SPATIAL ANALYTICITY OF SOLUTIONS OF A NONLOCAL PERTURBATION OF THE KDV EQUATION

Let H denote the Hilbert transform and 0. We show that if the initial data of the following problems ut +uux +uxxx + (Hux +Huxxx) = 0; u( ; 0) = ( ) andvt+ 1 (vx) 2 +vxxx+ (Hvx+Hvxxx) = 0; v( ; 0) = ( ) has an analytic continuation to a strip containing the real axis, then the solution has the same property, although the width of the strip might diminish with time. When > 0 and the initial data is complex-valued we prove local well- posedness of the two problems above in spaces of analytic functions, which implies the constancy over time of the radius of the strip of analyticity in the complex plane around the real axis.


Introduction
We are interested in studying spatial analyticity of solutions of the following problems: where H denotes the Hilbert transform given by Hf (x) = 1 π P ∞ −∞ f (y) y−x dy for f ∈ S(R) the Schwartz space of rapidly decreasing C ∞ (R) functions, P represents the principal value of the integral and the parameter η is an arbitrary nonnegative number.It is known that (Hf )(ξ) = isgn(ξ) f (ξ), for all f ∈ H s (R), where sgn (ξ) = −1, ξ < 0, 1, ξ > 0.
Equation (1) was derived by Ostrovsky et.al.(see [9] for more details) to describe the radiational instability of long non-linear waves in a stratified fluid caused by internal wave radiation from a shear layer; the fourth term corresponds to the wave amplification and the fifth term represents damping.It models the motion of a homogeneous finite-thickness fluid layer with density δ 1 , which moves at a constant speed U , slipping over an immobile infinitely deep stratified fluid with a density δ 2 > δ 1 .The upper boundary of the layer is supposed to be rigid and the lower one is contiguous to the infinitely deep fluid.Here u(x, t) is the deviation of the EJQTDE, 2005 No. 20, p. 1 interface from its equilibrium position.Let us remark that some numerical results for periodic and solitary-wave solutions of equation (1) were obtained by Bao-Feng Feng and T. Kawahara [4].
The Cauchy problems associated to (1) and (2) were studied in [1], where it was proved that problems (1) and ( 2) are globally well-posed in H s (R) for s ≥ 1, considering real-valued solutions.
In this paper we are interested in proving that if the initial condition of the problem (1) (resp.( 2)) is analytic and has an analytic continuation to a strip containing the real axis, then the solution of (1) (resp.( 2)) has the same property.
Section 2 is devoted to studying the case when the solutions are real-valued on the real axis at any time, and η ≥ 0. Hence, the results obtained in [1] about the initial value problems associated to (1) and (2) will be helpful.We use the method developed by Kato&Masuda [8] which estimates certain families of Liapunov functions for the solutions, to prove global spatial analyticity of the solutions, but the width of the strip might decrease with time.
Section 3 shows that problems (1) and ( 2) admit a Gevrey-class analysis.For η > 0, we prove local well-posedness of problem (1) (resp.( 2)) in X σ,s for σ > 0 and s > 1/2 (resp.s > 3/2); here, the initial data can be complex-valued.So, if the initial data of problem (1) (resp.(2)) is analytic and has an analytic continuation to a strip containing the real axis, then the solution of (1) (resp.( 2)) has the same property, maintaining the width of the strip in time.It should be mentioned that it was recently proved by Grujić&Kalisch [5] a result on local well-posedness of the generalized KdV equation (KdV is an abbreviation for Korteweg-de Vries) in spaces of analytic functions on a strip containing the real axis without shrinking the width of the strip in time; their proof uses space-time estimates and Bourgain-type spaces.
Here we do not make use of Bourgain spaces, we mainly use some properties of the Semigroup associated to the linear part of problem (1), namely Lemmas 3.1 and 3.2, to prove local well-posedness of problem (1) in X σ,s .Moreover, proceeding as in [2], where Bona&Grujić studied some KdV-type equations, we prove for real-valued solutions and η ≥ 0 that if the initial state belongs to a Gevrey class, then the solution of (1) (resp.( 2)) remains in this class for all time but the width of the strip of analyticity may diminish as a function of time.
Finally, in Section 4 we consider η ≥ 0 in (1) and complex-valued initial data in X r -spaces for r > 0. Similar as in [6], where analyticity of solutions of the KdV equation was studied, we use Banach's fixed point theorem in a suitable function space in order to find a local solution of problem (1) that is analytic and has an EJQTDE, 2005 No. 20, p. 2 analytic extension to a strip around the real axis although the radius of the strip of analyticity in the complex plane around the real axis may decrease with time. Notation: • f = Ff : the Fourier transform of f (F −1 : the inverse of the Fourier transform), where • • s , (•, •) s : the norm and the inner product respectively in • H: the Hilbert transform.
• B(X, Y ): set of bounded linear operators on X to Y .If X = Y we write B(X).
A(r): the set of all analytic functions f on S(r) such that f ∈ L 2 (S(r )) for each 0 < r < r and that f (x) ∈ R for x ∈ R.
• H p (r): the analytic Hardy space on the strip S(r).
< ∞}.• C(I; X) : set of continuous functions on the interval I into the Banach space X.
• C ω (I; X) : the set of weakly continuous functions from I to X.
• (z): the real part of the complex number z.

Real-valued initial data.
We deduce in this Section global analyticity (in space variables) of solutions of problems (1) and (2) when the initial data and the corresponding solution take real values on the real axis and supposing moreover that the initial data has an analytic continuation that is analytic in a strip containing the real axis.We will use the fact that problems (1) and (2) are globally well possed in H s (R) for s ≥ 1, when the solution of the two previously mentioned problems take real values on the real axis EJQTDE, 2005 No. 20, p. 3 at any time.More precisely we have the following two theorems (for real-valued solutions) which can be found in [1].
Theorem 2.3 (resp.2.4) states that if the initial state has an analytic continuation that belongs to A(r 0 ) for some r 0 > 0 then the solution u(t) (resp.v(t)) of problem (1) (resp.( 2)), with η ≥ 0, also has an analytic continuation belonging to A(r 1 ) for all t ∈ [0, T ], where r 1 might decrease with time.Theorem 2.3, below, is an application of the method developed by Kato&Masuda in [8] to study global analyticity (in space variables) of some partial differential equations.Similar as in the proof of Theorem 2 in [8] we consider H m+5 (R) ≡ Z ⊂ X ≡ H m+2 (R) and Φ σ;m (v) ≡ 1 2 v 2 σ,2;m defined on an appropriate open set O ⊂ Z, where Then there exist constants c, γ > 0 such that for every v ∈ H m+5 (R), where , and Q 0 (v) ≡ 0. By using Kato's inequality (K) in the Appendix we have that Then Now, using the Schwarz inequality and the formula We denote as in [8], . By using ( 6) it follows, as a particular case of Lemma 3.1 in [8], that By replacing the inequality ( 7) into (5), the Lemma follows.
Next, we enunciate Lemma 2.4 in [8] and we give, for expository completeness, a proof of this trivial result.

Gevrey Class Regularity.
Now, we make a Gevrey-class analysis of problems ( 1) and ( 2).Let us remark that, since each function f ∈ X σ,s = D(A s e σA ) with σ > 0 and s ≥ 0, satisfies , where σ > 0 is the radius of the strip of analyticity in the complex plane around the real axis.Theorem 3.1 (resp.Theorem 3.2) states that problem (1) (resp.(2)), for η > 0 and complex-valued initial data, is locally well-posed in X σ,s where σ > 0 is a fixed number and s is a suitable nonnegative number.Theorem 3.1 (resp.For t ≥ 0 and ξ ∈ R, let It is not difficult to see that |F η (t, ξ)| ≤ e ηt , for all t ≥ 0 and ξ ∈ R.
Proof.Similar to the proof of Lemma 2.1 in [1].
where φ ∈ X σ,s and c λ is a constant depending only on λ. Proof.
On the other hand, sup The lemma follows immediately from (12) and (13).
where in the last inequality we have used Lemmas 3.1 and 3.2 (with λ = 1) and the fact that X σ,s is a Banach algebra for s > 1/2 and σ ≥ 0 (Lemma 6 in [2]).Since τ − t ≥ t − t , for all t ∈ [0, t], it follows from Lemma 3.2 and from the triangle inequality that and the expression on the right hand side of the last inequality belongs to L 1 ([0, t], dt ).
Theorems 3.3 and 3.4, below, consider the case of real-valued solutions (on the real axis) to problems (1) and ( 2) respectively, for η ≥ 0. So, Theorems 2.1 and 2.2 will be required again.The following theorem is proved similarly to Theorem 11 in [2], where the rate of decrease of the uniform radius of analyticity for KdV-type equations of the form u t + G(u)u x − Lu x = 0 was also studied, where u = u(x, t), EJQTDE, 2005 No. 20, p. 9 for x, t ∈ R, G is a function that is analytic at least in a neighborhood of zero in C, but real-valued on the real axis, and L is a homogeneus Fourier multiplier operator defined by Lu(ξ) = |ξ| µ û(ξ), for some µ > 0. In the case of problem (1) we have that Lu(ξ) = [ξ 2 − ηi(sgn(ξ) − ξ|ξ|)]û(ξ).
Proof.Let η, T, σ 0 , and s be as in the hypothesis of the theorem.Let r ≡ s−1 > 3/2.
By the Remark at the end of this Section we have that φ ∈ A(σ 0 ).So, by using Lemma 2.2 in [8], we have that φ ∈ H ∞ (R).Then u ∈ C([0, T ]; H s (R)), which follows from Theorem 2.1 and from Corollary 4.7 in [7].Let v ≡ u x , then Let σ ∈ C 1 ([0, T ]; R) be a positive function such that σ < 0 and σ(0) = σ 0 .Then It follows from the last expression and from (17) that 1 2 where I 1 and I 2 are particular cases of the corresponding ones in [2], taking G(u) = u.For the sake of completeness we estimate them here.Since r > 1/2, by using Lemmas 6 and 9 in [2], we have that Since r > 3/2, by using Lemma 10 in [2], we obtain and similarly ).So, using the last three inequalities, it is not difficult to prove that It follows from the last inequality that Y r ⊂ X r .Now, let us consider the set Let g ∈ L 2,0 (R), then so g ∈ Y r .Now we will prove that L 2,0 (R) is a dense set in X r .Let f be an element of X r .For each n ∈ N, take f n ∈ L 2,0 (R) given by and . Then, by using Theorem 1 in [6], we have that f , f ∈ L r and moreover . EJQTDE, 2005 No. 20, p. 13 (ii.)By Lemma 4.1, Y r ⊂ X r .So, it follows from Lemma 2.1 in [6] that Xr .Now, using the last inequalities, we have that where in the last inequality we have used (27).
Proof.Using Leibniz's rule and Sobolev's inequality ( , the proof of the Lemma follows from the last two inequalities.
Corollary 4.1.There exists a polynomial ã1 with nonnegative coefficients such that Now, we state the main theorem of this Section.Proof.Let η ≥ 0, σ 0 > 0, and φ ∈ X σ0 .Let σ(t) = σ 0 e −At/σ0 , where the positive constant A will be conveniently chosen later.For T > 0, consider the space Let ρ = 4 φ Xσ 0 .For v ∈ B ρ (T ), we define the mapping M by u = M v, where u is the solution of the linearized problem More precisely u can be obtained as follows.Take the Fourier transform to (32), then Now, integrating the last expression between 0 and t, it follows that where the last integral is well defined, since and in the last inequality we have used the fact that Let us choose A, T > 0 such that Let v ∈ B ρ (T ).It will be proved that u = M v ∈ B ρ (T ), for suitably chosen T, A > 0. Now, it is not difficult to see that = 2σ (t) Using (34) and Corollary 4.1, it follows from the last inequality that 1 2 where a(•) is a polynomial with nonnegative coefficients.Now integrating the last inequality from 0 to t we get Then, By choosing T > 0 small enough such that we have that Now we take A, T > 0 such that Then, Multiplying both sides of the last equation by ξ 2j cosh(2σ(t)ξ)+ξsinh(2σ(t)ξ) ŵ(t, ξ), integrating in ξ, summing the terms for j = 0, 1 and taking the real part we obtain where the last inequality is a consequence of Lemma 4.4 and ã(•, •) is a polynomial with nonnegative coefficients.Integrating the last expression on [0, t] and applying the Cauchy Schwarz inequality, we get By using (35) in the last inequality we obtain If we take A, T > 0 such that 6ã(ρ, ρ) then M is a contraction.So, by choosing A, T > 0 such that (34)-(37) are satisfied, the mapping M has a unique fixed point u ∈ B ρ (T ) that is the solution of problem for all f ∈ Y σ(T ) .So, let f be an arbitrary but fixed element of Y σ(T ) .Let us denote by h j (ξ, T ) ≡ ξ 2j cosh(2σ(T )ξ) + ξ sinh(2σ(T )ξ) .First, let us remark that |û(t, ξ) − û(t , ξ)| → 0 as t ↑ t, for all ξ ∈ R. In fact, by using (33), we have that û(t, ξ) − û(t , ξ) = 3 j=1 I j (t, t , ξ), where EJQTDE, 2005 No. 20, p. 17 where the last convergence is a consequence of ), and the dominated convergence theorem.
On the other hand we have that where which, by the dominated convergence theorem, tends to zero as t ↑ t.
Now we estimate I 2 (t, t ) defined below.First we see that ≤ c(T ) u(τ where in the last inequalities we have used Lemmas 4.2 and 4.3, and U (τ ) is the analytic extension of u(τ ) on S(σ(T )).I 2 (t, t ) is given by I 2 (t, t ) ≡ The interchange of the integrals in (41) is a consequence of Fubini's Theorem since where the second and third last inequalities were consequence of ( 38  The interchange of the integrals in the last expression was a consequence of Fubini's Theorem (similar to (41)).So, by the dominated convergence theorem, we have that 3.2)) implies that if the initial condition of problem (1) (resp.(2)) is analytic and has an analytic continuation to a strip EJQTDE, 2005 No. 20, p. 6 containing the real axis, then the solution of (1) (resp.(2)) has the same property, without reducing the width of the strip in time.
if |ξ| ≤ n and | f (ξ)| ≤ n 0, otherwise.It follows easily from the definition of f n that | f n (ξ)| ≤ | f(ξ)|, for all n ∈ N and ξ ∈ R.Moreover, f n (ξ) → f (ξ) as n → ∞, for all ξ ∈ R. So, by the dominated convergence theorem, we have that f n − f Xr → 0 as n → ∞.This concludes the proof.Lemma 4.2.(i.) Suppose that F ∈ H 2,2 (r).Let f be the trace of F on the real line.Then f ∈ Y r and

Lemma 4 . 4 and
Corollary 4.1, below, are particular cases of Lemma 2.4 and its Corollary in Hayashi [6] respectively.Lemma 4.4.There exists a polynomial ã of which coefficients are all nonnegative with the following property: If r > 0 and f