## Abstract

We study the existence and scattering of global small amplitude solutions to modified improved Boussinesq equations in one dimension with nonlinear term behaving as a power as . Solutions are considered in space for all . According to the value of *s*, the power nonlinearity exponent *p* is determined. Liu (Liu 1996 *Indiana Univ. Math. J*. **45**, 797–816) obtained the minimum value of *p* greater than 8 at for sufficiently small Cauchy data. In this paper, we prove that *p* can be reduced to be greater than at and the corresponding solution *u* has the time decay, such as as . We also prove non-existence of non-trivial asymptotically free solutions for under vanishing condition near zero frequency on asymptotic states.

## 1. Introduction

We consider the following initial-value problem for the one-dimensional generalized IMBq equation (modified improved Boussinesq equation):(1.1)where in the real sense and , for and *p*>1. By Duhamel's principle, the solution *u* can be written as(1.2)Here andwhere is the Fourier transform of . Since and , hereafter we will use *ξ* instead of for *S*, and *T*.

The generalized IMBq equation governs the various physical models like nonlinear wave in weakly dispersive medium (in this case ; Makhankov 1978; Kano & Nishida 1986) and longitudinal variation wave in elastic rod ( or ; Clarkson *et al*. 1986), etc. For the local or global existence of solution of IMBq equations, see Chen *et al*. (1996), Liu (1996) and Wang & Chen (1998, 1999, 2002*a*,*b*), and for the small amplitude solution and scattering, see Liu (1996) and Wang & Chen (2002*b*).

In this paper, the small amplitude solution and scattering to the nonlinear problem (1.1) are considered in one-dimensional case. Our main concerns are to provide the lower bound of nonlinearity *p* for the global existence of solution and scattering according to the regularity of initial data, and also the upper bound of *p* for the non-existence of non-trivial asymptotically free solutions. The methods below can be applied to the high-dimensional case. For this, see §3*c*.

To state our main results, let us define a function space byand and by and , respectively, where . We use the usual Sobolev spaces and with the normsThe first result is on the following global existence for small data.

*Let* *be numbers, such that* ,*for* . *Suppose that the data* *satisfy the regularity condition**and the smallness condition*(1.3)*Then if δ is sufficiently small, then there exists a unique global solution u in* *of* *(1.2)* *and small positive number ρ depending only on* *, such that*

*Let u be the solution of* *(1.2)* *as in* *theorem 1.1*. *Then there exist functions* *and* *in* *, such that**where* *is the unique solution of linear homogeneous equation,*(1.4)

The minimum values of *p* can be chosen to be greater than at *r*=10 and then *α*=2 and . We also have that . If we choose *r*=6, then we can take the values as *p*>5, *α*=1 and , and also have that . Thus, theorems 1.1 and 1.2 contain the physical situation *p*=5 and also give slight improvements of the previous result (Liu 1996), in which the global existence and scattering was established for *p*>8 at .

For the purpose of improvement, we use the stationary phase method and Young's inequality for the dyadically localized kernel estimate of high frequency part of , *S* and *T* instead of integration estimate used in Liu (1996) and Wang & Chen (2002*b*). For the kernel estimate, the condition is used. We also use van der Corput type estimate for medium frequency part of the operators similar to the one in Liu (1996) and Wang & Chen (2002*b*). To obtain an estimate for low frequency part, the condition is necessary. For details, see §2*b*.

In view of theorem 1.1, if *r*=10 and and , then for , it can be easily shown that by the decay estimate and the scattering . On the other hand, the following theorem shows that there is no non-trivial asymptotically free solution *u* with , if *p* is small and .

*Let* *and suppose that* *for some positive constant c*. *Let u be a smooth solution to* *(1.1)**, with* *and* *be a pair of smooth functions with compact Fourier supports*. *Suppose that*(1.5)*for some ϵ*>0*, where* *is the free solution to the linear problem* *(1.4)*. *Then* .

Theorem 1.3 shows the invalidity of theorem 1.2 for the value of *p* less than or equal to 2 in the case that an initial datum vanishes near the zero frequency. But it remains open whether the theorem is true for *p*>2 in one-dimensional case or not.

For the proof, we use an analogous argument to the one of Glassey (1973, 1977) that is uniformly bounded but under the conditions stated in theorem 1.3 and hence a contradiction occurs. For related topics, see Matsumura (1976), Barab (1984) and Strauss (1989).

If not specified, throughout this paper, the notation and denote and , respectively. Positive constants *C* vary line by line and depend only on *r* and *f*. means that both and hold.

## 2. Preliminaries

### (a) Linear estimates

First, we introduce an estimate of oscillatory integral.

*For* *and* *, we have**where* *and* .

A direct application of van der Corput lemma (Stein 1993) yields readily the proof. For the case *m*=0, see lemma 4.3 in Liu (1996) or lemma 2.2 in Wang & Chen (2002*b*). ▪

Let us choose a Littlewood–Paley function *η* and define a frequency projection operator for a dyadic number *N* byAnd we also denote , and byWe choose *η*, so that .

*Let* *and* . *Then for any* *with* *, we have*

Taking to and using change of variable, we havewhereand . Since for sufficiently large *N* and , by the method of stationary and non-stationary phase (Stein 1993), we haveThus, using , we deduce that for large and any (2.1)Here is the Besov space with norm for byFor the last inequality, we used the well-known embedding , for and the fact (see the book of Bergh & Löfström (1976) for instance).

As for the medium frequency of , using lemma 2.1, we can easily show that for *t*>1(2.2)

By Hausdorff–Young's inequality, if , then we have(2.3)And if *r*>6, then(2.4)Now let us choose *ϵ* by . Then since for and for *r*>6, from (2.2)–(2.4), we have for (2.5)If , then since for *r*>2, by direct calculation, we haveCombining this estimate, (2.1) and (2.5), we obtain for ▪

*Let* *and* . *Then for any* *with* *, if* *and* *, if r*>6*, we have*

The proof for the high frequency part of is almost the same as the one for . Thus, we consider only the low and medium frequency parts. With *α* as above, we have

On the other hand, for the medium frequency we have from lemma 2.1 thatif *t*>1. Now if we choose for and for *r*>6, then since for and , we havefor large *t*. If *t* is small, then similarly to the estimate for , we havewhere for and *β*=*r* for *r*>6. We have just finished the proof of the lemma. ▪

As a corollary of lemmas 2.2 and 2.3, we have the following lemma.

*Let* *and* . *Then for any* , *we have*

The only difference between and consists in the lower frequency part. For this, we have

Thus, from the low and medium frequency estimate in the proof of lemma 2.2, we deducefor large *t*. This completes the proof. ▪

### (b) Remarks on the linear estimates

In lemmas 2.2 and 2.3, we used the condition and for some time decay of the supreme norm and uniform bound on time of Sobolev norm of , respectively. In Liu (1996), the condition was used. Actually, the condition is necessary for the energy conservation and momentum conservation. This type of condition implies at least that should be zero at *ξ*=0. This vanishing condition at zero frequency turns out to be inevitable for the uniform bound on *t* because of the following fact: if if and 2 if , then for large *t*

Moreover, the vanishing condition is inevitable for the time decay. To see this, let *ψ* be a smooth function, such that if and if . Then the limit exists for all *x* and the following holds:(2.6)For the proof let us choose a positive number *θ* smaller than . Then by lemma 2.1 with *m*=1 and , we have(2.7)as . Thus, for the proof of the estimate (2.6), it suffices to show that(2.8)uniformly on compact subsets of . Letting , by change of variable, we haveBy an integration by parts, we haveWe also haveuniformly on compact subsets of . From these two estimates, we deduce that it suffices to showSinceand , we haveuniformly on compact subsets of . This completes the proof of (2.6).

## 3. Proof of the theorems

### (a) Existence and scattering

The strategy of proof is to use the standard contraction mapping theorem. For this purpose, let us define a nonlinear mapping byWe will prove that for sufficiently small *ρ*, maps from to . To do this, we introduce generalized chain and Leibniz rules.

*For any* *, we have*(3.1)(3.2)

Now let and . Then from lemmas 2.2, 2.3 and 2.4, the condition (1.3) and the chain rule (3.1), it follows that for any Since , we have for sufficiently small *δ* and *ρ*(3.3)And also we have(3.4)Thus, maps from to .

Now for any , we can show from the chain rule (3.1) and Leibniz rule (3.2) that if *δ* and *ρ* are sufficiently small, thenSimilarly, we can also showThus, for small *ρ*, is a contraction mapping and hence there exists a unique solution to the problem .

Since the time derivative satisfies the following equation:by the same argument in §2, one can easily show that , provided *δ* and *ρ* are much smaller. This completes the proof of theorem 1.1.

Once the existence has been established, the proof of theorem 1.2 is rather straightforward. Let us define functions and byLet be the solution to the linear problem (1.4) with initial data . Then it can be represented bySince for and , we have from lemma 3.1

Similarly, we haveSince , we have just proved the theorem. For more details, see Wang & Chen (2002*b*).

### (b) Non-existence of non-trivial asymptotically free solutions

Let us define a bilinear form byThen is well defined and uniformly bounded on *t*>0 for .

Our strategy of proof is to use a contradiction to the uniform boundedness of *H*. Suppose that there are non-zero functions *u* and satisfying the condition of theorem 1.3. Then we obtain(3.5)Let . Then we haveNow using an argument of Glassey (1977) and Barab (1984), we prove that if *t* is sufficiently large,(3.6)for some positive constant *A* and depending on and and *β*>1 depending on *ϵ*. Here and after, every constant depends on and , if not specified. For the proof of (3.6), we first show that(3.7)Using Hölder inequality, (3.7) yields the required estimate (3.6). To obtain the lower bound, let us choose a cut-off function supported in , such thatwhere . Since is the solution to the linear problem (1.4), for the last integral, we have(3.8)By change of variable and Plancheral's theorem, we have for the first term

From the identity , we deduce thatBy the integration by parts, it follows from the Hölder inequality that(3.9)and hence(3.10)Now we claim that there exists a large number , such that(3.11)

For the proof of (3.11), we may assume that . Let us define a function by . Then from (3.10), we can find a positive number , such that for all . Using the integration by parts *m*-times, we get for ,

We then have for some *A* depending on . This gives us thatNow if we choose *m* and *β*, so that , then the claim (3.11) is proved, provided is sufficiently large.Similarly, we can prove thatas and hence by the same argument as above, we have the estimate(3.12)if for some large .

Finally, for the last term of (3.8), let us consider the integralThen by change of variable and Plancheral's theorem, is converted byHere we also used the identity . Similarly to the estimate (3.9), we have . With this estimate we prove that(3.13)Actually, by the integration by parts as above, we haveas .

Therefore, (3.13) together with (3.11) and (3.12) yields the lower bound estimate (3.7) and hence (3.6).

Since and have compact Fourier supports, it follows from the proof of lemmas 2.2 and 2.3 that for all (3.14)From the estimate (3.14) and the hypothesis (1.5), we readily have for ,(3.15)Thus, choosing *β*, such as and , we conclude from (3.6) that for large *t*. This is a contradiction to the uniform boundedness of *H*.

### (c) Remarks

The methods of proof for theorems 1.1 and 1.2 are applicable to the high-dimensional case with a slight modification of lemma 2.1. One can treat the high-dimensional version of lemma 2.1 by using a dyadic decomposition and the method of stationary phase in the case of non-vanishing Gaussian curvature of the phase. As for theorem 1.3, using the radial symmetry, one can carry out the integration by parts with respect to the radial derivatives and hence obtain a high-dimensional version of (3.7). The high-dimensional results, stated as above, will be pursued in the forthcoming paper.

In the proof of theorem 1.3, the assumption was necessary for the comparison between (3.6) and (3.15). For the proof of (3.6), it was inevitable to use unlike the Schrödinger case in Barab (1984), where *M* is just a large constant. In our problem, the dependence of *M* on *t* was caused by the reason that the norm converges to the norm , but the function itself does not converges to in . This is a difficulty different from other dispersive equations with well curved phase *ω* like the Schrödinger case and so on. It will be very interesting to prove the non-existence of scattering without decay assumption (1.5).

## Acknowledgments

The first author is JSPS Research Fellow. The authors thank the referees so much for their kind comments, which improve the presentation of the paper.

## Footnotes

- Received September 14, 2005.
- Accepted January 13, 2006.

- © 2006 The Royal Society