Asymptotics for the Ostrovsky-Hunter Equation in the Critical Case

We consider the Cauchy problem for the Ostrovsky-Hunter equation ∂x(∂tu−(b/3)∂3 xu−∂xKu3) = au, (t, x) ∈ R, u(0, x) = u0(x), x ∈ R, where ab > 0. Define ξ0 = (27a/b)1/4. Suppose that K is a pseudodifferential operator with a symbol ?̂?(ξ) such that ?̂?(±ξ0) = 0, Im?̂?(ξ) = 0, and |?̂?(ξ)| ≤ C. For example, we can take ?̂?(ξ) = (ξ2 − ξ2 0)/(ξ2 + 1). We prove the global in time existence and the large time asymptotic behavior of solutions.

We define the evolution group U() = F −1 F, where the multiplication factor  =  −Λ() , Λ() = / + (/3) 3 .It is well known that the operator J = U()U(−) is a useful tool for obtaining the L ∞ -time decay estimates of solutions and has been used widely for studying the asymptotic behavior of solutions to various nonlinear dispersive equations.We have where Λ  (−  ) =  −2  −  2  , and the antiderivative  −1  is defined by the Fourier transform such that Note that the commutators are true [J, L] = 0, [  , L] = 0, [J,   ] = −1, [ −1  ,   ] = − −1  , where L =   + Λ(−  ) =   −  −1   − (/3) 3  .However, it seems that J does not work well on the nonlinear terms.In order to avoid the derivative loss, when estimating the norm ‖  J‖ L 2 instead 2 International Journal of Differential Equations of the operators J we apply the modified dilation operator defined by Note that P acts well on the nonlinear terms as the firstorder differential operator and it almost commutes with L: [P, L] = −L.Also J and P are related via the identity where Note that [I, L] = 0.In order to get the estimate of   J, we will show the a priori estimates of , L, and I.Different point compared to the previous works is to consider the estimate of I since I contains the term  −1  with an additional explicit time growth.
When () =  2 , then (1) was introduced in [2] for modelling the small-amplitude long waves in a rotating fluid of finite depth.Therefore (1) with () =  2 is called the Ostrovsky equation.It was studied by many authors (see, e.g., [3][4][5] and references cited therein).When  = 0, (1) is called the reduced Ostrovsky equation.Equation (1) has some conservation quantities, when () = || −1 ,  ∈ R. One of them is the zero mass conservation law which is obtained by integrating in space  ∫  (, )  = 0 (7) under the restriction ∫  0 () = 0. Rewrite (1) as Multiplying both sides of (8) by , integrating in space, using (7), we obtain which is the conservation of the momentum.The same approach as in deriving (9) will be used for the high frequency part in order to avoid the derivative loss, when proving the existence of solutions of (1).
Local well-posedness for the Ostrovsky equation was shown in [5] in the case of the initial data by using the parabolic regularization technique and limiting arguments.Their method works also for the case of the generalized nonlinearity () = || −1  and also generalized reduced Ostrovsky equation (1), since the dispersive effects were not used in the proof.Thanks to the high frequency part   , the solutions to the linear equation (  −   )  =  obtain a smoothing property.By using this property, in [3], the local well-posedness for the Ostrovsky equation was shown under the condition The method of [3] depends on the linear part of the equation and also works for the nonlinearities of a general order.In [4,[6][7][8] the local well-posedness for the Ostrovsky equation was treated by the Fourier restriction norm method of [9] and in [4] the H −3/4+ local well-posedness was shown.We note here that the Sobolev space H −3/4 is considered as critical regularity concerning the Korteweg-de Vries equation.
Global well-posedness in the energy class was obtained for the Ostrovsky equation in [3] through the energy conservation law, when the initial data and  > 0. After their work, the global well-posedness in was proved in [4,6] due to the L 2 -conservation law.The global well-posedness, in the negative order Sobolev space H −3/10+ , was shown in [8] by using the  method of [10].
We now turn to the case of the reduced Ostrovsky equation.The local well-posedness was shown in the space H 2 in paper [11] and after that in H 3/2+ in [12].Their methods work also in the case of the general nonlinear dispersive equations with different nonlinearities.We also refer to [13,14] for the local well-posedness in the class However there are few works on the global well-posedness for the reduced Ostrovsky equation due to the lack of the smoothing property.The global well-posedness for reduced Ostrovsky equation (1) with  = 0 and cubic nonlinearity () =  3 (which is called the short pulse equation) was obtained in [15], when the initial data whereas for the quadratic nonlinearity () =  2 (which is called the reduced Ostrovsky equation or the Ostrovsky-Hunter equation; see [16,17]), it was shown in [18] when the initial data for all  ∈ R. The time decay properties of solutions to the corresponding linear problem can be studied if we assume that the initial data decay rapidly at infinity.So the global existence was shown in [12], for the nonlinearity () =   with an integer  ≥ 4, when the initial data are small and sufficiently regular: In [1,19,20], we considered the large time asymptotics for reduced Ostrovsky equation ( 1) with  = 0 and some conditions on the order of nonlinearity.
To state our results precisely we introduce Notation and Function Spaces.We denote the Lebesgue space by L  = { ∈ S  ; ‖‖ L  < ∞}, where the norm ‖‖ L  = (∫ |()|  ) We also use the notations H , = H , 2 , H  = H ,0 shortly, if they do not cause any confusion.Let C(I; B) be the space of continuous functions from an interval I to a Banach space B. Different positive constants might be denoted by the same letter .We define the free evolution group U() =  −Λ(−  ) = F −1 F, where the multiplication factor (, ) =  −Λ () .
We are now in a position to state our main result.
We organize the rest of our paper as follows.In Section 3, we state main estimates for the decomposition operators V() and V * () related to the evolution group U().We prove a priori estimates of solutions in Section 4. Section 5 is devoted to the proof of Theorem 1. where  = −1, 0, 1, 2, the phase function
In particular we have the estimate ‖ * ()‖ L ∞ ≤ . are valid for all  ≥ 1.
To estimate the second integral  2 we integrate by parts via the identity Using the estimate Changing  =  we obtain (57) International Journal of Differential Equations 7 Hence In the same manner changing  =  we get (61) ] −1 = 1, and Hence for  = −1, 0, 1.Thus we have for all  ≥ 1,  > 0, and  = −1, 0, 1.
For the case of  < 0 we integrate by parts using identity (53): Using the estimate in the domain  > 0,  < 0, we obtain Then as above we get We next consider the operator V * .
We need estimate of V  .

Asymptotics for the Nonlinearity.
We obtain the asymptotic representation for the nonlinear term.Define the norm Lemma 6.The asymptotics is true for all  ≥ 1,  ≥ 0, where φ() = FU(−)().
Proof.In view of (37) we find for the new dependent variable φ = FU(−)() where   = V()  φ.By Lemma 3 we have By Lemma 2 Therefore Also by the Leibnitz rule Then In the same manner Hence the result of the lemma follows.

A Priori Estimates
We define where J = U()U(−), φ = FU(−)(), and  > 0 is small.We have the local in time existence of solutions.

Theorem 7. Let the initial data
Then there exists a time  0 > 0 such that (1) has a unique solution  in X  0 .
To get the desired results, we prove a priori estimates of solutions uniformly in time.
is true for all  ≥ 1.
Proof.By continuity of the norm ‖‖ X  with respect to , arguing by the contradiction we can find the first time  ≥ 1 such that ‖‖ X  = .Consider a priori estimates of To avoid the derivative loss in (1) we apply the operator Define the high and short frequency projectors Q   = F −1   φ, where  1 () = 1 for || ≥ 1 and  1 () = 0 for || ≤ 1, and also  2 () = 1 −  1 ().Then we get and the integral equation Hence applying the energy method to the first equation we find and by the integral equation Next we estimate the norm ‖I‖ L 2 .Denote K3 () =   K() =  (1).Applying the operator I =   −  −1  to (1) via the commutator [I, L] = 0, we get where  = 2K(3   −1   +  3 ).Using the factorization formulas as in the derivation of (37) we find where we denote   = V()  φ.Then we get Next using identity (36) we find with Ω() = Λ() − 3Λ(/3).Next using the relations where Then we represent where We need to estimate the derivative V *  .We have 2 and also V * () − V * () = V * ()A 0 () we find for the second summand Since    0 = V  φ + Vφ  , we obtain Thus we get Hence Then as the above using the projectors Q 1 and Q 2 we find We have and then we get   .Therefore we find the desired estimate ‖⟨⟩ 1/2 φ‖ L ∞ ≤ .This is the desired contradiction.Lemma 8 is proved.

Proof of Theorem 1
The follows by a standard continuation argument from Lemma 8 and local existence Theorem 7. We need only to prove asymptotic formula (20).We need to compute the asymptotics of the function φ(, ).As in the proof of Lemma  ( This completes the proof of asymptotics (20).Theorem 1 is proved.