Gevrey regularity for the generalized Kadomtsev-Petviashvili I (gKP-I) equation

Abstract: The task of our work is to consider the initial value problem based on the model of the generalized Kadomtsev-Petviashvili I equation and prove the local well-posedness in an anisotropic Gevrey spaces and then global well-posedness which improves the recent results on the well-posedness of this model in anisotropic Sobolev spaces [17]. Also, wide information about the regularity of the solution in the time variable is provided.


Introduction and position of problem
The study of nonlinear wave processes in real media with dispersion, despite the significant progress in this area in recent years, for example, [1][2][3] and numerous references in these works are still relevant. This, in particular, concerns the dynamics of oscillations in cases where high-energy particle fluxes occur in the medium, which significantly change such parameters of propagating wave structures, such as their phase velocity, amplitude and characteristic length. In recent and earlier years, a fairly large number of works have been devoted to studies of this kind of relativistic effects (see [12][13][14]).
Recently, the great interest on the KP equation has led to the construction and the study of many extensions to the KP equation . These new extended models propelled greatly the research that directly resulted in many promising findings and gave an insight into some novel physical features of scientific and engineering applications. Moreover, lump solutions, and interaction solutions between lump waves and solitons, have attracted a great amount of attentions aiming to make more progress in solitary waves theory. Lump solutions, have been widely studied by researchers for their significant features in physics and many other nonlinear fields [18][19][20].
(1.1) with D α x u(x, y, t) = 1 (2π) 3 2 R 3 |ξ| α Fu(ξ, µ, τ)e ixξ+iyµ+itτ dξdµdτ This equation belongs to the class of Kadomtsev-Petviashvili equations, which are models for the propagation of long dispersive nonlinear waves which are essentially unidirectional and have weak transverse effects. Due to the asymmetric nature of the equation with respect to the spatial derivatives, it is natural to consider the Cauchy problem for (1.1) with initial data in the anisotropic Sobolev spaces H s 1 ,s 2 (R 2 ), defined by the norm u H s 1 ,s 2 (R 2 ) = ξ 2s 1 η 2s 2 | u(ξ, η)| 2 dξdη Many authors have investigated the Cauchy problem for Kadomtsev-Petviashvili equations as in, for instance [4,8,16]. Yan et al. [17] established the local well-posedness of the Cauchy problem for the Kadomtsev-Petviashvili I equation in anisotropic Sobolev spaces H s 1 ,s 2 (R) with s 1 > − α−1 4 , s 2 ≥ 0 with α ≥ 4 and globally well-posed in H s 1 ,0 (R) with if 4 ≤ α ≤ 5 also proved that the Cauchy problem is globally well-posed in H s 1 ,0 (R) with s 1 > − α(3α−4) 4(5α+4) if α > 5. The authors in [8] proposed the problem 2) and proved that it is globally well-posed for given data in an anisotropic Gevrey space G σ 1 ,σ 2 (R 2 ), σ 1 , σ 2 ≥ 0, with respect to the norm With initial data in anisotropic Gevrey space and κ 1, we will consider the problem (1.1). The spaces G σ 1 ,σ 2 ,κ s 1 ,s 2 can be defined as the completion of the Schwartz functions with respect to the norm In addition to the holomorphic extension property, Gevrey spaces satisfy the embeddings G σ 1 ,σ 2 The main aim to consider initial data in these spaces is because of the Paley-Wiener Theorem.
if and only if it is the restriction to the to the real line of a function F which is holomorphic in the strip {x + iy ∈ C : |y| < σ}, and satisfies

Notation
We will also need the full space time Fourier transform denoted bŷ In both cases, we will denote the corresponding inverse transform of a function To simplify the notation, we introduce some operators. We first introduce the operator A σ 1 ,σ 2 κ , which we define as Then, we may then define another useful operator For x ∈ R n , we denote x = (1 + |x| 2 ) 1/2 . Finally, we write a b if there exists a constant C > 0 such that a ≤ Cb, and a ∼ b if a b a. If the constant C depends on some quantity q, we denote this by a q b.

Function spaces
Since our proofs rely heavily on the theory developed by Yan et al., let us state the function spaces they used explicitly, so that we can state their useful properties which we will exploit in our modifications of their spaces. The main function spaces they used are the so-called anisotropic Bourgain spaces,adapted to the generalized Kadomtsev-Petviashvili I, whose norm is given by with m(ξ, η) = ξ|ξ| α + η 2 ξ . Furthermore, we will also need a hybrid of the anlytic Gevrey and anisotropic Bourgain spaces, designated X σ 1 ,σ 2 ,κ s 1 ,s 2 ,b (R 3 ) and defined by the standard It is well-known that these spaces satisfy the embedding Thus, solutions constructed in X σ 1 ,σ 2 ,κ s 1 ,s 2 ,b belong to the natural solution space. When considering local solutions, it is useful to consider localized versions of these spaces. For a time interval I and a Banach space Y, we define the localized space Y(I) by the norm

Main results
The first result related to the short-term persistence of analyticity of solutions is given in the next Theorem. Moreover the solution depends continuously on the data f . In particular, the time of existence can be chosen to satisfy for some constants c 0 > 0 and γ > 1. Moreover, the solution u satisfies The second main result concerns the evolution of the radius of analyticity for the x-direction is given in the next Theorem. Here X σ 1 ,0,1 s 1 ,0,b = X σ 1 ,0 s 1 ,b , s 2 , σ 2 = 0 and κ = 1.
The method used here for proving lower bounds on the radius of analyticity was introduced in [15] in the study of the non-periodic KdV equation. It was applied to the the higher order nonlinear dispersive equation in [9] and the system of mKdV equation in [10]. Our last aim is to show the regularity of the solution in the time. A non-periodic function φ(x) is the Gevrey class of order κ i.e, φ(x) ∈ G κ , if there exists a constant C > 0 such that Here we will show that for x, y ∈ R, for every t ∈ [0, δ] and j, l, n ∈ {0, 1, 2, . . . }, there exist C > 0 such that, given by Theorem 2.1, belongs to the Gevrey class G (α+1)κ in time variable.
given by Theorem 2.2, belongs to the Gevrey class G (α+1) in time variable.
The rest of the paper is organized as follows: In section 3, we present all the auxiliary estimates that will be employed in the remaining sections. We prove Theorem 2.1 in subsection 4.1 using the standard contraction method and Theorem 2.2 in subsection 4.2. Finally, in section 5, we prove G (α+1) regularity in time.

Auxiliary estimates
To begin with, let us consider the related linear problem By Duhamel's principle the solution can be written as We localize it in t by using a cut-off function satisfying ψ ∈ C ∞ 0 (R), with ψ = 1 in [−1, 1] and suppψ ⊂ [−2, 2].
We consider the operator Φ given by where ψ δ (t) = ψ( t δ ). To this operator, we apply the following estimates.
Proof. The proofs of (3.3) and (3.4) for σ 1 = σ 2 = 0 can be found in Lemma 2.1 of [17]. These inequalities clearly remain valid for σ 1 , σ 2 > 0, as one merely has to replace f by The final preliminary fact we must state is the following bilinear estimate, which is Lemma 3.1 of [17].
To this result, we apply the following Lemma, which is a corollary of Lemma 3.2.
Proof. It is not hard to see that By Lemma 3.2, we get

Local well-posedness in an anisotropic Gevrey space
The above Lemmas will be used without somtimes mention to prove Theorem 2.1.
Proof. Combining Lemma 3.3 and Lemma 3.1 with the fixed point Theorem. We define Then, we have We choose δ such that δ < 1 Here we choose δ such that δ < 1 We choose the time of existence where For appropriate choice of c 0 , this will satisfy inequalities (4.1) and (4.2).
From Lemma 4.1, we see that for initial data f (x, y) ∈ G σ 1 ,σ 2 ,κ 1 then the map Φ(u) is a contraction on a small ball centered at the origin in X σ 1 ,σ 2 ,κ s 1 ,s 2 ,b . Hence, the map Φ(u) has a unique fixed point u in a neighborhood of 0 with respect to the norm · X σ 1 ,σ 2 ,κ The rest of the proof follows the standard argument.

Global well-posedness
In this section, we prove Theorem 2.2. The first step is to obtain estimates on the growth of the norm of the solutions. For this end, we need to prove the following approximate conservation law.
with ∈ [0, 3 4 ) if α = 4, and = 1 if α = 6, 8, 10, . . . . Before we may show the proof, let us first state some preliminary Lemmas. The first one is an immediate consequence of Lemma 12 in [15]. This will be used to prove the following key estimate.  3 4 ) if α = 4 and = 1 if α = 6, 8, 10, ... Proof. We first observe that the inequality in Lemma 3.2, is equivalent to where we denote φ(τ, ξ, η) = τ + m(ξ, η) . With this, we observe that the left side of the inequality in Lemma 4.4 can be estimated by Lemma 4.3 as If we apply Lemma 3.2 with s 1 = − , s 2 = 0, it will follow, from the comments above, that Proof of Theorem 4.2. Begin by applying the operator A σ 1 ,0 1 to Eq (1.1). If we let U = A σ 1 ,0 1 u, then Eq (1.1) becomes If we apply integration by parts, we may rewrite the left-hand side as which can then be rewritten as By noticing that ∂ j x U(x, t) → 0 as |x| → ∞ (see [15]) we obtaining ∂ t U 2 (x, y, t) dxdy = 2 U(x, y, t)N σ 1 ,0 1 (u)(x, y, t) dxdy.
Integrating with respect to time yields Applying Cauchy-Schwarz and the definition of U, we obtain . We now apply Lemma 4.4 and the fact that b = 1 2 + , we can further estimate this by If T * = ∞, there is nothing to prove, so let us assume that T * < ∞. In this case, it suffices to prove that for all T > T * . To show that this is the case, we will use Theorem 2.1 and Theorem 4.2 to construct a solution which exists over subintervals of width δ, using the parameter σ 1 to control the growth of the norm of the solution. We first prove the case s = 0 and then we will generalize the case.

The case s = 0
The desired result will follow from the following proposition.
and sup for some constant C > 0.
Proof. For fixed T ≥ T * , we will prove, for sufficiently small σ 1 > 0, that We will use the Theorem 2.1 and Theorem 4.2 with the time step The smallness conditions on σ 1 will be where C > 0 is the constant in Theorem 4.2. Proceeding by induction, we will verify that for n ∈ {1, · · ·, m + 1}, where m ∈ N is chosen, so that T ∈ [mδ, (m + 1)δ). This m does exist, since by Theorem 2.1 and the definition of T * , we have We cover now, the interval [0, δ], and by Theorem 4.2, we have where we used that This verifies (4.12) for n = 1 and now, (4.13) follows using again as well as Cσ 1 f G σ 0 ,0 0 ≤ 1. Next, assuming that (4.12) and (4.13) hold for some n ∈ {1, · · ·, m}, we will prove that they hold for n + 1. We estimate , verifying (4.12) with n replaced by n + 1. To get (4.13) with n replaced by n + 1, it is then enough to have (n + 1)Cσ 1 2 3 f G σ 0 ,0 0 ≤ 1.

The general case
For general s, we have u 0 ∈ G σ 0 ,0 The case s = 0 already being proved, we know that there is a T 1 > 0, such hat where ς > 0 depends on f, σ 0 and ς. We now conclude that The proof of Theorem 2.2 is now completed.

Gevrey's regularity in time
We follow the methods found in [5][6][7]11] to treat the regularity in time in Gevrey sens for unique solution of (1.1).

Conclusions
We have discussed the local well-posedness for a generalized Kadomtsev-Petviashvili I equation in an anisotropic Gevrey space. We proved the existence of solutions using the Banach contraction mapping principle. This was done by using the bilinear estimates in anisotropic Gevrey-Bourgain. We used this local result and a Gevrey approximate conservation law to prove that global solutions exist. These solutions are Gevrey class of order (α + 1)κ in the time variable. The results of the present paper are new and significantly contribute to the existing literature on the topic.