## Abstract

Let with *p* being the ratio of an even to an odd integer. For the generalized Kadomtsev–Petviashvili equation, coupled with Benjamin–Ono equations, in the formit is proved that the solutions blow up in finite time even for those initial data with positive energy. As a by-product, it is proved that for all , the solitary waves are strongly unstable if . This result, even in a special case , improves a previous work by Liu (Liu 2001 *Trans. AMS* **353**, 191–208) where the instability of solitary waves was proved only in the case of .

## 1. Introduction

This paper is concerned with the following generalized Kadomtsev–Petviashvili (KP) equation and two-dimensional Benjamin–Ono (BO) equations(1.1)(1.2)where , *p*>0 and is the Hilbert operator defined byHere, p.v. denotes the Cauchy principal value.

System (1.1) has been raised in many branches of physics. For example, when and *p*=1, (1.1) is the well known KP-I equation (Kadomtsev & Petviashvili 1970; Ablowitz & Clarkson 1991). When and *p*>0, (1.1) is called the generalized KP (GKP) equation. Mathematical interests are concentrated on the existence of solitary waves (i.e. solutions of (1.1) with the form ), existence of solutions of Cauchy problem (1.1) and (1.2) as well as the possible blow-up solutions of (1.1) and (1.2). But one of the most important and challenging issues is to investigate the existence of solutions that blow up in a finite time.

Recently, Guo & Han (1996) proved the existence of blow-up solutions of (1.1) and (1.2) if and the initial energyThis is one motivation of the present paper. Another motivation is the recent results for (1.1) and (1.2) in the case of . When , Liu & Wang (1997) and Liu (2001) proved the existence of solitary waves of (1.1) for and the existence of blow-up solutions of (1.1) and (1.2) for , as well as the stability of solitary waves of (1.1) for and the instability of solitary waves of (1.1) for . The main ingredients used in Liu & Wang (1997) and Liu (2001) are based on solving several complicated minimization problems and constructing several invariant sets (e.g. Liu & Wang 1997, theorem 1 and 3; Liu 2001, lemma 3.1). In particular, they use essentially the property of invariance of the problem under scaling and . But the case of is quite different from those of since the scaling invariance does not hold any more due to the term . The method used in Liu (2001) seems not to be applied in the case of .

There are two main purposes of the present paper. One is to prove the existence of solitary waves of (1.1) in the case of and , which covers and generalizes the previous works in de Bouard & Saut (1997) and Liu & Wang (1997). In particular, we do not require the wave speed *c* satisfying , which is usually assumed in finding solitary waves of the BO equations. The other is to prove that (1.1) and (1.2) has a solution that has positive initial energy and blows up in a finite time. As a by-product, it is proved that the solitary wave is strongly unstable for , and . This result improves a previous work by Liu (2001) where the strong instability of a solitary wave is proved only in the case of and .

This paper is organized as follows. In §2, we use the Nehari method to prove that for any *c*>0, (1.1) has a solitary wave solution . In §3, we use several minimization problems with multiple constraints to construct sets and that are invariant under the flow generated by the Cauchy problem (1.1) and (1.2). In §4, we prove that the solutions of (1.1) and (1.2) with initial data in must blow up in a finite time. In §5, we prove that for any , the solitary wave is strongly unstable in the sense of definition 5.1.

We end this introduction by notations where and will denote the norm in and Sobolev space . Throughout this paper, denotes the Fourier transform of *f*, defined as . All integrals are taken over unless stated otherwise and d*x*d*y* will be omitted if no confusion occurs. Let *Y* be the closure of under the normfor and , where , . is the dual space of *Y*. We also have by a choice of . Letwith the norm , where ‘∨’ is the Fourier inverse transform. We define the operator by is the Riesz potential of order *s* defined by . is the Schwartz class in .

Throughout this paper, we only consider the case of and , where *n*_{1} is any even integer and *n*_{2} any odd integer so that . Different positive constants might be denoted by the same letter *C* or *C _{j}*. If necessary, by we denote the constant depending only on the quantities appearing in parentheses.

## 2. Solitary wave solutions

In this section, we will use the Nehari method to prove that (1.1) possesses a solitary wave solution. Before doing this, we recall the following well-known properties of the Hilbert operator (Guo & Tan 1992):(I)(II)(III)(IV)(V)(VI)For any *c*>0, a solitary wave of (1.1) with wave speed *c* is a solution of (1.1) of the form and decaying to zero at infinity. More precisely, *φ* is a solution of(2.1)We say that is a solution of (2.1) if, and only if, , where with andLetThen whenever *φ* is a solution of (2.1). Here, and after, denotes the dual product between *Y*′ and *Y*.

We say that *φ* is a groundstate solution of (2.1) if and for any *ψ* satisfying there holds .

Next we will prove that the following minimum(2.2)is achieved by some *φ*, which is a groundstate solution of (2.1).

*Suppose that both* (*u _{n}*)

*and*(

*u*)

_{nx}*are bounded in*.

*Assuming*

*u*→

_{n}*u a.e. in*,

*then*(2.3)

*Here, and after*,

*goes to zero as n goes to infinity*.

We firstly claim that if in , then in . Indeed, for any ,proving the claim.

Since in and in , we have, from the properties of , thatThe proof is complete. ▪

*If c*>0 *and* , *then for any* , *there exists a unique* *such that* . *Moreover if* , *then* .

For any , define Direct computations show thatClearly, from the expression of *θ _{u}* we know that if , then . ▪

*Suppose that c*>0 *and* . *Then*

*and**is manifold. Moreover, there exists α*>0*such that for any*,*and**if u achieves the minimum d, then u is a critical point of L*._{c}in Y, i.e.

follows from lemma 2.3. For any ,this implies that is manifold. For any , denote , then using the anisotropic Sobolev embedding theorem (Besov *et al*. 1978, p. 323) we havewhich implies that . Thus (i) is proved.

Since *u* achieves the minimum *d*, we know that there exists such that . Noticing that andwe have that . It follows that . Thus (ii) is proved. ▪

Next we prove that *d* is achieved.

*Suppose that c*>0 *and* . *Then the minimum d is achieved by some* , *which is a groundstate solution of* *(2.1)*.

If is a solution of (2.1), then and hence . The definition of *d* implies that for any *φ* satisfying and .

From remark 2.6 and (ii) of lemma 2.4, it suffices to prove that *d* is achieved by some . Let be such that and . We know from (i) of lemma 2.4 thatSince the functionals *L _{c}* and

*N*are invariant under translation, i.e. for any , and , we obtain from concentration compactness lemma of Lions (1984

*a*,,

*b*) (see also Ambrosetti & Wang (2003) and Willem (1996)) that there exist such that satisfies and . Moreover, a.e. in and .

If , then by lemma 2.3 there exists such that . Using and the Fatou lemma, we have that(2.4)Since , (2.4) implies that , which is a contradiction because .

If , then using lemma 2.2 and the Brezis–Lieb lemma (Brezis & Lieb 1983) we have thatwith and hence . From lemma 2.3 we know that there are such that . Moreover, we claim that . Indeed, if otherwise there exists a subsequence still denoted by such that , then . That is a contradiction, since(2.5)one has again for large *n*.

Thus . Using lemma 2.2 and the Brezis–Lieb lemma again, we easily get that , i.e. in *Y*. The proof is complete. ▪

When , de Bouard & Saut (1997) proved that any solitary wave solutions of (1.1) belong to for (see de Bouard & Saut 1997, p. 227, theorem 4.1.).

Observing the proofs of de Bouard & Saut (1997, p. 227, theorem 4.1), we know that if we replace the by in the multipliers

*Φ*_{1},*Φ*_{2}and*Φ*_{3}, then all the proofs contained in de Bouard & Saut (1997, p. 227, theorem 4.1) hold provided . Therefore if , then the solitary wave of (1.1) belongs to for .

We end this section by another characterization of the minimum *d*.

*Let* *and*(2.6)*Then* .

Clearly . To see , it suffices to prove that for any and any there holds(2.7)Since *X _{s}* is dense in

*Y*, we find a sequence such thatFrom , and in

*Y*, we know that and . The proof of lemma 2.3 implies that there exists a sequence (

*ρ*) such that and .

_{n}Next, denoting , we compare with . Since and strongly in *Y*, we have immediately thatWe now calculate the termIndeed, using the fact that in , in , and interpolation inequality (VI), we know that all other terms go to zero as but . Hence we get that for large *n*Thereforeand equation (2.7) holds. The proof is complete. ▪

## 3. Invariant sets

In this section, we will use the minimization problem defined in (2.1) and the property of the groundstate solution *φ* to construct several sets that are invariant under the flow generated by the Cauchy problem (1.1) and (1.2). We emphasize that the method used in Liu (2001) seems not to be applied here due to the term . Hence we need to develop further the techniques in Chen & Guo (submitted) to get the invariant sets. First, we have the following local well-posed result.

*Suppose* *and* . *Then T*>0 *such that* *(1.1) and (1.2)* *have a unique solution**and if* , *one has**Moreover there holds*(3.1)(3.2)Next, we define another two functionals(3.3)(3.4)We denote thatWe set that

*If c*>0 *and* , *then* .

Keep the definition of , and in mind. For any , we want to find such that . Firstly, for any , and . We denote that , thenNoticing that as and as , we find a such that , i.e. . Therefore .

Secondly, we want to prove that for any . In fact, fromwe get that implies that(3.5)where and . Since for as , as and for , we know that for . Similar arguments arrive at for . Thus for any . In particular, . Taking , we conclude the proof. ▪

*If c*>0 *and* , *then* .

The idea is similar to those used in the proof of lemma 3.2, but we need a quite different scaling argument. Our purpose is to show that for any , there is such that . Firstly, for any , and . We denote that , thenNoticing that as and as , we find a such that , i.e. . Therefore .

Secondly, we prove that for any . In fact, fromand , we obtain that(3.6)where and . Using an argument similar to those in the proofs of for any , we have that () for any . Thus for all . In particular, and we complete the proof. ▪

Now we are in a position to define several invariant sets.

*For any c*>0 *and* , *and* *are invariant under the flow generated by the Cauchy problem* *(1.1) and (1.2)*.

We only prove that is invariant under the flow generated by the Cauchy problem (1.1) and (1.2) since the proofs of the others are similar. Let be the solution of (1.1) and (1.2) with initial data . We simply denote the function by . First, by lemma 3.1, we know that if , then .

Second, we show that for . If this is not true, from the continuity, such that . Then since , we know that . This contradicts with for all . Therefore for .

Finally, we show that for . If this is not true, from the continuity, such that . Since we have proved that , we can have that . So that , which contradicts with for all . Therefore for . ▪

## 4. Improved blow up

Recalling that, in a different setting, if is a solution of the nonlinear Schrödinger equationthen it is shown that either exists globally in the energy space or blows up in a finite time in the energy space, i.e. such that(e.g. Berestycki & Cazenave 1981; Weinstein 1983; Zhang 2001). For the problem considered here, we know from lemma 3.4, (3.1) and (3.2) that if , then the solution with satisfies . Consequently, exists globally in *Y* for and hence blow up cannot occur in a finite time in *Y*. But as we will see below, we do have a blow-up result that is only due to the transverse dispersion.

Guo & Han (1996) proved that the solution of (1.1) and (1.2) blows up in a finite time if the initial data satisfies and . However, using the invariant sets constructed in §3, we are able to extend this blow-up result to allow the energy to be positive and .

*If* , *and* , *then the solution* *of* *(1.1) and (1.2)* *with initial data* *blows up at finite time*.

For any , we have that . Now, for any fixed *t*, we simply denote by *u*. Thus , , and . For any , we denote that , then similar to the computations used in the proof of lemma 3.3, we have that as and as . Hence there is such that and for . For , has the following three possibilities:

for ,

, and

there is such that and .

For cases (i) and (ii), we all have that and . It follows that . Moreover we have that(4.1)For case (iii), we have and , i.e. . Thus and we can also obtain from a similar computation thatIn all cases, we have thatNote that from Guo & Han (1996, theorem 2.4), as long as . Moreover by Guo & Han (1996, theorem 3.1), for , there holdsOn the other hand, since . Thus we find that there exists such that . The conserved momentum and the classical inequality imply that there exists a blow-up time such that ▪

If and , then

## 5. Instability of solitary waves

In this section, we will prove that the solitary wave of (1.1) is strongly unstable. We assume throughout this section that and if is a solitary wave of (1.1), then .

Suppose for *c*>0, is a groundstate solution of (2.1) (of course a solitary wave of (1.1) with wave speed *c* in the direction). We say that is strongly unstable if for any there is (), with such that the solution *u* of (1.1) with initial data blows up in a finite time. More precisely, there is such that(5.1)

*Suppose that* *is a solitary wave solution of* *(1.1)* *corresponding to the groundstate solution of* *(2.1)* *and* . *If* , *then* *is strongly unstable in the sense of* *definition 5.1*.

Before proving theorem 5.2, we need the following lemmas.

*Suppose that* *is the groundstate solution of* *(2.1)* *obtained in* *§2*. *Then* *and* .

Since is a groundstate solution of (2.1), with and with . Direct computations arrive at and . The proof is complete. ▪

*There exists* *and* *such that for* ,

Let be the groundstate solution of (2.1). We denote that . Then(5.2)Lemma 5.3 implies that(5.3)(5.4)From (5.2) and (5.4), we have that(5.5)By (5.3) and (5.4), we get that(5.6)Let . We want to find and such that(5.7)Since(5.8)(5.9)(5.10)(5.11)we have from (5.3), (5.5), (5.6) and delicate computations that condition (5.7) is equivalent to conditions(5.12)(5.13)(5.14)(5.15)Conditions (5.12) and (5.15) are equivalent to(5.16)Taking (where but is still small enough) and withwe know that (5.7) holds for . The proof is complete. ▪

*For any fixed* *with* , *there exist positive constants* () *independent of w, such that*(5.17)(5.18)(5.19)(5.20)

We prove (5.19) in detail and indicate the differences in the proofs of the others. First using the elementary inequalityand the anisotropic Sobolev imbedding theorem (Besov *et al*. 1978), we have that(5.21)Second using the properties (I)–(V) of the Hilbert operator , we have that(5.22)Finally, we have thatproving (5.19). The proofs of (5.17) and (5.18) are almost the same and omitted here. For (5.20), firstly using the properties (I)–(V) of the Hilbert operator , we have thatSecondly, using the anisotropic Sobolev imbedding theorem andwe have thatThusWe complete the proof of lemma 5.5. ▪

Now we are in a position to prove theorem 5.2.

For any , we choose sufficiently small ,such thatwhere is defined as in lemma 5.4. Since is dense in *Y*, we find such thatwhere () are chosen as in lemma 5.5 and is defined asThereforeand we obtain from lemmas 5.4 and 5.5 thatSince is arbitrary and , we have that . Theorem 4.1 implies that the solution of (1.1) with initial data blows up in a finite time. Hence for any , the solitary wave is strongly unstable in the sense of definition 5.1. ▪

## Acknowledgments

The authors would like to thank the unknown referees for valuable comments that improved the manuscript. Supported in part by the Youth Foundation of NSFC (no. 10501006) and by the program for NCETFJ.

## Footnotes

- Received May 1, 2007.
- Accepted September 19, 2007.

- © 2007 The Royal Society