution to equation (1.1) with initial data satisfies
for any .
The paper is organized as follows. In Section 2, we give some preliminaries. In Section 3, we study the existence and stability of solitary waves of equation (1.1) with some special nonlinearities . Section 4 is devoted to studying equation (1.1) with general nonlinearities satisfying the assumption (A). We shall consider the existence and stability of solitary waves of equation (1.1) with in Section 5.
2 Some preliminaries
At first, we give some notations. The set of all integers and the set of all real numbers are written as ℤ and ℝ, respectively. And all the integrals will be taken over ℝ unless specified. denotes the usual Lebesgue space with the norm given by
The Sobolev space is defined by
whose norm is given by
Now, we give Lemma 2.1 and Lemma 2.2 which will be used to study the behavior of the minimizing sequence for the problem (). Lemma 2.2 is due to Lions [7, 8].
Lemma 2.1 Suppose that and are given. Then there exists such that if with and , then
Proof We have
Therefore, there exists some such that
Applying the Sobolev embedding theorem [9], there exists a constant A such that
Thus, we obtain
Taking , it follows that
□
Lemma 2.2 Let be a bounded sequence in such that
for some . Then in for .
Proof Let . Without loss of generality, we may assume . It follows from the interpolation inequalities that
where is a constant independent of u.
Covering ℝ by a family of intervals such that each point of ℝ is contained in at most two such intervals and summing this inequality over this family of intervals, we get
Since is bounded in and as , applying the above inequality, we know that in for . □
Next, we establish a convergence result that will be used in the proof of Theorem 1.3.
Lemma 2.3 Let and suppose that
where . If in and a.e. on ℝ, then
where .
Proof Let . Applying the mean value theorem, we have
where is dependent on x and R. Now we write
It follows from (2.1), the mean value theorem and the Hölder inequality that
and
Since the embedding () is compact, the inequalities (2.3) and (2.4) imply that
Similarly, by the Hölder inequality, the Sobolev embedding theorem and (2.1), we get
Since is bounded in , we see that
Hence, combining (2.2), (2.5), (2.6) and (2.7), we obtain
□
3 The case of special nonlinearity
In this section, we only consider the case of and , where . Correspondingly,
At first, we commence by studying some properties of the functional and the behavior of the minimizing sequences for the problem ().
Lemma 3.1 For any ,
-
(i)
;
-
(ii)
If is a minimizing sequence for the problem (), there exists a constant such that for all n;
-
(iii)
If is a minimizing sequence for the problem (), there exist a positive constant δ and a sequence of real numbers such that
for sufficiently large n.
Proof (i) Choose any function such that and . For any , define . Then we have
and
For , by taking sufficiently small, we get .
Next we prove . Let such that . By the Sobolev embedding theorems and interpolation inequalities, we get
where A denotes various constants which are independent of u. Using the Young inequality, we derive from (3.1)
where is arbitrary and depends on ε and q, but not on u. Therefore,
Choosing , we obtain the lower bound of the functional E
which implies .
-
(ii)
Let be a minimizing sequence for the problem (). Then, by (3.1), we have
where A denotes various constants which are independent of n. Since , the existence of the desired bound B follows.
-
(iii)
Let be a minimizing sequence for the problem (). Then we claim: there exists a constant such that for all sufficiently large n. We argue by contradiction: if no such exists, then . Hence
which contradicts (i). So, the claim is achieved.
Combining (ii) and Lemma 2.1, there exist a positive constant δ and a sequence of real numbers such that
for sufficiently large n. The proof of Lemma 3.1 is completed. □
The next lemma will establish a subadditivity inequality which will be a crucial step in the proof of the existence minimizer for the problem ().
Lemma 3.2 For all , .
Proof For given , , , let , where . Then it follows that
and
Hence we get
Now, from (3.2) and Lemma 3.1, we obtain for all ,
□
Now we formulate the following two theorems, which are special cases corresponding to Theorem 1.1 and Theorem 1.2, and give their proof with the aim of Lemma 3.1 and Lemma 3.2.
Theorem 3.1 Let and , where . For any , the set is not empty. Moreover, if is a minimizing sequence for the problem (), then there exist a sequence and such that contains a subsequence converging strongly in to g, and
Theorem 3.2 Let and , where . For any , the set is -stable with respect to equation (1.1), i.e., for any , there exists such that if
then the solution to equation (1.1) with initial data satisfies
for any .
Proof of Theorem 3.1 From (3.2), it is easy to check that is continuous on . Let be a minimizing sequence for the problem (). By Lemma 3.1, there exist a positive constant δ and a sequence of real numbers such that
for sufficiently large n.
Let us define . Hence , , as , and
Since is bounded in , by Lemma 3.1, we may assume going, if necessary, to a subsequence
Hence, by (3.3), we get . And applying the Brezis-Lieb lemma [10], we have
Now we show that . In the contrary case, . By (3.5), we obtain . Then it follows from (3.5) and (3.6) that
Since is continuous on , letting , we get , which contradicts Lemma 3.2. Therefore . It then follows from (3.5) that
Applying the interpolation inequality, (3.4) and (3.8), we get
Using the weak low semi-continuity of the norm in , we know that
Letting , by (3.8) and (3.9), we obtain . On the other hand, it follows from that . Therefore , which implies that g is a minimizer of the problem () (i.e., ). Then it follows from (3.7), (3.8) and (3.9) that
We prove with an argument by contradiction. Assume that there exist and a subsequence of such that
for all . With the result of the above proof, we obtain that there exist a subsequence of , denoted again by , and such that
Since ,
which contradicts (3.10). □
An immediate consequence of Theorem 3.1 is that forms a stable set for the initial-value problem for equation (1.1).
Proof of Theorem 3.2 We prove Theorem 3.2 with an argument by contradiction. Assume that the set is not -stable. Then there exist , and a sequence of times such that
and
for all n, where solves equation (1.1) with .
Equation (3.11) implies that
Choose such that for all n. Thus as . Hence the sequence satisfies and
Therefore is a minimizing sequence for the problem (). By Theorem 1.1, there exists such that
for sufficiently large . Since and is bounded, we derive from (3.12) and (3.13)
for sufficiently large . (3.14) is a contradiction. Therefore, the set is -stable with respect to equation (1.1). □
4 The case for more general nonlinearities
In this section, we consider (1.1) with and more general nonlinearities f satisfying condition (A). At first, we study the properties of the functional and the minimizing sequence of the problem ().
Lemma 4.1
-
(i)
For any , is finite and continuous on . Moreover, each minimizing sequence for () is bounded;
-
(ii)
for any .
Proof (i) According to assumption (A), we observe that for each there exists such that
where . By the Sobolev embedding theorems and interpolation inequalities, we obtain
and
where is independent of u. Then using the Young inequality, we can derive from (4.2) and (4.3) that for all , there exists such that
and
Let such that . It follows from (4.1), (4.4) and (4.5) that
where is a positive constant dependent only on ε and η for given . Choosing and such that , we see that .
Since for , it is easy to check that is continuous on .
Let be a minimizing sequence for the problem (). From (4.6) and the fact that is finite, we know that is bounded in .
-
(ii)
For given such that , let for . We obtain that
(4.7)
and
Combining (4.1), (4.7) and (4.8), we obtain
□
Lemma 4.2 Suppose that for some . Then the following two properties hold:
-
(i)
is non-increasing on and ;
-
(ii)
there exists such that
Proof First we observe that if and with and , then and
Consequently, for and , we have
If , then for each , there exists with such that
This inequality yields for .
Since for all , we see that
We claim that . Letting , , from (4.9), there exists with such that
It follows from (4.6) and (4.10) that
where and are constants independent of . Hence we obtain , which implies
where is dependent only on . Combining (4.1), (4.4), (4.5) and (4.11), we also get
where is dependent only on .
We claim that
Indeed, if there exist and such that , then by (4.11) and (4.12), we obtain
which contradicts . Therefore (4.13) is achieved and this implies that and, consequently,
This shows that .
-
(2)
We observe that , which implies (ii). □
Then we establish a subadditivity inequality similar to Lemma 3.2 with the aim of Lemma 4.2.
Lemma 4.3 Suppose that for some . Then there exists such that for .
Proof According to Lemma 4.2, the set
is nonempty. We define
It follows from the continuity of and that ,
Therefore,
for all . □
Now we give the proof of Theorem 1.1 with the aim of Lemma 4.1, Lemma 4.2 and Lemma 4.3. Since the proof of Theorem 1.1 is similar to that of Theorem 3.1, we only give the sketch of the proof.
Proof of Theorem 1.1 Let be a minimizing sequence of , where is defined in (4.14). Since is bounded, we may assume
First, we consider the case . In this case, by Lemma 2.2, either
-
(a)
in for , or
-
(b)
there exists a sequence such that
In the case (a), combining Lemma 2.2 and condition (A), we obtain
and, consequently,
which contradicts Lemma 4.1. Hence (b) holds. Then it follows from Lemma 2.3 and Lemma 4.3 that g is the minimizer for the problem () (i.e., ) and the result of Theorem 1.1 holds. The proof is similar to that of Theorem 3.1, we omit the details.
If , we repeat the previous argument in the case (b) to obtain the result of Theorem 1.1. □
Proof of Theorem 1.2 Theorem 1.2 is an immediate result of Theorem 1.1. We can prove it with an argument similar to that of Theorem 3.2. Here, we omit the details of the proof. □
5 The case for
In this section, we only consider the case of , i.e., we consider the existence and stability of solitary waves for the fractional Korteweg-de Vries equations with general nonlinearities. At first, we give the definition of . The fractional order Sobolev space is defined by
whose norm is given by
Since the functionals
and
are two conserved quantities with (1.1), for studying the existence of solitary wave solutions to (1.1), by the variational methods, the solitary wave solutions to the equation (1.1) will be founded as minimizers of
where . Denote the set of minimizers of the problem () by
Similar to Lemma 4.1 and Lemma 4.3, we obtain the following two lemmas.
Lemma 5.1
-
(i)
and are finite and continuous on ; moreover, for any , each minimizing sequence for the problem () or () is bounded in ;
-
(ii)
for any .
Lemma 5.2 If for some , then there exists , , such that
Applying the above two lemmas and commutator estimates [[5], Lemma 2.5], we prove Theorem 1.3 and Theorem 1.4 by similar steps to those given in Section 4. Here we omit the details of Theorem 1.3 and Theorem 1.4.
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Acknowledgements
The work was supported in part by Special Fund of Fundamental Scientific Research Business Expense for Higher School of Central Government (projects for young teachers, No. ZY20110226) and the National Natural Science Foundation of China (No. 41276020).
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Hong, M. Existence and stability of solitary waves for the generalized Korteweg-de Vries equations. Bound Value Probl 2013, 121 (2013). https://doi.org/10.1186/1687-2770-2013-121
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DOI: https://doi.org/10.1186/1687-2770-2013-121