Norm-inflation results for purely BBM-type Boussinesq systems

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Norm-inflation results for purely BBM-type Boussinesq systems © 2022 the Authors Published version Bautista, George J.; Potenciano-Machado, Leyter Bautista, G. J., & Potenciano-Machado, L. (2022). Norm-inflation results for purely BBM-type Boussinesq systems. Journal of Mathematical Analysis and Applications, 514(1), Article 126254. https://doi.org/10.1016/j.jmaa.2022.126254 2022 J. Math. Anal. Appl. 514 (2022) 126254 Contents lists available at ScienceDirect Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa Regular Articles Norm-inflation results for purely BBM-type Boussinesq systems George J. Bautista a, Leyter Potenciano-Machado b,∗ aUniversidad Privada del Norte, Campus Breña, Av. Tingo María 1122, Lima, Peru bUniversity of Jyväskylä, Department of Mathematics and Statistics, Jyväskylä, Finland a r t i c l e i n f o a b s t r a c t Article history: Received 5 June 2021 Available online 26 April 2022 Submitted by S.-M. Sun Keywords: Boussinesq system Benjamin-Bona Mahony equation Spectral analysis Fourier series Norm inflation Picard’s iteration This article is concerned with the norm-inflation phenomena associated with a periodic initial-value abcd-Benjamin-Bona-Mahony type Boussinesq system. We show that the initial-value problem is ill-posed in the periodic Sobolev spaces H−s p(0, 2π) ×H−s p(0, 2π)for all s >0. Our proof is constructive, in the sense that we provide smooth initial data that generates solutions arbitrarily large in H−s p(0, 2π) ×H−s p(0, 2π)-norm for arbitrarily short time. This result is sharp since in [13]the well-posedness is proved to holding for all positive periodic Sobolev indexes of the form Hs p(0, 2π) ×Hs p(0, 2π), including s =0. © 2022 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). 1. Introduction The physical phenomena are usually modeled by equations involving differential operators. In the study of partial differential equations (PDEs), it is crucial to know if the equation or system are well-posed in the Hadamard’s sense: existence, uniqueness, and continuous dependence of the solutions with respect to the initial data. The lack of the latter condition represents one of the main obstacles to tackle any further analysis for the underlying PDE. In particular, it would cause incorrect solutions or non meaningful solutions at all. As consequence one can not address, for instance, the numerical implementation of the solutions [8–10]or controllability properties of the PDE [3]. One way to prove ill-posedness is to evidence the lack of continuous dependence with respect to the initial data by showing that small initial data could generate arbitrarily large solutions. This phenomena is so-called norm-inflation phenomena by obvious reasons. The main purpose of this article is to study ill-posedness for a family of Boussinesq systems proposed by J. L. Bona, M. Chen and J.-C. Saut in [5,6]: *Corresponding author. E-mail addresses: geo[email protected] (G.J. Bautista), leyter.m.p[email protected] (L. Potenciano-Machado). https://doi.org/10.1016/j.jmaa.2022.126254 0022-247X/© 2022 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). 2G.J. Bautista, L. Potenciano-Machado / J. Math. Anal. Appl. 514 (2022) 126254 ηt+wx+(ηw)x+awxxx −bηxxt =0. wt+ηx+wwx+cηxxx −dwxxt =0.(1) We prove that the system satisfies the norm-inflation phenomena for certain constant parameters a, b, c, d ∈ R. Thus, the third Hadarmard’s condition is violated. Here ηand ware real-valued functions whose physical interpretation will be described in next lines. System (1) approximates the motion of small amplitude long waves on the surface of an ideal fluid under the force of gravity in situations where the motion is sensibly two-dimensional. In (1), the variable xis proportional to the distance in the direction of propagation, while t is proportional to elapsed time. The quantity η(x, t) +h0corresponds to the liquid’s total depth at the point xand at time t, where h0is the undisturbed water depth. The variable w(x, t) represents the horizontal velocity at the point (x, y) =(x, θh0), at time t, where yis the vertical coordinate, with y= 0 corresponding to the channel bottom or sea bed. Thus, wis the horizontal velocity field at the height θh0, where θis a fixed constant in the interval [0, 1]. According to the choice of the constants a, b, cand d, we can distinguish several Boussinesq systems. In all these cases, the parameters must satisfy the following consistency conditions a+b=1 2(θ2−1 3),c+d=1 2(1 −θ2)≥0.(2) In particular, one always has a+b+c+d=1/3. A detailed study on local well-posedness of system (1)on the real line was initially addressed in [5,6]. For results on ill/well-posedness in periodic domains, depending on the sign of the abcd-parameters, we refer the reader to [1]and [13]. In contrast with other classical wave models like Korteweg-de Vries (KdV) systems [11], Boussinesq system (1)does not assume the uni-directional propagation of shallow water waves but describing the bi-directional propagation of such waves. Its two-way propagation feature seems to have a wide range of applications in different physical and mathematical branches. Indeed, among other studies, system (1)has recently been addressed from the control theory point of view, see for example [2–4,12–14], and the references contained therein. In the articles above, the well-posedness property is necessary to prove their controllability and stability results by introducing appropriate dissipative mechanisms into the system. In this work we focus on studying ill-posedness of the system (1)-(2)posed in a periodic domain. It will follows by showing that the system posses the norm-inflation phenomena. We restrict the abcd-parameters to the cases a=0,c=0,b>0andd>0.(3) The underlying system with such those restrictions is called purely BBM-type Boussinesq system, which in turn is an instance of weakly dispersive systems, see e.g. [6, Sections 2.1 and 2.2]. To be more precise, the system (1)–(3) becomes ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ηt+wx−bηtxx +(ηw)x=0 forx∈(0,2π),t>0, wt+ηx−dwtxx +wwx=0 forx∈(0,2π),t>0, η(0,x)=η0(x)forx∈(0,2π), w(0,x)=w0(x)forx∈(0,2π), (4) with periodic boundary conditions η(t, 0) = η(t, 2π); ηx(t, 0) = ηx(t, 2π)fort>0, w(t, 0) = w(t, 2π); wx(t, 0) = wx(t, 2π)fort>0.(5) G.J. Bautista, L. Potenciano-Machado / J. Math. Anal. Appl. 514 (2022) 126254 3 The main result of this work reads as follows. Theorem 1.1. Let s >0be given. There exist two sequences, one consisting of periodic initial data η0 ν w0 νν∈N ∈C∞ p([0,2π])2satisfying η0 ν w0 ν˙ H−s p(0,2π)×˙ H−s p(0,2π) →0as ν→+∞ and another one (Tν)ν∈Nof positive times, tending to zero as ν→∞, such that if ην wνis the solution of (4)-(5)coming up from the initial data η0 ν w0 ν, then ην wν˙ H−s p(0,2π)×˙ H−s p(0,2π) →+∞as ν→+∞. Here ˙ H−s p(0, 2π) stands for the homogeneous version of H−s p(0, 2π), whose definition can be found in Section 2. The precise definition of the periodic Sobolev spaces we shall use throughout this article is given in Section 2. Furthermore, unless otherwise stated, we reserve the letter s to indicate a non-negative real number standing for the periodic Sobolev index in Hs(0, 2π). We remark that Theorem 1.1 is sharp since the system (3)-(5)is well-posed in Hβ(0, 2π) ×Hβ(0, 2π)for β≥0as showed in [13, Theorem 3.2]. The analogous of the latest result in the real line was proved in [6, Theorem 2.1]. Moreover, Theorem 1.1 adds one extra family to the ill-posedness result in [1, Theorem 5.2]. The scalar version of Theorem 1.1, in a periodic domain, was obtained by Bona and Dai in [7]. A similar result for the scalar case in the whole real line Rwas proved by Phantee in [15]. The proof of Theorem 1.1 closely follows the ideas from [7]. Bona and Dai constructed initial periodic data and proved that the corresponding solution blows up in H−s p(0, 2π)-norm when time is sufficiently short. One of the main ingredients in their proof relies on the solutions’ knowledge of the forward linear problem. In our case, thanks to the periodic framework and the spectral analysis carried out in [2]—to study controllability and stability issues— we also have the explicit expressions for the solutions to the linear counterpart of (4)-(5). This was done in [2]by combining tools from Fourier analysis with the well known Duhamel’s principle. See Section 2for details. In fact, the authors in [2]made a more refined spectral analysis to deduce the asymptotic behavior of the eigenvalues associated with (4)-(5), which is essential for proving their exact and approximate controllability results. On the other hand, apart from mentioned above, we show that a sequence of initial periodic data (constructed by hand) generates another sequence of solutions to (4)-(5), which in turn can be decomposed as the sum of three terms. The decomposition is closely linked to a Picard’s iteration of second order applied to sequence of solutions to (4)-(5). The result then follows by showing that the first and third terms of the expansion remain bounded in H−s p(0, 2π) ×H−s p(0, 2π)-norm while the second one can be arbitrarily large and eventually goes to infinity when times goes to zero. This paper is structured as follows. In Section 2, we state the well-posedness property of the linear version of (4)-(5). We also collect some useful results from the spectral analysis made in [2]. In Section 3, we analyze the Picard’s iteration method to solutions to (4)-(5). As a consequence, we prove Theorem 1.1, the main result of this work. Finally, the Appendix is dedicated to describing computations needed for intermediate steps in proving Theorem 1.1. 4G.J. Bautista, L. Potenciano-Machado / J. Math. Anal. Appl. 514 (2022) 126254 2. Preliminaries In this section we state some well-posedness results for both linear and nonlinear Boussinesq systems. A remarkable fact is the knowledge of the solution associated with the linear case, see Theorem 2.2. 2.1. Linear systems We first analyze the linearized version of (4)-(5), that is, we consider the following linear system ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ηt+wx−bηtxx =0 forx∈(0,2π),t>0, wt+ηx−dwtxx =0 forx∈(0,2π),t>0, η(t, 0) = η(t, 2π); ηx(t, 0) = ηx(t, 2π)fort>0, w(t, 0) = w(t, 2π); wx(t, 0) = wx(t, 2π)fort>0, η(0,x)=η0(x)forx∈(0,2π), w(0,x)=w0(x)forx∈(0,2π), (6) where b >0and d >0as in (3). Its well-posedness was derived from the spectral analysis done in [13] by using a Fourier approach. For the sake of completeness, we include such those results here. Firstly, we introduce a few notations. Given any v∈L2(0, 2π)and k∈Z, we denote by vkthe k-Fourier coefficient of v, defined by vk=1 2π 2π 0 v(x)e−ikx dx, and, for any m ∈N, we define the space Hm p(0,2π)=v∈L2(0,2π)v= k∈Zvkeikx, k∈Z |vk|2(1 + k2)m<∞, which is a Hilbert space with respect to the inner product (v,w)m= k∈Zvkwk(1 + k2)m.(7) The norm associated with (7)is denoted by || ||m. It can be seen that Hm p(0,2π)=v∈Hm(0,2π) ∂rv ∂xr(0) = ∂rv ∂xr(2π),0≤r≤m−1, where Hm(0, 2π) stands for the classical Sobolev space of index min the interval (0, 2π). We can extend the definition of Hm p(0, 2π)to the case m =s ≥0, a non-negative real number, by setting Hs p(0,2π)=v= k∈Zvkeikx ∈Hs(0,2π) k∈Z |vk|2(1 + k2)s<∞. For any nonnegative real number s, Hs p(0, 2π)can also be seen as a Hilbert space with respect to the inner product defined by (7)with mreplaced by s. In particular, for any v∈Hs p(0, 2π), v2 s= k∈Z |vk|2(1 + k2)s. G.J. Bautista, L. Potenciano-Machado / J. Math. Anal. Appl. 514 (2022) 126254 5 For s >0, we define the space H−s p(0, 2π)as the topological dual of Hs p(0, 2π): H−s p(0,2π)=Hs p(0,2π). Similarly, and for a given β∈R, one can define the homogeneous version of Hβ p(0, 2π)as ˙ Hβ p(0,2π)=v= k∈Zvkeikx ∈Hβ(0,2π) k∈Z |k|2β|vk|2<∞ with norm v2 ˙ Hβ p= k∈Z |k|2β|vk|2. On the other hand, for α>0, let ψα(Dx)be the Fourier multiplier operator given in terms of the Fourier transform by  ψα(Dx)u(k)= k 1+αk2u(k),D x:= −i∂x. Given β∈R, let us introduce the Hilbert space Vβ=Hβ p(0,2π)×Hβ p(0,2π), endowed with the inner product defined by f1 f2,g1 g2=b(f1,g 1)β+d(f2,g 2)β, where (·, ·)βdenotes the inner product given in (7)with mreplaced by β. One can see that system (6)can be rewritten in the following vectorial form i⎛ ⎜ ⎝ η w⎞ ⎟ ⎠t (t)+A⎛ ⎜ ⎝ η w⎞ ⎟ ⎠(t)=⎛ ⎜ ⎝ 0 0⎞ ⎟ ⎠,⎛ ⎜ ⎝ η w⎞ ⎟ ⎠(0) = ⎛ ⎜ ⎝ η0 w0⎞ ⎟ ⎠, where Ais a linear and compact operator in Vβ, see for instance [13], defined by A=⎛ ⎜ ⎝ 0ψb(Dx) ψd(Dx)0 ⎞ ⎟ ⎠.(8) Thus, if we assume that the initial data in (6)are given by η0 w0= k∈Zη0 k w0 keikx, then the solution of (6)can be formally written as η w(t, x)= k∈Zηk(t) wk(t)eikx, 6G.J. Bautista, L. Potenciano-Machado / J. Math. Anal. Appl. 514 (2022) 126254 where the pair (ηk(t), wk(t)) fulfill the following initial-value ODE with T>0 ⎧ ⎪ ⎨ ⎪ ⎩ (1 + bk2)(ηk)t+ik wk=0,t∈(0,T), (1 + dk2)( wk)t+ikηk=0,t∈(0,T), ηk(0) = η0 k,wk(0) = w0 k. (9) Then, we have the following result: Lemma 2.1. (see [13]) Let λ± k=±ikσ(k); σ(k)= 1 (1 + bk2)(1 + dk2),(k∈Z\{0}).(10) The solution (ηk(t), wk(t)) of (9)is given by ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ηk(t)=1 2η0 k+1+dk2 1+bk2w0 ke−λ+ kt+η0 k−1+dk2 1+bk2w0 ke−λ− kt , wk(t)=1 21+bk2 1+dk2η0 k+w0 ke−λ+ kt−1+bk2 1+dk2η0 k−w0 ke−λ− kt , (11) if k=0and η0(t)=η0 0, w0(t)= w0 0.(12) Thanks to Lemma 2.1, one can prove that the operator Agenerates an analytic group in Vβ. Theorem 2.1. (see [13]) The family of linear operators (S(t))t≥0defined by S(t)η0 w0= k∈Zηk(t) wk(t)eikx,η0 w0∈Vβ, where the coefficients ηk(t) wk(t)are given by (11)-(12), is a group of isometries in Vβ, for each β∈R. Moreover, its infinitesimal generator is the operator (D(A), A), where D(A) =Vβand Ais given by (8). From Theorem 2.1 and standard techniques from semigroup theory, we also have the following global well-posedness result: Theorem 2.2. (see [13]) Let T>0and β∈R. For each η0 w0∈Vβand f g∈L10,T;Vβ, there exists a unique solution (η, w) ∈W1,1[0,T]; Vβof the system η wt (t)+Aη w(t)=f g,η w(0) = η0 w0, which verifies the constant variation formula G.J. Bautista, L. Potenciano-Machado / J. Math. Anal. Appl. 514 (2022) 126254 7 η w(t)=S(t)η0 w0+ t 0 S(t−s)f g(s)ds. Moreover, if f g≡0 0it follows that η w∈C ω(R, Vβ), the class of analytic functions in t ∈Rwith values in Vβ. 2.2. Nonlinear systems Consider now the nonlinear Boussinesq system ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ηt+wx−bηtxx +(ηw)x=ffor x∈(0,2π),t>0, wt+ηx−dwtxx +wwx=gfor x∈(0,2π),t>0, η(t, 0) = η(t, 2π); ηx(t, 0) = ηx(t, 2π)fort>0, w(t, 0) = w(t, 2π); wx(t, 0) = wx(t, 2π)fort>0, η(0,x)=η0(x)forx∈(0,2π), w(0,x)=w0(x)forx∈(0,2π). As in the linear case, we can write it as iη wt (t)=Aη w(t)+Nη w(t)+iψb(f) ψd(g),η w(0) = η0 w0,(13) where N:Vs→Vsis the nonlinear operator defined by Nη w=ψb(Dx)(ηw) ψd(Dx)!w2 2".(14) We have the well-posedness result for the nonlinear system for s ≥0: Theorem 2.3. (see [13]) Assume that b >0and d >0. Let T>0and s ≥0be given. Then, there exists a constant M>0, depending on T, such that for any η0 w0∈Vsand any f g∈L1(0, T; Vs−2)satisfying η0 w0Vs ≤Mand f gL1(0,T ;Vs−2) ≤M, the system (13)-(14)admits a unique mild solution η w∈C([0, T]; Vs). The next section is then dedicated to proving ill-posedness for the nonlinear system in case of negative Sobolev indexes. 3. Picard’s iteration method and norm-inflation result Picard’s iteration method is a useful tool to prove, for instance, existence of solutions for differential equations. The method requires a starting point and later one makes an iterative procedure. It generates a 8G.J. Bautista, L. Potenciano-Machado / J. Math. Anal. Appl. 514 (2022) 126254 sequence of elements whose convergence (in a suitable Hilbert space) is the primary purpose of the method. If the sequence converges, then the limit is usually the desired solution to the underlying differential equation. We use a truncated version of this method up to second order. One can easily prove that the solution of (13)-(14), with the homogeneous source f g=0 0, can be written in (0, T) ×(0, 2π)with T>0, as follows η(t, x) w(t, x)=v(t, x) u(t, x)+ξ(t, x) ϕ(t, x)+y(t, x) z(t, x),(15) where v u(t)=S(t)η0 w0, ξ ϕ(t)= t 0 S(t−τ)NS(τ)η0 w0dτ, (16) and y z(t)= t 0 S(t−τ)# Ny z(τ)dτ. (17) The operator # Nis defined as # Ny z=ψb(Dx)(v(ϕ+z)+u(ξ+y)+(ϕ+z)(ξ+y)) ψd(Dx)1 2ϕ2+z2+2(uϕ +uz +ϕz) . As in [7], the functions sine and cosine will be involved in our construction. The explicit computations below shall be useful in our analysis. Remark 3.1. For k∈N, we have •S(t) cos(kx) 0=cos(kx +kσ(k)t)+cos(kx −kσ(k)t) 0, •S(t) sin(kx) 0=sin(kx +kσ(k)t)+sin(kx −kσ(k)t) 0, •S(t) 0 cos(kx)=0 cos(kx +kσ(k)t)+cos(kx −kσ(k)t), •S(t) 0 sin(kx)=0 sin(kx +kσ(k)t)+sin(kx −kσ(k)t), where (S(t))t≥0is the group defined in Theorem 2.1, and σ(k)is given by (10). Remark 3.2. A straightforward computation combined with Remark 3.1 yield t 0 S(t−τ)sin(kx −lτ) 0dτ =M1(t, x)+M2(t, x) 0, G.J. Bautista, L. Potenciano-Machado / J. Math. Anal. Appl. 514 (2022) 126254 15 N4(t, x) 0= t 0 S(t−τ)L4(τ,x) 0dτ =−ik2 1+b(2k2)2N1 4(t, x)+N2 4(t, x) 0, with N1 4(t, x)=cos (2k2x−2k2σ(k1)t)−cos (2k2x+2k2σ(2k2)t) k1σ(k2)+k2σ(2k2), N2 4(t, x)=cos (2k2x2k2σ(k2)t)−cos (2k2x+2k2σ(2k2)t) k2σ(k2)−k2σ(2k2), N5(t, x) 0= t 0 S(t−τ)L5(τ,x) 0dτ =−ik1 1+b(2k1)2N1 5(t, x) 0, with N1 5(t, x)=cos (2k1x−2k1σ(2k1)t)−cos (2k1x+2k1σ(2k1)t) 2k1σ(2k1), N6(t, x) 0= t 0 S(t−τ)L6(τ,x) 0dτ =i(k1−k2) 2(1+b(k1−k2)2)N1 6(t, x)+N2 6(t, x)−i(k1+k2) 2(1+b(k1+k2)2)N3 6(t, x)+N4 6(t, x) 0, with N1 6(t, x)=cos ((k1−k2)x+(k1−k2)σ(k1−k2)t)−cos ((k1−k2)x+(k1σ(k1)−k2σ(k2))t) k1σ(k1)−k2σ(k2)−(k1−k2)σ(k1−k2), N2 6(t, x)=cos ((k1−k2)x−(k1−k2)σ(k1−k2)t)−cos ((k1−k2)x+(k1σ(k1)−k2σ(k2))t) k1σ(k1)−k2σ(k2)+(k1−k2)σ(k1−k2), N3 6(t, x)=cos ((k1+k2)x+(k1+k2)σ(k1+k2)t)−cos ((k1+k2)x+(k1σ(k1)+k2σ(k2))t) k1σ(k1)+k2σ(k2)−(k1+k2)σ(k1+k2), N4 6(t, x)=cos ((k1+k2)x−(k1+k2)σ(k1+k2)t)−cos ((k1+k2)x+(k1σ(k1)+k2σ(k2))t) k1σ(k1)+k2σ(k2)+(k1+k2)σ(k1+k2), N7(t, x) 0= t 0 S(t−τ)L7(τ,x) 0dτ =i(k1−k2) 2(1+b(k1−k2)2)N1 7(t, x)+N2 7(t, x)−i(k1+k2) 2(1+b(k1+k2)2)N3 7(t, x)+N4 7(t, x) 0, with N1 7(t, x)=cos ((k1−k2)x+(k1−k2)σ(k1−k2)t)−cos ((k1−k2)x+(k1σ(k1)+k2σ(k2))t) k1σ(k1)+k2σ(k2)−(k1−k2)σ(k1−k2), N2 7(t, x)=cos ((k1−k2)x−(k1−k2)σ(k1−k2)t)−cos ((k1−k2)x+(k1σ(k1)+k2σ(k2))t) k1σ(k1)+k2σ(k2)+(k1−k2)σ(k1−k2), N3 7(t, x)=cos ((k1+k2)x+(k1+k2)σ(k1+k2)t)−cos ((k1+k2)x+(k1σ(k1)−k2σ(k2))t) k1σ(k1)−k2σ(k2)−(k1+k2)σ(k1+k2), 16 G.J. Bautista, L. Potenciano-Machado / J. Math. Anal. Appl. 514 (2022) 126254 N4 7(t, x)=cos ((k1+k2)x−(k1+k2)σ(k1+k2)t)−cos ((k1+k2)x+(k1σ(k1)−k2σ(k2))t) k1σ(k1)−k2σ(k2)+(k1+k2)σ(k1+k2), N8(t, x) 0= t 0 S(t−τ)L8(τ,x) 0dτ =i(k1−k2) 2(1+b(k1−k2)2)N1 8(t, x)+N2 8(t, x)−i(k1+k2) 2(1+b(k1+k2)2)N3 8(t, x)+N4 8(t, x) 0, with N1 8(t, x)=cos ((k1−k2)x−(k1σ(k1)+k2σ(k2))t)−cos ((k1−k2)x+(k1−k2)σ(k1−k2)t) k1σ(k1)+k2σ(k2)+(k1−k2)σ(k1−k2), N2 8(t, x)=cos ((k1−k2)x−(k1σ(k1)+k2σ(k2))t)−cos ((k1−k2)x−(k1−k2)σ(k1−k2)t) k1σ(k1)+k2σ(k2)−(k1−k2)σ(k1−k2), N3 8(t, x)=cos ((k1+k2)x−(k1σ(k1)−k2σ(k2))t)−cos ((k1+k2)x+(k1+k2)σ(k1+k2)t) k1σ(k1)−k2σ(k2)+(k1+k2)σ(k1+k2), N4 8(t, x)=cos ((k1+k2)x−(k1σ(k1)−k2σ(k2))t)−cos ((k1+k2)x−(k1+k2)σ(k1+k2)t) k1σ(k1)−k2σ(k2)−(k1+k2)σ(k1+k2), N9(t, x) 0= t 0 S(t−τ)L9(τ,x) 0dτ =i(k1−k2) 2(1+b(k1−k2)2)N1 9(t, x)+N2 9(t, x)−i(k1+k2) 2(1+b(k1+k2)2)N3 9(t, x)+N4 9(t, x) 0, with N1 9(t, x)=cos ((k1−k2)x−(k1σ(k1)−k2σ(k2))t)−cos ((k1−k2)x+(k1−k2)σ(k1−k2)t) k1σ(k1)−k2σ(k2)+(k1−k2)σ(k1−k2), N2 9(t, x)=cos ((k1−k2)x−(k1σ(k1)−k2σ(k2))t)−cos ((k1−k2)x−(k1−k2)σ(k1−k2)t) k1σ(k1)−k2σ(k2)−(k1−k2)σ(k1−k2), N3 9(t, x)=cos ((k1+k2)x−(k1σ(k1)+k2σ(k2))t)−cos ((k1+k2)x+(k1+k2)σ(k1+k2)t) k1σ(k1)+k2σ(k2)+(k1+k2)σ(k1+k2), N4 9(t, x)=cos ((k1+k2)x−(k1σ(k1)+k2σ(k2))t)−cos ((k1+k2)x−(k1+k2)σ(k1+k2)t) k1σ(k1)+k2σ(k2)−(k1+k2)σ(k1+k2), N10(t, x) 0= t 0 S(t−τ)L10(τ,x) 0dτ =−ik2 1+b(2k2)2N1 10(t, x) 0, with N1 10(t, x)=cos (2k2x−2k2σ(2k2)t)−cos (2k2x+2k2σ(2k2)t) 2k2σ(2k2). Thanks to the real mean value theorem applied to the cosine function, we obtain  cos(kx −ω1t)−cos(kx −ω2t) ω1−ω2≤t, ω1=ω2,t≥0. G.J. Bautista, L. Potenciano-Machado / J. Math. Anal. Appl. 514 (2022) 126254 17 From all the above identities, we get t 0 S(t−τ)ψb!η0w0" 0(τ,x)dτ = t 0 S(t−τ)⎛ ⎜ ⎝ 10  l=1 Ll(τ,x) 0 ⎞ ⎟ ⎠dτ =⎛ ⎜ ⎝ 10  l=1 Nl(t, x) 0 ⎞ ⎟ ⎠, (23) where Nl(t, x)∼ik−1 1t, 1≤l≤5; l=10, i 1+bt−ik−1 1t, 6≤l≤9. Proceeding in a similar way, we obtain t 0 S(t−τ)⎛ ⎝0 1 2ψd%!w0"2&⎞ ⎠(τ,x)dτ =⎛ ⎜ ⎝ 0 10  l=1 Rl(t, x)⎞ ⎟ ⎠,(24) where Rl(t, x)∼ik−1 1t, 1≤l≤5; l=10, i 1+dt−ik−1 1t, 6≤l≤9. Combining (22), (23)and (24), ξ ϕcan be written as ξ ϕ(t, x)=k2γ 1 t 0 S(t−τ)NS(τ)η0 w0dτ =k2γ 1⎡ ⎣ t 0 S(t−τ)ψb!η0w0" 0(τ,x)dτ + t 0 S(t−τ)⎛ ⎝0 1 2ψd%!w0"2&⎞ ⎠(τ,x)dτ⎤ ⎦ =⎛ ⎜ ⎜ ⎜ ⎜ ⎝ k2γ 1 10  l=1 Nl(t, x) k2γ 1 10  l=1 Rl(t, x) ⎞ ⎟ ⎟ ⎟ ⎟ ⎠, where now k2γ 1 10  l=1 Nl(t, x)∼ik2γ 1t, k2γ 1 10  l=1 Rl(t, x)∼ik2γ 1t. Therefore, one gets the following estimate for the second term in Picard’s iteration ξ ϕ(t, ·)V−s ∼k2γ 1t. (25) 18 G.J. Bautista, L. Potenciano-Machado / J. Math. Anal. Appl. 514 (2022) 126254 Acknowledgments G. J. B was partially supported by the Universidad Privada del Norte, Lima-Perú. L. P-M thanks the Department of Mathematics and Statistics of the University of Jyväskylä, Finland, for providing an excellent environment to prepare this manuscript. References [1] D.M. Ambrose, J.L. Bona, T. Milgrom, Global solutions and ill-posedness for the Kaup system and related Boussinesq systems, Indiana Univ. Math. J. 68 (4) (2019) 1173–1198. [2] G.J. Bautista, A.F. Pazoto, Decay of solutions for a dissipative higher-order Boussinesq system on a periodic domain, Commun. Pure Appl. Anal. 19 (2020) 747–769. [3] G.J. Bautista, A.F. Pazoto, On the controllability of a Boussinesq system for two-way propagation of dispersive waves, J. Evol. Equ. 20 (2020) 607–630. [4] G.J. 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