\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2015 (2015), No. 141, pp. 1--24.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2015 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2015/141\hfil Cauchy problems for fifth-order KdV equations] {Cauchy problems for fifth-order KdV equations in weighted Sobolev spaces} \author[E. Bustamante, J. Jim\'enez, J. Mej\'ia \hfil EJDE-2015/141\hfilneg] {Eddye Bustamante, Jos\'e Jim\'enez, Jorge Mej\'ia } \address{Eddye Bustamante M. \newline Departamento de Matem\'aticas, Universidad Nacional de Colombia, A. A. 3840 Medell\'in, Colombia} \email{eabusta0@unal.edu.co} \address{Jos\'e Jim\'enez U. \newline Departamento de Matem\'aticas, Universidad Nacional de Colombia, A. A. 3840 Medell\'in, Colombia} \email{jmjimene@unal.edu.co} \address{Jorge Mej\'ia L. \newline Departamento de Matem\'aticas, Universidad Nacional de Colombia, A. A. 3840 Medell\'in, Colombia} \email{jemejia@unal.edu.co} \thanks{Submitted March 29, 2015. Published May 21, 2015.} \subjclass[2010]{35Q53, 37K05} \keywords{Nonlinear dispersive equations; weighted Sobolev spaces} \begin{abstract} In this work we study the initial-value problem for the fifth-order Korteweg-de Vries equation $$ \partial_{t}u+\partial_{x}^{5}u+u^k\partial_{x}u=0, \quad x,t\in \mathbb{R}, \; k=1,2, $$ in weighted Sobolev spaces $H^s(\mathbb{R})\cap L^2(\langle x \rangle^{2r}dx)$. We prove local and global results. For the case $k=2$ we point out the relationship between decay and regularity of solutions of the initial-value problem. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction}\label{int} In this article we consider the initial-value problem (IVP) \begin{equation} \begin{gathered} \partial_{t}u+\partial_{x}^{5}u+u^k\partial_{x}u=0, \quad x,t\in \mathbb{R}\\ u(0)=u_0, \end{gathered} \label{KdV} \end{equation} with $k=1,2$. When $k=1$ we refer to this problem as the IVP for the fifth-order Korteweg-de Vries (KdV) equation. When $k=2$ we refer to this problem as the IVP for the modified fifth-order KdV equation. For $k=1$ the equation was proposed by Kakutani and Ono as a model for magneto-acoustic waves in plasma physics (see \cite{KO}). The equations that we study are included in the class \begin{align}\label{horder} \partial_t u+\partial_x^{2j+1} u+P(u,\partial_x u,\dots, \partial_x^{2j}u)=0,\quad x,t\in\mathbb{R},\quad j\in \mathbb Z^+, \end{align} where $P:\mathbb{R}^{2j+1}\to\mathbb{R}$ (or $P: \mathbb C^{2j+1}\to\mathbb C$) is a polynomial having no constant or linear terms, i.e. $$ P(z)=\sum_{|\alpha|=l_0}^{l_1} a_\alpha z^\alpha\quad \text{with $l_0\geq 2$ and $z=(z_1,\dots, z_{2j+1})$}. $$ The class in \eqref{horder} generalizes several models, arising in both mathematics and physics, of higher-order nonlinear dispersive equations. For many years the well-posedness of these IVP has been studied in the context of the classical Sobolev spaces $H^s(\mathbb{R})$. In particular, fifth-order KdV equations with more general non-linearities, than those we are considering, were studied in \cite{GKK,KP, Kw,Kw1,L,P}. In 1983 Kato \cite{K} studied the IVP for the generalized KdV equation in several spaces, besides the classical Sobolev spaces. Among them, Kato considered weighted Sobolev spaces. In this work we are concerned with the well-posedness of \eqref{KdV} in weighted Sobolev spaces. This type of spaces arises in a natural manner when we are interested in determining if the Schwartz space is preserved by the flow of the evolution equation in \eqref{KdV}. Kenig, Ponce and Vega \cite{KPVHigher} studied the IVP associated with equation \eqref{horder} in weighted Sobolev spaces $H^s(\mathbb{R})\cap L^2(|x|^m dx)$, with $m$ positive integer. Pilod \cite{Pi} study the case of higher-order dispersive models in the context of weighted Besov and Sobolev spaces. Some relevant nonlinear evolution equations as the KdV equation, the non-linear Schr\"odinger equation and the Benjamin-Ono equation, have also been studied in the context of weighted Sobolev Spaces (see \cite{CP,FLP,FLP1,FLP2,FP,IO1,IO,J,N,NP1,NP2} and references therein). We study real valued solutions of \eqref{KdV} in the weighted Sobolev spaces $$ Z_{s,r}:=H^s(\mathbb{R})\cap L^2(\langle x\rangle^{2r} dx), $$ where $\langle x \rangle:=(1+x^2)^{1/2}$, and $s,r\in\mathbb{R}$. The relation between the indices $s$ and $r$ for \eqref{KdV} can be found, after the following considerations, contained in the work by Kato: Suppose we have a solution $u\in C([0,\infty); H^s(\mathbb{R}))$ to \eqref{KdV} for some $s\geq 2$. We want to estimate $(pu,u)$, where $p(x):=\langle x\rangle^{2r}$ and $(\cdot,\cdot)$ is the inner product in $L^2(\mathbb{R})$. Proceeding formally we multiply the equation in \eqref{KdV} by $up$, integrate over $x\in\mathbb{R}$ and apply integration by parts to obtain \begin{equation} \frac{d}{dt} (pu,u)=5(p^{(1)} \partial_x^2 u, \partial_x^2 u) -5(p^{(3)}\partial_x u,\partial_x u)+(p^{(5)}u,u) +\frac 2{k+2}(p^{(1)} u^{k+2},1).\label{p} \end{equation} To see that $(pu,u)$ is finite and bounded in $t$, we must bound the right-hand side in \eqref{p} in terms of $(pu,u)$ and $\| u\|_{H^s}^2$. The most difficult term to control in the right-hand side in \eqref{p} is $5(p^{(1)}\partial_x^2 u,\partial_x^2 u)$. Using the interpolation Lemma \ref{uno} (see section 2), for $\theta\in [0,1]$ and $u\in Z_{s,r}$ we have $$ \| \langle x \rangle^{(1-\theta)r} u \|_{H^{\theta s}} \leq C \| \langle x \rangle^r u \|_{L^2}^{1-\theta}\|u \|_{H^s}^\theta. $$ The term $5(p^{(1)}\partial_x^2 u,\partial_x^2 u)$ can be controlled when $\theta s=2$ if $p^{(1)}(x)\sim \langle x \rangle^{2(1-\theta)r}$. Taking into account that $p^{(1)}(x)\sim \langle x\rangle^{2r-1}$, we must require that $2r-1=2(1-\theta)r$ and $\theta s=2$, which leads to $s=4r$. In this way the natural weighted Sobolev space to study \eqref{KdV} is $Z_{4r,r}$. Now, we describe the main results of this work. With respect to \eqref{KdV} with $k=1$ we establish local well-posedness (LWP) in $Z_{4r,r}$ for $\frac 5{16}0$ such that in two different times $t_0,t_1\in\mathbb{R}$ $$ |x|^\alpha u(t_0), |x|^\alpha u(t_1)\in L^2(\mathbb{R}), $$ then $u\in C(\mathbb{R},H^{2\alpha}(\mathbb{R}))$. To achieve this goal, they chose a functional setting, where the norm $\|\partial_x u \|_{L^\infty(\mathbb{R}; L^2([0,T]))}$ of the solution $u$ depends continuously on the initial data in $L^2(\mathbb{R})$. Following \cite{ILP}, and taking into account that the norm $\|\partial_x^4 u \|_{L^\infty(\mathbb{R}; L^2([0,T]))}$ of the solution $u$ of \eqref{KdV} with $k=2$, depends continuously on the initial data in $Z_{2,1/2}$, we prove that if $u\in C([0,T];Z_{2,1/2})$ is a solution of \eqref{KdV} with $k=2$ and, for some $\alpha>0$, there exist two different times $t_0,t_1\in[0,T]$ such that $|x|^{1/2+\alpha} u(t_0)$ and $|x|^{1/2+\alpha} u(t_1)$ are in $L^2(\mathbb{R})$ then $u\in C([0,T];H^{2+4\alpha}(\mathbb{R}))$. Before stating in a precise manner the main results of this article, let us explain the notation for mixed space-time norms. For $f:\mathbb{R}\times [0,T]\to\mathbb{R}$ (or $\mathbb C$) we have $$ \|f \|_{L^p_x L^q_T}:=\Big( \int_{\mathbb{R}} \Big( \int_0^T |f(x,t)|^q dt\Big)^{p/q} dx \Big)^{1/p}. $$ When $p=\infty$ or $q=\infty$ we must do the obvious changes with the essential supremum. When in the space-time norm appears $t$ instead of $T$, the time interval is $[0,+\infty)$. Our results read as follows: \begin{theorem}\label{prin1} Let $r>\frac{5}{16}$ and $u_0\in Z_{4r,r}$. Then there exist $T>0$ and a unique $u$, solution of \eqref{KdV} with $k=1$ such that \begin{gather} u\in C([0,T];Z_{4r,r}),\label{con1}\\ \| \partial_x u \|_{L^4_T L^\infty_x}<\infty,\label{con2}\\ \|D_x^{4r}\partial_x u\|_{L_x^{\infty}L_T^2}<\infty,\quad\text{and}\label{con3}\\ \|u\|_{L_x^2L_T^{\infty}}<\infty.\label{H1} \end{gather} Moreover, for any $T'\in(0,T)$ there exists a neighborhood $V$ of $u_0$ in $Z_{4r,r}$ such that the data-solution map $\tilde{u}_0\mapsto\tilde{u}$ from $V$ into the class defined by \eqref{con1}-\eqref{H1} with $T'$ instead of $T$ is Lipschitz. When $5/160$ and a unique $u$, solution of \eqref{KdV} for the modified fifth-order KdV equation ($k=2$), such that \begin{gather} u\in C([0,T];Z_{2,1/2})\,,\label{mdes11}\\ \| \partial^4_x u\|_{L^\infty_x L^2_T}<\infty\,, \label{mdes12}\\ \| u\|_{L^{16/5}_x L^\infty_T}<\infty\,, \label{mdes13}\\ \| u\|_{L^4_x L^\infty_T}<\infty. \label{mdes14} \end{gather} Moreover, for any $T'\in(0,T)$ there exists a neighborhood $V$ of $u_0$ in $Z_{2,1/2}$ such that the data-solution map $\tilde{u}_0\mapsto\tilde{u}$ from $V$ into the class defined by \eqref{mdes11}-\eqref{mdes14} with $T'$ instead of $T$ is Lipschitz. \end{theorem} \begin{theorem}\label{tmkdv2} The initial-value problem \eqref{KdV} for the modified fifth-order KdV equation ($k=2$) is globally well-posed in $Z_{2,1/2}$. \end{theorem} \begin{theorem} \label{tmkdv3} For $T>0$ let $u\in C([0,T];Z_{2,1/2})$ be the solution of the modified fifth-order KdV equation ($k=2$), obtained in Theorems \ref{tmkdv1} and \ref{tmkdv2}. Let us suppose that for $\alpha>0$ there exist two different times $t_0,t_1\in [0,T]$, with $t_0\frac 34$ and $s>\frac 54$ there exists $C$ such that \begin{equation} \| W(t)u_0\|_{L^2_x L^\infty_T}\leq C(1+T)^\rho \|u_0 \|_{H^s}.\label{le4} \end{equation} \item[(v)] There exists a constant $C$ such that, for $u_0\in H^{1/4}(\mathbb{R})$ (see \cite{KR}), \begin{equation} \| W(t)u_0\|_{L^4_x L^\infty_T}\leq C\| D^{1/4} u_0\|_{L^2}. \label{le5} \end{equation} By interpolation it follows, from \eqref{le4} and \eqref{le5}, that for $\rho>\frac 34$ and $s>\frac 54$, \begin{equation} \| W(t)u_0\|_{L^{16/5}_x L^\infty_T} \leq C(1+T)^\rho \|u_0 \|_{H^s}.\label{le6m} \end{equation} \item[(vi)] There exists a constant $C$ such that \begin{equation} \| D_x^{3/4} W(t)u_0 \|_{L^4_t L^\infty_x} \leq C\| u_0\|_{L^2}. \label{le6} \end{equation} \end{itemize} Using \eqref{le1}, \eqref{le4} and \eqref{le5}, and proceeding as in the proofs of \cite[Theorem 2.1]{KPVWell} and \cite[Theorem 1.1]{KPVOsc}, it can be established the following theorem. \begin{theorem}\label{principal} Let $s>5/4$. Then for any $u_0\in H^s(\mathbb{R})$ there exist a positive value $T=T(\|u_0\|_{H^s})$ (with $T(\rho)\to\infty$ as $\rho\to 0$) and a unique solution $u$ of \eqref{KdV} with $k=1$, satisfying \begin{gather} u\in C([0,T];H^s(\mathbb{R}))\,,\label{con}\\ \| \partial_xu \|_{L^4_T L^\infty_x}<\infty\,,\\ \|D_x^s \partial_x u \|_{L_x^{\infty}L_T^2}<\infty\,,\label{g1d}\\ \|u\|_{L_x^2L_T^{\infty}}<\infty\,.\label{H} \end{gather} Moreover, for any $T'\in(0,T)$ there exists a neighborhood $V$ of $u_0$ in $H^s(\mathbb{R})$ such that the data-solution map $\tilde{u_0}\mapsto\tilde{u}$ from $V$ into the class defined by \eqref{con}-\eqref{H} with $T'$ instead of $T$ is Lipschitz. Also, if $u_0\in H^{s'}$ with $s'>s$ then the above results hold with $s'$ instead of $s$ in the same time interval $[0,T]$ (regularity property). \end{theorem} Let us observe the gain of two derivatives in $x$ in the linear estimate \eqref{le1}. However, the condition \eqref{g1d} only uses the gain of one derivative in $x$. One of the main tools for establishing local well-posedness of \eqref{KdV} with $k=1$ in weighted Sobolev spaces with low regularity is the following pointwise formula, proved by Fonseca, Linares, and Ponce in \cite{FLP}: \begin{itemize} \item[(vii)] For $r\in(0,1)$ and $u_0\in Z_{4r,r}$ we have for all $t\in\mathbb{R}$ and for almost every $x\in \mathbb{R}$: \begin{equation} |x|^r [W(t)u_0](x)=W(t) (|x|^r u_0)(x)+W(t)\{\Phi_{t,r} (\widehat {u_0}) \} {}^\vee (x), \label{FW} \end{equation} where \begin{equation} \| (\Phi_{t,r} (\widehat{u_0})(\xi)) {}^\vee \|_{L^2} \leq C_r (1+|t|) (\| u_0\|_{L^2}+\|D_x^{4r} u_0\|_{L^2}). \label{FW1} \end{equation} \end{itemize} With respect to the weight $\langle x\rangle:=(1+x^2)^{1/2}$, for $N\in\mathbb N$, we will consider a truncated weight $w_N$ of $\langle x \rangle$, such that $w_N\in C^\infty(\mathbb{R})$, \begin{equation} w_N(x)= \begin{cases} \langle x\rangle &\text{if } |x|\leq N, \\ 2N &\text{if } |x|\geq 3N, \end{cases} \label{peso} \end{equation} The function $w_N$ is non-decreasing in $|x|$ and for $j\in\mathbb{N}$ and $x\in\mathbb{R}$, the derivatives $w_N^{(j)}$ of order $j$ of $w_N$ satisfy \begin{equation} |w_N^{(j)}(x)|\leq\frac{c_j}{w_N^{j-1}(x)}\,,\label{cota} \end{equation} where the constant $c_j$ is independent from $N$. Fonseca and Ponce \cite{FP} deduced the following interpolation inequality, related to the weights $\langle x \rangle$ and $w_N$. \begin{lemma}\label{uno} Let $a,b>0$ and $f\in Z_{a,b}\equiv H^a (\mathbb{R})\cap L^2(\langle x\rangle^{2b }dx)$. Then for any $\theta\in(0,1)$ \begin{equation} \|J^{\theta a}(\langle x\rangle^{(1-\theta)b}f)\|_{L^2} \leq C\|\langle x\rangle^bf\|_{L^2}^{1-\theta}\|J^af\|_{L^2}^{\theta}, \label{inter} \end{equation} where $J^a f:=(1-\partial_x^2) ^{a/2} f$. Moreover, inequality \eqref{inter} is still valid with $w_N(x)$ as in \eqref{peso} instead of $\langle x\rangle$ with a constant $C$ independent of $N$. \end{lemma} Finally, in our arguments we will use the following standard estimate, concerning the weights $\langle x\rangle$ and $w_N$. \begin{lemma}\label{dos} Let $b>0$ and $n\in\mathbb{N}$. Suppose that $J^n(\langle x\rangle^{b}u_0)\in L^2(\mathbb{R})$. Then \begin{equation} \|\langle x\rangle^{b}\partial_x^nu_0\|_{L^2} \leq C(b,n)\|J^n(\langle x\rangle^{b}u_0)\|_{L^2}\,.\label{leibniz} \end{equation} Moreover, the inequality \eqref{leibniz} is still valid with $w_N(x)$ as in \eqref{peso} instead of $\langle x\rangle$ with a constant $C(b,n)$ independent of $N$. \end{lemma} The proof of the above lemma follows by induction on $n$ and the Leibniz formula. \section{Well-posedness of \eqref{KdV} with $k=1$} \subsection{Proof of Theorem \ref{prin1}} We consider two cases. \noindent\textbf{Case: $5/16 \frac 34$}.\label{N4} \end{gather} Additionally, we introduce \begin{equation} \lambda_5^T(u):=\| |x|^r u\|_{L^\infty_T L^2_x}.\label{N5} \end{equation} Let us consider \begin{gather} \Lambda^T(u):=\max_{1\leq j\leq 5} \lambda_j^T(u), \label{N} \\ X_T:=\{u\in C([0,T];H^{4r}(\mathbb{R})):\Lambda^T(u)<\infty \}.\label{XT} \end{gather} Using the linear estimates \eqref{le6}, \eqref{le1} and \eqref{le4}, Kenig, Ponce and Vega \cite{KPVWell}, showed that for $u_0\in H^{4r} (\mathbb{R})$, $T>0$ and $1\leq i\leq 4$, \begin{align} \lambda_i^T (W(t)u_0)\leq C \| u_0\|_{H^{4r}}.\label{Gi} \end{align} On the other hand, from \eqref{FW} and \eqref{FW1}, it follows that, for $t\in[0,T]$, \begin{equation} \lambda_5^T (W(t)u_0)\leq \| |x|^r u_0\|_{L^2} +C_r (1+T)(\| u_0\|_{L^2}+\| D_x^{4r}u_0 \|_{L^2}).\label{G5} \end{equation} In consequence, for $u_0\in Z_{4r,r}$, the estimates \eqref{Gi} and \eqref{G5} imply that \begin{equation} \Lambda^T(W(t)u_0)\leq \| |x|^r u_0 \|_{L^2}+C(1+T) \| u_0\|_{H^{4r}}.\label{LG} \end{equation} Let us denote by $u:=\Phi (v)\equiv \Phi_{u_0}(v)$ the solution of the linear inhomogeneous IVP \begin{equation} \begin{gathered} \partial_{t}u+\partial_{x}^{5}u+v\partial_{x}v=0, \\ u(0)=u_0, \end{gathered}\label{KdV1} \end{equation} where $v\in X_T^a:=\{ w\in X_T: \Lambda^T(w)\leq a\}$, for $a>0$. By Duhamel's formula: \[ \Phi(v)(t)\equiv u(t)=W(t)u_0-\int_0^t W(t-t')(v\partial_x v)(t') dt'. \label{DF} \] Taking into account that $$ \Lambda^T(u)\leq \Lambda^T(W(t)u_0) +\int_0^T \Lambda^T(W(t-t')(v\partial_x v(t'))) dt', $$ from \eqref{LG} it follows that \begin{equation} \begin{aligned} \Lambda^T(u)&\leq \| |x|^r u_0\|_{L^2}+C(1+T) \| u_0\|_{H^{4r}} +C(1+T)(\| v\partial_x v \|_{L^1_T L^2_x}\\ &\quad + \| D_x^{4r} (v\partial_x v)\|_{L^1_T L^2_x}) +\| |x|^r v\partial_x v\|_{L^1_T L^2_x}. \end{aligned} \label{LDF} \end{equation} In \cite{KPVWell1} (see proof of Lemma 4.1) it was proved that \begin{equation} \begin{aligned} &\| v\partial_x v\|_{L^1_T L^2_x}+\| D^{4r}_x (v\partial_x v)\|_{L^1_T L^2_x}\\ &\leq CT^{1/2} (1+T)^\rho \lambda_4^T (v)\lambda_3^T(v)+CT^{3/4}\lambda_2^T(v) \lambda_1^T(v) +CT(\lambda_1^T(v))^2\\ &\leq C(T^{1/2}(1+T)^\rho+T^{3/4}+T)(\Lambda^T(v))^2,\label{LDF1} \end{aligned} \end{equation} and let us observe that \begin{equation} \begin{aligned} \| |x|^r v\partial_x v\|_{L^1_T L^2_x} &\leq CT^{3/4} \| |x|^r v\partial_x v\|_{L^4_T L^2_x}\\ &\leq CT^{3/4} \| |x|^r v\|_{L^\infty_T L^2_x} \| \partial_x v \|_{L^4_T L^\infty_x} \\ &\leq CT^{3/4} \lambda_5^T (v)\lambda_2^T(v) \leq C T^{3/4}(\Lambda^T(v))^2. \end{aligned} \label{LDF2} \end{equation} From \eqref{LDF}-\eqref{LDF2} it follows that $$ \Lambda^T(u)\leq \| |x|^r u_0\|_{L^2} +C (1+T) \|u_0 \|_{H^{4r}}+C(1+T)(T^{1/2}(1+T)^\rho+T^{3/4}+T)(\Lambda^T(v))^2. $$ Taking $a:=2(\| |x|^r u_0\|_{L^2} +C (1+T) \|u_0 \|_{H^{4r}})$ and $T$ sufficiently small in order to have $$ C(1+T)(T^{1/2}(1+T)^\rho+T^{3/4}+T)a<\frac 12, $$ it can be seen that $\Phi: X_T^a \to X_T^a$. Reasoning as in \cite{KPVWell} (proof of Theorem 2.1), for $T>0$ small enough, $\Phi: X_T^a \to X_T^a$ is a contraction. In consequence, there exists a unique $u\in X_T^a$ such that $\Phi(u)=u$. In other words, for $t\in[0,T]$: $$ u(t)=W(t)u_0-\int_0^t W(t-t')(u\partial_x u)(t') dt'. $$ To conclude the proof of this case we reason in the same manner as it was done at the end of the proof of \cite[Theorem 2.1]{KPVWell}. \smallskip \noindent\textbf{Case: $r\geq1/2$.} By Theorem \ref{principal} there exist $T=T(\|u_0 \|_{H^{4r}})$ and a unique $u$ in the class defined by the conditions \eqref{con}-\eqref{H} with $s=4r$, which is a solution of \eqref{KdV} with $k=1$. Let $\{u_{0m}\}_{m\in\mathbb N}$ be a sequence in $C_0^\infty(\mathbb{R})$ such that $u_{0m}\to u_0$ in $H^{4r}(\mathbb{R})$ and let $u_m\in C([0,T];H^\infty(\mathbb{R}))$ be a solution of the equation in \eqref{KdV} corresponding to the initial data $u_{0m}$. (Without loss of generality we can suppose that $u_m$ is defined in the same interval $[0,T]$ (see regularity property in Theorem \ref{principal})). By Theorem \ref{principal} $u_m\to u$ in $C([0,T];H^{4r}(\mathbb{R}))$. We multiply the equation \begin{equation} \partial_t u_m +\partial_x^5 u_m +u_m\partial_x u_m=0\label{ecnm} \end{equation} by $u_m w_N^{2r}$, where $w_N$ is the truncated weight defined in \eqref{peso}, and for a fixed $t\in[0,T]$, we integrate in $\mathbb{R}$ with respect to $x$ and use integration by parts to obtain \begin{equation} \begin{aligned} &\frac d{dt} (u_m(t),u_m(t)w_N^{2r})\\ &= 5(\partial_x^2u_m(t),\partial_x^2u_m(t)(w_N^{2r})^{(1)}) -5(\partial_x u_m(t),\partial_x u_m(t)(w_N^{2r})^{(3)}) \\ &\quad +(u_m(t),u_m(t)(w_N^{2r})^{(5)}) +\frac 23 (1,u_m(t)^3(w_N^{2r})^{(1)}), \end{aligned} \label{derum} \end{equation} where $(\cdot,\cdot)$ denotes the inner product in $L^2(\mathbb{R})$. Integrating the above equation with respect to the time variable in the interval $[0,t]$, we have \begin{equation} \begin{aligned} &(u_m(t),u_m(t)w_N^{2r})\\ &= (u_{0m},u_{0m}w_N^{2r})+5\int_0^t (\partial_x^2 u_m(t'), \partial_x^2 u_m(t')(w_N^{2r})^{(1)})dt' \\ &\quad -5\int_0^t (\partial_xu_m(t'),\partial_x u_m (t')(w_N^{2r})^{(3)})dt' +\int_0^t (u_m(t'),u_m(t')(w_N^{2r})^{(5)})dt' \\ &\quad +\frac 23 \int_0^t (1,u_m(t')^3(w_N^{2r})^{(1)})dt'. \end{aligned} \label{intum} \end{equation} Since $u_m\to u$ in $C([0,T]; H^{4r}(\mathbb{R}))$, with $4r\geq 2$, and the weight $w_N^{2r}$ and its derivatives are bounded functions, it follows from equality \eqref{intum}, after passing to the limit when $m\to\infty$, that \begin{equation} \begin{aligned} &(u(t),u(t)w_N^{2r})\\ &=(u_{0},u_{0}w_N^{2r})+5\int_0^t (\partial_x^2 u(t'),\partial_x^2 u(t') (w_N^{2r})^{(1)})dt'\\ &\quad -5\int_0^t (\partial_xu(t'),\partial_x u (t')(w_N^{2r})^{(3)})dt'\\ &\quad +\int_0^t (u(t'),u(t')(w_N^{2r})^{(5)})dt' +\frac 23 \int_0^t (1,u(t')^3(w_N^{2r})^{(1)}) dt' \\ &\equiv I+II+III+IV+V. \end{aligned}\label{intu} \end{equation} Let us estimate the terms on the right-hand side of \eqref{intu}. First of all \begin{equation} I\leq \| u_0 \|_{L^2(\langle x \rangle^{2r}dx)}^2.\label{cotI} \end{equation} With respect to the term $II$, using Lemmas \ref{dos} and \ref{uno}, we have \begin{align} |II| &\leq 10r \int_0^t (\partial_x^2 u(t'),\partial_x^2 u(t') w_N^{2r-1} |(w_N)^{(1)}|)dt' \\ &\leq C \int_0^t (\partial_x^2 u(t'),\partial_x^2 u(t') w_N^{2r-1})dt' \leq C \int_0^t \|J^2( w_N ^{r-\frac 12} u(t')) \|_{L^2}^2 dt' \\ &\leq C \int_0^t \|J^{4r} u(t') \|_{L^2}^{1/r} \| w_N^r u(t') \|_{L^2}^{2-1/r} dt' \leq C \int_0^t \|w_N^r u(t')\|_{L^2}^{2-1/r}dt' \\ &\leq C\int_0^t (1+\|w_N^r u(t') \|_{L^2}^2) dt' \leq Ct +C\int_0^t (u(t'),u(t')w_N^{2r})dt'. \label{cotII} \end{align} Using inequality \eqref{cota} for the derivatives of $w_N$ it can be seen that \begin{equation} |(w_N^{2r})^{(3)}|\leq Cw_N^{2r-3}\quad\text{and}\quad |(w_N^{2r})^{(5)}|\leq Cw_N^{2r-5}\,. \end{equation} In this manner we can bound the term III as follows: \begin{equation} |III|\leq C\int_0^t (\partial_xu(t'),\partial_x u (t')w_N^{2r-3})dt'\,. \end{equation} If $2r-3\leq 0$, since $u\in C([0,T];H^{4r}(\mathbb{R}))$ with $4r\geq 2$, it is clear that \begin{equation} |III|\leq Ct\,.\label{cotaIII} \end{equation} If $2r-3>0$, we apply Lemmas \ref{dos} and \ref{uno} to conclude that \begin{equation} \begin{aligned} |III| &\leq C \int_0^t \|J( w_N^{r-\frac 32} u(t')) \|_{L^2}^2 dt' \\ &\leq C \int_0^t \|J^{4r} u(t') \|_{L^2}^{\frac1{2r}} \| w_N^\frac{r(r-3/2)}{(r-1/4)} u(t') \|_{L^2}^{\frac{4r-1}{2r}} dt' \\ &\leq C \int_0^t \|w_N^\frac{r(r-3/2)}{(r-1/4)} u(t')\|_{L^2}^{2-\frac1{2r}}dt' \leq C \int_0^t \|w_N^{r} u(t')\|_{L^2}^{2-\frac1{2r}}dt' \\ &\leq C\int_0^t (1+\|w_N^r u(t') \|_{L^2}^2) dt' \leq Ct +C\int_0^t (u(t'),u(t')w_N^{2r})dt'. \label{cotIII} \end{aligned} \end{equation} In any case the estimate \eqref{cotIII} holds. In a similar manner it can be shown the following estimate for the term IV: \begin{equation} |IV|\leq C\int_0^t (u(t'),u(t')w_N^{2r})dt'. \label{cotIV} \end{equation} With respect to the term V we have: \begin{equation} \begin{aligned} |V| &\leq C\int_0^t\|u(t')\|_{L^{\infty}}(u(t'),u(t')w_N^{2r-1})dt'\\ &\leq C\int_0^t\|u(t')\|_{H^{4r}}(u(t'),u(t')w_N^{2r})dt' \\ &\leq C\int_0^t(u(t'),u(t')w_N^{2r})dt'\,. \end{aligned} \label{cotV} \end{equation} From equality \eqref{intu} and the estimates \eqref{cotI}-\eqref{cotV} it follows that, for $t\in[0,T]$, \[ (u(t),u(t)w_N^{2r})\leq \|u_0\|_{L^2(\langle x\rangle^{2r}dx)}^2+Ct +C\int_0^t(u(t'),u(t')w_N^{2r})dt'\,. \] Gronwall's inequality enables us to conclude that, for $t\in[0,T]$, \begin{equation} \begin{aligned} &(u(t),u(t)w_N^{2r})\\ &\leq \|u_0\|_{L^2(\langle x\rangle^{2r}dx)}^2+Ct +C\int_0^t(\|u_0\|_{L^2(\langle x\rangle^{2r}dx)}^2+Ct')e^{C(t-t')}dt'\,. \end{aligned} \label{Gron} \end{equation} Passing to the limit in \eqref{Gron} when $N\to\infty$ we obtain, for $t\in[0,T]$, \begin{equation} \begin{aligned} &\|u(t)\|_{L^2(\langle x\rangle^{2r}dx)}^2\\ &\leq \|u_0\|_{L^2(\langle x\rangle^{2r}dx)}^2 +Ct+C\int_0^t(\|u_0\|_{L^2(\langle x\rangle^{2r}dx)}^2+Ct')e^{C(t-t')}dt' \leq C(T), \end{aligned}\label{Gron1} \end{equation} which implies that $u\in L^{\infty}([0,T];L^2(\langle x\rangle^{2r}dx))$. Now let us see that $u\in C([0,T];L^2(\langle x\rangle^{2r}dx))$. For that we follow an argument contained in \cite{CP} and \cite{IO}. From \eqref{Gron1} it is clear that there is a positive constant $M$ such that, for all $t\in[0,T]$, \begin{equation} \|u(t)\|_{L_w^2}^2\leq \|u_0\|_{L_w^2}^2+Mt, \label{M} \end{equation} where the notation $L_w^2:=L^2(\langle x\rangle^{2r}dx)$ was used. Taking into account that $u\in C([0,T];L^2)$ and using \eqref{M}, it can be seen that, for $\phi\in L_w^2$, the function $t\mapsto(\phi,u(t))_{L_w^2}$ is continuous from $[0,T]$ to $\mathbb C$. From this fact and \eqref{M} it follows that \begin{align*} \|u(t)-u(0)\|_{L_w^2}^2 &=\|u(t)\|_{L_w^2}^2+\|u(0)\|_{L_w^2}^2-2\operatorname{Re}(u(0),u(t))_{L_w^2} \\ &\leq \|u(0)\|_{L_w^2}^2+Mt+\|u(0)\|_{L_w^2}^2 -2\operatorname{Re}(u(0),u(t))_{L_w^2}\to 0 \end{align*} as $t\to 0^+$, which proves that $u:[0,T]\to L^2(\langle x\rangle^{2r}dx)$ is continuous at $t=0$. The continuity of $u$ at a point $t_0\in(0,T]$ is a consequence from the continuity of $u$ at $t=0$ and from the fact that the functions $v_1(x,t):=u(x,t_0+t)$ and $v_2(x,t):=u(-x,t_0-t)$ are also solutions of the fifth-order KdV equation. In this manner, we had proved that if $u_0\in Z_{4r,r}$ $(r\geq\frac12)$ there exist $T=T(\|u_0 \|_{H^{4r}})>0$ and a unique $u\in C([0,T]; Z_{4r,r})$, solution of \eqref{KdV}, with $k=1$, belonging to the class defined by the conditions \eqref{con}-\eqref{H} with $s=4r$. Finally, let us prove that if $\widetilde u_m\in C([0,T]; Z_{4r,r})$ is the solution of the fifth-order KdV equation, corresponding to the initial data $\widetilde u_{m0}$, where $\widetilde u_{m0}\to u_0$ in $Z_{4r,r}$ when $m\to\infty$, then $\widetilde u_m\to u$ in $C([0,T]; Z_{4r,r})$. By Theorem \ref{principal} we have that $\widetilde u_m\to u$ in $C([0,T];H^{4r})$. In consequence we only must prove that $\widetilde u_m\to u$ in $C([0,T];L^2(\langle x\rangle^{2r}dx))$. Let $v_m:=\widetilde u_m -u$ and $v_{m0}:=\widetilde u_{m0}-u_0$. Proceeding in a similar manner as it was done when we established that $u\in L^{\infty}([0,T];L^2(\langle x\rangle^{2r}dx))$ and taking into account that $v_m\to 0$ in $C([0,T];H^{4r})$ it can be seen that, for $t\in[0,T]$, \[ \|v_m(t)\|_{L^2(w_N^{2r}dx))}^2\leq \|v_{m0}\|_{L^2(\langle x\rangle^{2r}dx)}^2 +C_mt+C\int_0^t\|v_m(t')\|_{L^2(w_N^{2r}dx))}^2dt'\,, \] where $\lim_{m\to\infty} C_m=0$. Hence, by Gronwall's inequality, we have for $t\in[0,T]$ and $N\in\mathbb N$ that \[ \|v_m(t)\|_{L^2(w_N^{2r}dx))}^2\leq (\|v_{m0}\|_{L^2(\langle x\rangle^{2r}dx)}^2 +C_mT)e^{CT}\,. \] From this inequality it follows, after passing to the limit when $N\to\infty$, that \[ v_m\to 0\quad\text{in }C([0,T];L^2(\langle x\rangle^{2r}dx))\,. \] The proof of Theorem \ref{prin1} is complete. \subsection{Proof of Theorem \ref{prin2}} Taking into account Remarks \ref{R1}(a) and \ref{R1}(b) it is sufficient to show that \eqref{KdV} for the fifth-order KdV equation is globally well-posed in $H^2(\mathbb{R})$. To see this, first of all, we prove that if $u\in C([0,T];H^2(\mathbb{R}))$ is a solution of \eqref{KdV} then, for all $t\in[0,T]$, \begin{equation} \| u(t) \|_{H^2}^2\leq K\equiv K(\|u_0 \|_{H^2}),\label{EstAprio} \end{equation} where $K$ depends only on $\| u_0\|_{H^2(\mathbb{R})}$. Let us observe that \begin{align} \int_\mathbb{R} (\partial_x u)^2(t) dx\leq \frac 12 \Big[ \int_\mathbb{R} (\partial_x^2 u)^2(t) dx+ \int_\mathbb{R} u^2(t)dx\Big]. \label{des1} \end{align} Using the definition of the $H^2$-norm, inequality \eqref{des1} and the conservation laws \eqref{cl1} and \eqref{cl2} it follows that \begin{equation} \begin{aligned} \| u(t) \|^2_{H^2} &=\int_\mathbb{R} u^2(t) dx+ \int_\mathbb{R} (\partial_x u)^2(t) dx +\int_\mathbb{R} (\partial^2_x u)^2 (t) dx \\ &\leq \frac 32 \int_\mathbb{R} u^2(t) dx +\frac 32 \int_\mathbb{R} (\partial^2_x u)^2 (t) dx \\ &=\frac 32 I_1(t)+3 I_2^1(t)-\frac 12 \int_\mathbb{R} u^3(t) dx \\ &= \frac 32 \| u_0 \|_{L^2}^2 +3\Big[\frac 12 \|\partial_x^2 u_0 \|^2_{L^2} +\frac 16 \int_\mathbb{R} u_0^3 dx \Big] -\frac 12 \int_\mathbb{R} u^3(t) dx. \end{aligned}\label{des2} \end{equation} Now, from the Sobolev lemma, we have \begin{align} \int_\mathbb{R} u_0^3 dx\leq \|u_0 \|_{L^\infty} \int_\mathbb{R} u_0^2 dx\leq C \|u_0 \|_{H^2}^3\label{des3}. \end{align} On the other hand, the Sobolev lemma, the conservation law \eqref{cl1} and Young's inequality imply that \begin{equation} \begin{aligned} \big|\int_\mathbb{R} u^3(t) dx\big| &\leq \|u(t) \|_{L^\infty} \|u(t) \|^2_{L^2}\\ &\leq C\|u(t) \|_{H^1} \|u(t) \|_{L^2}^2=C\|u(t) \|_{H^1}\|u_0 \|^2_{L^2} \\ &\leq \frac 12 \| u(t)\|^2_{H^1}+\frac {C^2}2 \|u_0 \|^4_{L^2}\\ &\leq \frac 12 \| u(t)\|^2_{H^1}+\frac {C^2}2 \|u_0 \|^4_{H^2}. \end{aligned} \label{des4} \end{equation} Therefore, from \eqref{des2}--\eqref{des4}, we have $$ \| u(t) \|^2_{H^2} \leq \frac 32 \| u_0\|^2_{H^2}+C \|u_0 \|^3_{H^2} + \frac{C^2}4 \| u_0\|^4_{H^2} +\frac 14 \|u(t) \|^2_{H^2}, $$ and from the above inequality \begin{equation} \| u(t) \|^2_{H^2} \leq C (\| u_0\|^2_{H^2}+\|u_0 \|^3_{H^2}+\|u_0 \|^4_{H^2})\equiv K,\label{des5} \end{equation} which proves \eqref{EstAprio}. Now we show how to extend the local solution $u$ to any time interval. From the proof of Theorem \ref{principal} it can be seen that the size of the time interval of the solution $u\in C([0,T];H^2(\mathbb{R}))$ of \eqref{KdV} is such that $$ T\geq \min\big\{ 1, \frac 1{C\|u_0 \|^2_{H^2}}\big\}. $$ Reasoning as in the proof of Theorem \ref{principal} we obtain a solution $u\in C([T,T+t_0];H^2(\mathbb{R}))$ of the IVP \begin{gather*} \partial_{t}v+\partial_{x}^{5}v+v\partial_{x}v=0, \quad x,t\in \mathbb{R}\\ v(T)=u(T) , \end{gather*} such that $$ t_0\geq \min\big\{ 1, \frac 1{C\|u(T)\|^2_{H^2}}\big\}. $$ In this manner we obtain a solution $u\in C([0,T+t_0];H^2(\mathbb{R}))$ of \eqref{KdV}. By the a priori estimate \eqref{EstAprio} we have that $$ \frac 1{\|u(t) \|_{H^2}^2}\geq \frac 1{K}, $$ for $t\in[0,T+t_0]$, and therefore $$ t_0\geq \min\big\{1,\frac 1{CK} \big\}. $$ We repeat this argument $n+1$ times to obtain a solution $u\in C([0,T+t_0+\dots+t_n];H^2(\mathbb{R}))$ with $$ t_j\geq \min\big\{ 1, \frac 1{CK}\big\}, \quad j=0,\dots, n. $$ Since $\sum_{j=0}^n t_j\to\infty$ as $n\to \infty$ then we can extend the solution to any time interval. The proof is complete. \section{Well-posedness of \eqref{KdV} with $k=2$} \subsection{Proof of Theorem \ref{tmkdv1}} For $T>0$, let us define the space \begin{align} Y_T:=\{ u\in C([0,T];H^2(\mathbb{R})): \|\partial_x^4 u \|_{L^\infty_x L^2_T}<\infty,\, \| u\|_{L^{16/5}_x L^\infty_T}<\infty,\, \| u\|_{L^4_x L^\infty_T}<\infty \}, \label{mY} \end{align} and, for $u\in Y_T$, let us consider the norms \begin{gather} \lambda_1^T(u):= \max_{[0,T]} \| u(t)\|_{H^2},\label{mN1}\\ \lambda_2^T(u):= \| \partial^4 _x u\|_{L^\infty_x L^2_T}, \label{mN2}\\ \lambda_3^T(u):= \| u \|_{L^{16/5}_x L^\infty_T},\label{mN3}\\ \lambda_4^T(u):= \| u \|_{L^4_xL^\infty_T}, \label{mN4}\\ \Lambda^T(u):= \max_{1\leq i\leq 4} \lambda^T_i (u). \label{mN} \end{gather} For $a>0$, let $Y_T^a$ be the closed ball in $Y_T$ defined by \begin{align} Y_T^a:= \{ u\in Y_T : \Lambda^T(u)\leq a\}. \label{mYa} \end{align} We shall prove that there exist $T>0$ and $a>0$ such that the operator $\Psi: Y_T^a \to Y_T ^a$ defined by \[ \Psi(v)=W(t)u_0-\int_0^t W(t-t') (v^2 \partial_x v)(t') dt' \] is a contraction. Also the linear estimates in section \ref{pre}, we will need some nonlinear estimates in order to prove that $\Psi$ is a contraction. First of all we establish these nonlinear estimates. Let $u\in Y_T$: \begin{itemize} \item[(i)] Using interpolation we have \begin{equation} \begin{aligned} \| u^2\partial_x u\|_{L^1_xL^2_T} &\leq \|u^2 \|_{L^{8/5}_x L^\infty_T} \| \partial_x u\|_{L^{8/3}_x L^2_T}\\ &\leq \| u\|^2_{L^{16/5}_x L^\infty_T} \| u\|^{3/4}_{L^2_x L^2_T} \| \partial_x^4 u\|^{1/4}_{L^\infty_x L^2_T} \\ &\leq C T^{3/4} \| u\|^2_{L^{16/5}_x L^\infty _T} \| u \|^{3/4}_{L^\infty_T L^2_x} \| \partial_x^4 u\|^{1/4}_{L^\infty_x L^2_T}. \end{aligned} \label{nle1} \end{equation} \item[(ii)] By \eqref{le2} and \eqref{nle1} it follows that \begin{equation} \begin{aligned} &\| \partial^4_x \int_0^t W(t-t') (u^2\partial_x u)(t')dt' \|_{L^\infty_x L^2_T}\\ &\leq C\| u^2 \partial_x u\|_{L^1_x L^2_T}\\ &\leq C T^{3/4} \| u\|^2_{L^{16/5}_x L^\infty_T} \| u\|^{3/4}_{L^\infty_T L^2_x} \| \partial_x^4 u\|^{1/4}_{L^\infty_x L^2_T}. \end{aligned} \label{nle2} \end{equation} \item[(iii)] By \eqref{le3} and \eqref{nle1} it follows that \begin{equation} \begin{aligned} &\| \partial^2_x \int_0^t W(t-t') (u^2\partial_x u)(t')dt' \|_{L^\infty_T L^2_x}\\ &\leq C \| u^2\partial_x u\|_{L^1_x L^2_T}\\ &\leq C T^{3/4} \| u\|^2_{L^{16/5}_x L^\infty_T} \| u\|^{3/4}_{L^\infty_T L^2_x} \| \partial_x^4 u\|^{1/4}_{L^\infty_x L^2_T}. \end{aligned}\label{nle3} \end{equation} \end{itemize} Now we prove that there exist $T>0$ and $a>0$ such that $\Psi(Y_T^a)\subset Y_T^a$. Let $v\in Y_T^a$. Then by \eqref{nle3}, \begin{equation} \begin{aligned} &\lambda_1^T(\Psi(v))\\ &\leq \lambda_1^T(W(t)u_0) +\lambda_1^T\Big(\int_0^t W(t-t')(v^2\partial_x v)(t') dt'\Big)\\ &\leq \|u_0 \|_{H^2} +C \Big( \sup_{[0,T]} \|\int_0^t W(t-t')(v^2\partial_x v)(t') dt' \|_{L^2}\\ &\quad +\sup_{[0,T]} \|\partial_x^2 \int_0^t W(t-t') (v^2\partial_x v)(t')dt' \|_{L^2} \Big)\\ &\leq \| u_0 \|_{H^2}+CT\sup_{[0,T]} \| v(t) \|^3_{H^2} +CT^{3/4}\|v \|^2_{L_x^{16/5}L_T^\infty} \|v \|^{3/4}_{L^\infty_T L^2_x} \| \partial_x^4 v\|^{1/4}_{L^\infty_x L^2_T}\\ &\leq \|u_0 \|_{H^2}+CT^{3/4}(T^{1/4}+1)(\Lambda^T(v))^3. \end{aligned} \label{mdes1} \end{equation} From \eqref{le1} and \eqref{nle2} it follows that \begin{equation} \begin{aligned} \lambda_2^T(\Psi(v)) & \leq \| \partial_x^4 W(t)u_0 \|_{L^\infty_x L^2_T} +\|\partial_x^4 \int_0^t W(t-t') (v^2\partial_x v)(t') dt'\|_{L^\infty_x L^2_T}\\ &\leq C \| \partial_x^2 u_0 \|_{L^2} +CT^{3/4} \| v\|^2_{L^{16/5}_x L^\infty_T} \| v\|^{3/4}_{L^\infty_T L^2_x}\|\partial_x^4 v \|^{1/4}_{L^\infty_x L^2_T}\\ &\leq C \|u_0 \|_{H^2}+CT^{3/4} (\Lambda^T(v))^3. \end{aligned} \label{mdes2} \end{equation} Using \eqref{le6m}, the Leibniz rule and interpolation, we obtain \begin{align} &\lambda_3^T(\Psi(v)) \nonumber\\ &\leq \| W(t)u_0\|_{L^{16/5}_x L^\infty_T}+\|\int_0^t W(t-t')(v^2 \partial_x v)dt' \|_{L^{16/5}_x L^\infty_T} \nonumber\\ &\leq C(1+T)^\rho \|u_0 \|_{H^2}+C(1+T)^{\rho}\int_0^T \| v^2\partial_x v(t') \|_{H^2} dt' \nonumber\\ &\leq C(1+T)^\rho \|u_0 \|_{H^2}+C(1+T)^{\rho}\int_0^T \| v^2\partial_x v(t')\|_{L^2} dt' \nonumber\\ &\quad+C(1+T)^{\rho}\int_0^T \| \partial_x^2 (v^2\partial_x v) (t')\|_{L^2} dt' \nonumber\\ &\leq C(1+T)^\rho \| u_0\|_{H^2}+C(1+T)^\rho T(\Lambda^T(v))^3+C(1+T)^\rho \Big( \int_0^T \| (\partial_x v)^3(t') \|_{L^2} dt' \nonumber\\ &\quad + \int_0^T \| (v \partial_x v\partial_x^2 v)(t')\|_{L^2} dt' +\int_0^T \| (v^2\partial_x^3 v)(t')\|_{L^2} dt' \Big) \nonumber\\ &\leq C(1+T)^\rho \| u_0\|_{H^2}+C(1+T)^\rho T(\Lambda^T (v))^3 +C(1+T)^\rho T^{1/2} \| v^2 \partial_x^3 v \|_{L^2_T L^2_x} \nonumber\\ &\leq C (1+T)^\rho \| u_0\|_{H^2}+C(1+T)^\rho T(\Lambda^T (v))^3 \nonumber\\ &\quad +C(1+T)^\rho T^{1/2} \|v^2 \|_{L_x^{16/7}L^8_T} \| \partial_x^3 v\|_{L_x^{16}L_T^{8/3}} \nonumber\\ &\leq C (1+T)^\rho \| u_0\|_{H^2}+C(1+T)^\rho T(\Lambda^T (v))^3 \nonumber\\ &\quad +C(1+T)^\rho T^{1/2} \|v \|_{L^{16/5}_x L^\infty_T} \| v\|_{L^8_x L^8_T} \| v\|^{1/4}_{L^4_x L^\infty_T} \| \partial_x^4 v\|^{3/4}_{L^\infty_x L^2_T} \nonumber\\ & \leq C (1+T)^\rho \| u_0\|_{H^2}+C(1+T)^\rho T(\Lambda^T (v))^3 \nonumber\\ &\quad +C(1+T)^\rho T^{1/2} \lambda_3^T (v) T^{1/8} \lambda_1^T (v) \lambda_4^T (v)^{1/4}\lambda_2^T (v)^{3/4}\nonumber \\ &\leq C (1+T)^\rho \| u_0\|_{H^2}+C(1+T)^\rho T(\Lambda^T (v))^3+C(1+T)^\rho T^{5/8} (\Lambda^T(v))^3. \label{mdes3} \end{align} Applying \eqref{le5} we have that \begin{equation} \begin{aligned} &\lambda_4^T(\Psi(v))\\ &\leq \|W(t)u_0 \|_{L^4_x L^\infty_T}+\int_0^T \| W(t-t') (v^2 \partial_x v)(t') \|_{L^4_x L^\infty_T} dt'\\ &\leq C \| D_x^{1/4} u_0 \|_{L^2}+C \int_0^T \| D^{1/4}_x (v^2 \partial_x v(t'))\|_{L^2_x} dt'\\ &\leq C \| D_x^{1/4} u_0 \|_{L^2}+C \int_0^T \| (v^2\partial_x v)(t')\|_{L^2}dt' +C\int_0^T \| \partial_x (v^2 \partial_x v)(t')\|_{L^2} dt'\\ &\leq C \| u_0 \|_{H^2}+CT (\Lambda^T(v))^3. \end{aligned} \label{mdes4} \end{equation} From \eqref{mdes1}--\eqref{mdes4} we obtain \begin{equation} \begin{aligned} \Lambda^T(\Psi(v)) &\leq C(1+T)^\rho \| u_0\|_{H^2} +C T^{5/8} [T^{1/8}(T^{1/4}+1)\\ &\quad +(1+T)^\rho(1+T^{3/8}) +T^{3/8}] (\Lambda^T(v))^3. \end{aligned} \label{mdes5} \end{equation} Let us take $a:=2C(1+T)^\rho \| u_0\|_{H^2}$ and $T$ in such a way that \begin{equation} C T^{5/8} [T^{1/8}(T^{1/4}+1)+(1+T)^\rho(1+T^{3/8})+T^{3/8}] a^2 \leq \frac 12. \label{mdes6} \end{equation} Since $\Lambda^T(v)\leq a$, from \eqref{mdes5} and \eqref{mdes6}, we have that $$ \Lambda^T(\Psi(v))\leq\frac a2+\frac 12(\Lambda^T(v))\leq \frac a2+\frac a2=a, $$ i.e. $\Psi(Y_T^a)\subset Y_T^a$. Now we find an additional condition on the size of $T$ in order to guarantee that the operator $\Psi: Y_T^a\to Y_T^a$ is a contraction. Let $v,w\in Y_T^a$. Then \begin{align*} \Psi(w)-\Psi(v) &=\int_0^t W(t-t')(v^2\partial_x (v-w))(t') dt'\\ &\quad +\int_0^t W(t-t')((v+w)(v-w)\partial_x w)(t') dt'. \end{align*} Therefore, proceeding as before, \begin{equation} \begin{aligned} &\lambda_1^T(\Psi(w)-\Psi(v))\\ &\leq \lambda_1^T\Big(\int_0^t W(t-t')(v^2\partial_x (v-w))(t') dt'\Big)\\ &\quad +\lambda_1^T\Big(\int_0^t W(t-t')((v+w)(v-w)\partial_x w)(t') dt'\Big)\\ &\leq CT(\lambda_1^T(v))^2\lambda_1^T(v-w)+CT^{3/4}(\lambda_3^T(v))^2 \lambda_1^T(v-w)^{3/4} \lambda_2^T(v-w)^{1/4}\\ &\quad + CT \lambda_1^T(v+w) \lambda_1^T(v-w)\lambda_1^T(w)\\ &\quad +CT^{3/4}\lambda_3^T(v+w)\lambda_3^T(v-w)\lambda_1^T (w)^{3/4}\lambda_2^T(w)^{1/4}\\ &\leq C[(T+T^{3/4})\Lambda^T(v)^2+T \Lambda^T(v+w)\Lambda^T(w)\\ &\quad +T^{3/4}\Lambda^T(v+w)\Lambda^T(w)]\Lambda^T(v-w)\\ &\leq C(T+T^{3/4})(\Lambda^T(v)^2+\Lambda^T(w)^2)\Lambda^T(v-w), \end{aligned}\label{mdes7} \end{equation} \begin{equation} \begin{aligned} &\lambda_2^T(\Psi(w)-\Psi(v))\\ &\leq CT^{3/4}\|v \|^2_{L^{16/5}_x L^\infty_T}\|v-w \|^{3/4}_{L^\infty_T L^2_x} \| \partial_x^4(v-w)\|^{1/4}_{L^\infty_x L^2_T}\\ &\quad+CT^{3/4}\| v+w\|_{L^{16/5}_x L^\infty_T}\|v-w \|_{L^{16/5}_x L^\infty_T} \| w\|_{L^\infty_T L^2_x}^{3/4}\|\partial_x^4w \|_{L^\infty_x L^2_T}^{1/4}\\ &\leq CT^{3/4}\Lambda^T (v)^2 \Lambda^T(v-w)+CT^{3/4}\Lambda^T(v+w)\Lambda^T(v-w) \Lambda^T(w)\\ &\leq CT^{3/4} (\Lambda^T(v)^2+\Lambda^T(w)^2) \Lambda^T(v-w), \end{aligned}\label{mdes8} \end{equation} \begin{equation} \begin{aligned} &\lambda_3^T(\Psi(w)-\Psi(v))\\ &\leq C(1+T)^\rho T\lambda_1^T(v)^2 \lambda_1^T(v-w)\\ &\quad +C(1+T)^\rho T^{1/2}\|v^2 \|_{L^{16/7}_x L^8_T} \| \partial_x^3 (v-w)\|_{L^{16}_x L^{8/3}_T}\\ &\quad +C(1+T)^\rho T \lambda_1^T(v+w)\lambda_1^T(v-w)\lambda_1^T(w)\\ &\quad +C(1+T)^\rho T^{1/2}\| (v+w)(v-w)\|_{L^{16/7}_x L^8_T} \| \partial_x^3 w\|_{L^{16}_x L^{8/3}_T}\\ &\leq C(1+T)^\rho T\lambda_1^T(v)^2 \lambda_1^T(v-w)\\ &\quad +C(1+T)^\rho T^{1/2} \lambda_3^T(v)T^{1/8}\lambda_1^T (v)\lambda_4^T(v-w)^{1/4}\lambda_2^T(v-w)^{3/4}\\ &\quad +C(1+T)^\rho T\lambda_1^T(v+w)\lambda_1^T(v-w)\lambda_1^T(w)\\ &\quad +C(1+T)^\rho T^{1/2} \lambda_3^T(v+w)T^{1/8}\lambda_1^T(v-w) \lambda_4^T(w)^{1/4}\lambda_2^T(w)^{3/4}\\ &\leq CT^{5/8}(1+T)^\rho(1+T^{3/8})(\Lambda^T(v)^2 +\Lambda^T(w)^2)\Lambda^T(v-w), \end{aligned}\label{mdes9} \end{equation} \begin{equation} \begin{aligned} &\lambda_4^T(\Psi(w)-\Psi(v))\\ &\leq CT\lambda_1^T(v)^2\lambda_1^T(v-w)+CT\lambda_1^T (v+w)\lambda_1^T(v-w)\lambda_1^T(w)\\ &\leq CT(\Lambda^T(v)^2+\Lambda^T(w)^2)\Lambda^T(v-w). \end{aligned} \label{mdes10} \end{equation} From \eqref{mdes7} to \eqref{mdes10}, it follows that \begin{align*} &\Lambda^T(\Psi(w)-\Psi(v))\\ &\leq C(2T+2T^{3/4}+T^{5/8}(1+T)^\rho(1+T^{3/8}))(\Lambda^T(v)^2 +\Lambda^T(w)^2)\Lambda^T(v-w)\\ &\leq C(2T+2T^{3/4}+T^{5/8}(1+T)^\rho(1+T^{3/8})) 2a^2\lambda^T(v-w). \end{align*} If $a:=2C(1+T)^\rho \| u_0\|_{H^2}$ and $T$ are taken satisfying \eqref{mdes6} and the additional condition \[ C(2T+2T^{3/4}+T^{5/8}(1+T)^\rho(1+T^{3/8})) 2a^2<1, \] then $\Psi:Y_T^a\to Y_T^a$ is a contraction. Hence, there exists a unique $u\in Y_T^a$ such that $\Psi(u)=u$. From this point, proceeding in a similar way as it was done in \cite{L} we conclude that, given $u_0\in H^2(\mathbb{R})$, there exist $T=T(\|u_0 \|_{H^2})>0$ and a unique $u$, solution of \eqref{KdV} for the modified fifth-order KdV equation ($k=2$), such that \begin{equation} u\in C([0,T]; H^2(\mathbb{R})) \label{uinc} \end{equation} and $u$ satisfies the conditions \eqref{mdes12}, \eqref{mdes13} and \eqref{mdes14}. Moreover, for any $T'\in(0,T)$ there exists a neighborhood $V_1$ of $u_0$ in $H^2(\mathbb{R})$, such that the data-solution map $\tilde u_0\mapsto u_0$ from $V_1$ into the class defined by \eqref{uinc}, \eqref{mdes12}, \eqref{mdes13} and \eqref{mdes14} with $T'$ instead of $T$ is Lipschitz. If additionally, we have that $u_0\in Z_{2,1/2}$, then reasoning as in the proof of Theorem \ref{prin1} (case $r\geq 1/2$) we obtain that $u\in C([0,T];L^2(\langle x\rangle) dx)$, and that there exists a neighborhood $V$ of $u_0$ in $Z_{2,1/2}$ such that the data-solution map $\tilde u_0\mapsto u_0$ from $V$ into the class defined by \eqref{mdes11} to \eqref{mdes14} with $T'$ instead of $T$ is Lipschitz. The proof is complete. \subsection{Proof of Theorem \ref{tmkdv2}} From Theorem \ref{tmkdv1} to prove that \eqref{KdV} (with $k=2$) is globally well-posed in $Z_{2,1/2}$, it is sufficient to establish that this IVP is globally well-posed in $H^2(\mathbb{R})$. Reasoning as in the proof of Theorem \ref{prin2} it is enough to show that if $u\in C([0,T];H^2(\mathbb{R}))$ is a solution of \eqref{KdV} with $k=2$, then for every $t\in [0,T]$ \begin{equation} \| u(t) \|_{H^2}^2\leq K\equiv K(\|u_0 \|_{H^2}),\label{mEstAprio} \end{equation} where $K$ only depends on $\| u_0\|_{H^2(\mathbb{R})}$. From \eqref{des1} and the conservation law \eqref{cl1} it is clear that \eqref{mEstAprio} holds if we prove that, for every $t\in[0,T]$, $$ \| \partial_x^2 u(t)\|^2_{L^2}\leq K\equiv K(\|u_0 \|_{H^2}). $$ By the conservation law \eqref{mcl2} we have that $$ \| \partial_x^2 u(t) \|_{L^2}^2=\frac 1{12} \int_\mathbb{R} u_0^4 dx +\int_\mathbb{R} (\partial_x^2 u_0)^2 dx-\frac 1{12}\int_\mathbb{R} u^4(t) dx, $$ and, since the last term in the right-hand side of the above equality is non-positive, we obtain that \[ \| \partial_x^2 u(t)\|_{L^2}^2 \leq \frac 1{12}\int_\mathbb{R} u_0^4 dx +\int_\mathbb{R} (\partial_x^2 u_0)^2 dx \leq \frac 1{12} \| u_0\|_{L^\infty}^2 \int_\mathbb{R} u_0^2 dx +\int_\mathbb{R} (\partial_x^2 u_0)^2 dx. \] Taking into account that $\| u_0\|_{L^\infty}^2\leq C\|u_0 \|_{H^2}^2$, we have \begin{align*} \| \partial_x^2 u(t)\|_{L^2}^2 &\leq C\| u_0\|_{H^2}^2 \int_\mathbb{R} u_0^2 dx +\int_\mathbb{R} (\partial_x^2 u_0)^2 dx\\ &\leq C\|u_0 \|_{H^2}^2 \| u_0\|^2_{H^2}+\| u_0\|^2_{H^2}\equiv K(\| u_0\|_{H^2}). \end{align*} The proof is complete. \section{Relation between decay and regularity the solutions of \eqref{KdV} with $k=2$ (Proof of Theorem \ref{tmkdv3})} First we assume that $\alpha\in (0,1/8]$. The general case follows by an iterative argument as it was done by Isaza, Linares and Ponce in \cite{ILP}. Let us suppose that $t_0=0$ and let $u_0=u(0)$. For $x\geq 0$ and $N\in\mathbb N$ let us define $\varphi_{N,\alpha}\in C^5([0,\infty))$ such that $$ \varphi_{N,\alpha}(x)= \begin{cases} (1+x^2)^{\alpha+1/2}-1 & \text{if } x\in[0,N],\\ (2N^2)^{\alpha+1/2} & \text{if } x\geq10N, \end{cases} $$ $\varphi^{(1)}_{N,\alpha}(x)\geq 0$ and $|\varphi_{N,\alpha}^{(j)}(x)|\leq C$ for $j=2,3,4,5$, with $C$ independent of $N$. Let $\phi_N\equiv \phi_{N,\alpha}$ be the odd extension of $\varphi_{N,\alpha}$ to $\mathbb{R}$. Since $C_0^\infty(\mathbb{R})$ is dense in $Z_{2,1/2}$, there exist a sequence $\{u_{0m} \}_{m\in\mathbb N}$ in $C_0^\infty(\mathbb{R})$ such that \begin{align} \| u_0-u_{0m}\|_{Z_{2,1/2}}\to 0\label{seq} \end{align} as $m\to\infty$. Let $u_m$ be the solution of the modified fifth-order KdV equation such that $u_m(0)=u_{0m}$. By Theorems \ref{tmkdv1} and \ref{tmkdv2} we have that \begin{gather} \| u_m -u\|_{C([0,T];Z_{2,1/2})} \to 0, \label{seq2}\\ \|\partial_x^4 u_m-\partial_x^4 u \|_{L^\infty_x L^2_T} \to 0, \label{seq3} \end{gather} as $m\to\infty$. Since $u_{0m}\in H^s(\mathbb{R})$ for each $s\in\mathbb{R}$, it can be seen (regularity property), that $u_m(t)\in H^s(\mathbb{R})$ for each $s\in\mathbb{R}$ and each $t\in[0,T]$. Now we multiply the equation $\partial_t u_m+\partial_x^5 u_m+u_m^2 \partial_x u_m=0$ by $u_m \phi_N$, integrate in $x$ over $\mathbb{R}$ and apply integration by parts to obtain \begin{equation} \begin{aligned} &\frac d{dt} \int_\mathbb{R} u_m^2 \phi_N dx - 5\int_\mathbb{R} (\partial^2_x u_m)^2 \phi_N^{(1)} dx +5\int_\mathbb{R} (\partial_x u_m)^2 \phi_N^{(3)}dx\\ &-\int_\mathbb{R} u_m^2 \phi_N^{(5)}dx -\frac 12 \int_\mathbb{R} u_m^4 \phi_N^{(1)} dx=0. \end{aligned}\label{mdes15} \end{equation} (In the above equation we use the notation $u_m\equiv u_m(t)$). From convergence in \eqref{seq2}, since $\alpha\leq 1/2$, it is clear that, for $t\in[0,T]$, \begin{equation} \begin{aligned} \big|- \frac 12 \int_\mathbb{R} u_m^4(t) \phi_N^{(1)} dx\big| &\leq C\| u_m(t)\|^2_{L^\infty} \int_\mathbb{R} u_m^2(t) \langle x \rangle^{2\alpha}dx\\ &\leq C\| u_m(t)\|^2_{H^2} \int_\mathbb{R} u_m^2(t) \langle x \rangle^{2\alpha}dx\\ &\leq C\sup_{t\in[0,T]}\| u_m(t)\|^2_{H^2} \| u_m(t)\|^2_{L^2(\langle x\rangle dx)} \leq C. \end{aligned} \label{mdes16} \end{equation} On the other hand, it is also clear that \begin{equation} \big|-\int_\mathbb{R} u_m^2 \phi_N^{(5)}dx\big| \leq C\quad \text{and}\quad \big|5\int_\mathbb{R} (\partial_x u_m)^2 \phi_N^{(3)}dx\big|\leq C. \label{mdes17} \end{equation} Integrating \eqref{mdes15} in $t$ over the interval $[0,t_1]$ and taking into account the inequalities \eqref{mdes16} and \eqref{mdes17} we can conclude that $$ 5\int_0^{t_1} \int_\mathbb{R} (\partial_x^2 u_m)^2 \phi_N^{(1)} \,dx\,dt \leq \| u_m^2(t_1)\phi_N\|_{L^1}+\|u_m^2(0) \phi_N \|_{L^1}+Ct_1. $$ Hence \begin{equation} \limsup_{m\to \infty}\int_0^{t_1} \int_\mathbb{R} (\partial_x^2 u_m)^2 \phi_N^{(1)} \,dx\,dt \leq \frac 15 \| u^2(t_1)\phi_N\|_{L^1}+\frac 15\|u^2(0) \phi_N \|_{L^1} +Ct_1\leq M, \label{mdes18} \end{equation} where $M=M(\| \langle x \rangle^{\frac 12 +\alpha}u(0)\|_{L^2} +\| \langle x \rangle^{\frac 12 +\alpha}u(t_1)\|_{L^2})$. Taking into account that $\phi_N^{(1)}$ is a bounded function, the convergence in \eqref{seq2} implies that $$ \int_0^{t_1}\int_\mathbb{R} (\partial_x^2 u_m)^2 \phi_N^{(1)} \,dx\, dt \to \int_0^{t_1}\int_\mathbb{R} (\partial_x^2 u)^2\phi_N^{(1)} \,dx\,dt,$$ as $m\to\infty$. Therefore, from \eqref{mdes18}, we obtain that \begin{equation} \int_0^{t_1}\int_\mathbb{R} (\partial_x^2 u)^2 \phi_N^{(1)} \,dx\,dt\leq M. \label{mdes19} \end{equation} Since $\phi_N^{(1)}$ is an even function, $\phi_N^{(1)}(x)\geq 0$ and, for $x>1$, $$ \phi_N^{(1)}(x)\to (2\alpha+1)(1+x^2)^{\alpha-\frac 12}x \sim \langle x \rangle^{2\alpha}, $$ as $N\to\infty$, applying Fatou's Lemma in \eqref{mdes19}, we have that $$ \int_0^{t_1}\int_{|x|\geq 1} (\partial_x^2 u)^2\langle x \rangle^{2\alpha} \,dx\,dt \leq CM, $$ and taking into account that $$ \int_0^{t_1} \int_{|x|\leq 1} (\partial_x^2 u)^2 \langle x \rangle^{2\alpha} \,dx\,dt\leq C, $$ we obtain that \begin{equation} \int_0^{t_1} \int_\mathbb{R} (\partial_x^2 u)^2 \langle x\rangle^{2\alpha} \,dx\,dt \leq C+CM<\infty.\label{mdes20} \end{equation} From \eqref{mdes20} it follows that \begin{equation} \partial_x^2 u(t)\in L^2(\langle x \rangle^{2\alpha}dx),\quad \text{a.e. } t\in[t_0,t_1]. \label{mdes21} \end{equation} Let us define, for $t\in[t_0,t_1]$, $w(t):=\partial_x^2 u(t)$ and $w_m(t):=\partial_x^2 u_m(t)$. Then we have that \begin{equation} \partial_t w_m+\partial_x^5 w_m+u_m^2\partial_x w_m +6 u_m\partial_x u_m w_m+2(\partial_x u_m)^3=0.\label{mdes22} \end{equation} For $x\geq 0$ and $N\in\mathbb N$ let us define $\tilde \varphi_{N,\alpha}\in C^5([0,\infty))$ such that $$ \tilde\varphi_{N,\alpha}(x)=\begin{cases} (1+x^2)^\alpha-1 & \text{if } x\in[0,N],\\ (2N^2)^\alpha & \text{if } x\geq 10N, \end{cases} $$ $\tilde\varphi_{N,\alpha}^{(1)}(x)\geq 0$ and $|\tilde\varphi_{N,\alpha}^{(j)}(x)|\leq C$ for $j=1,\dots, 5$, with $C$ independent of $N$. Let $\tilde\phi_N\equiv \tilde\phi_{N,a}$ be the odd extension of $\tilde\varphi_{N,\alpha}$ to $\mathbb{R}$. Multiplying equation \eqref{mdes22} by $w_m\tilde\phi_N$, integrating in $x$ over $\mathbb{R}$, using integration by parts, and then integrating in $t$ over an interval $[t_0^*,t_1^*]\subset [t_0,t_1]$ such that $\partial_x^2 u(t_0^*), \partial_x^2 u(t_1^*) \in L^2(\langle x\rangle^{2\alpha} dx)$, we obtain \begin{equation} \begin{aligned} &\int_{t_0^*}^{t_1^*} \int_\mathbb{R} (\partial_x^2 w_m)^2 \tilde\phi_N^{(1)}\,dx\,dt\\ &=\frac 15\int_\mathbb{R} w_m^2(t_1^*) \tilde\phi_N dx -\frac 15 \int_\mathbb{R} w_m^2(t_0^*) \tilde\phi_N dx\\ &\quad +\int_{t_0^*}^{t_1^*}\int_\mathbb{R} (\partial_x w_m)^2(t) \tilde\phi_N^{(3)} \,dx\,dt -\frac 15 \int_{t_0^*}^{t_1^*}\int_\mathbb{R} w_m^2(t) \tilde\phi_N^{(5)} \,dx\,dt\\ &\quad -\frac 15 \int_{t_0^*}^{t_1^*} \int_\mathbb{R} u_m^2(t) w_m^2(t) \tilde\phi_N^{(1)} \,dx\,dt\\ &\quad +2\int_{t_0^*}^{t_1^*}\int_\mathbb{R} u_m(t)\partial_x u_m(t) w_m^2(t) \tilde\phi_N \,dx\,dt\\ &\quad +\frac 45 \int_{t_0^*}^{t_1^*} \int_\mathbb{R} (\partial_x u_m)^3(t) w_m(t) \tilde\phi_N \,dx\,dt. \end{aligned}\label{mdes23} \end{equation} From \eqref{mdes23} we shall prove that $$ \int_{t_0^*}^{t_1^*}\int_\mathbb{R} (\partial_x^2 w)^2(t) \langle x \rangle^{2\alpha-1}\,dx\,dt <\infty. $$ Let us observe that \begin{align*} &\int_{t_0^*}^{t_1^*}\int_\mathbb{R} (\partial_x w_m)^2(t) \tilde\phi_N^{(3)} \,dx\,dt\\ &=-\int_{t_0^*}^{t_1^*} \int_\mathbb{R} w_m(t)\partial_x^2 w_m(t) \tilde\phi_N^{(3)}\,dx\,dt +\frac 12 \int_{t_0^*}^{t_1^*}\int_\mathbb{R} w_m^2(t)\tilde\phi_N^{(5)}\,dx\,dt. \end{align*} Let $K$ be a constant independent of $N$ such that $|\tilde\phi_N^{(3)}|\leq K\tilde\phi_N^{(1)}$. Then \begin{equation} \begin{aligned} &\int_{t_0^*}^{t_1^*} \int_\mathbb{R}(\partial_x w_m)^2(t) \tilde\phi_N^{(3)} \,dx\,dt\\ &\leq \frac 12 \int_{t_0^*}^{t_1^*} \int_\mathbb{R}(K w_m^2(t) +\frac{(\partial_x^2 w_m)^2(t)}K)|\tilde\phi_N^{(3)}|\,dx\,dt\\ &\quad +\frac 12 \int_{t_0^*}^{t_1^*} \int_\mathbb{R} w_m^2(t) \tilde\phi_N^{(5)} \,dx\,dt\\ &\leq \frac K2 \int_{t_0^*}^{t_1^*} \int_\mathbb{R} w_m^2(t) |\tilde\phi_N^{(3)}|\,dx\,dt+\frac 12 \int_{t_0^*}^{t_1^*} \int_\mathbb{R}(\partial_x^2 w_m)^2(t) \tilde\phi_N^{(1)}\,dx\,dt\\ &\quad +\frac 12 \int_{t_0^*}^{t_1^*} \int_\mathbb{R} w_m^2(t) \tilde\phi_N^{(5)} \,dx\,dt. \end{aligned} \label{mdes24} \end{equation} Taking into account that the fifth term on the right-hand side of \eqref{mdes23} is not positive, from \eqref{mdes23} and \eqref{mdes24} it follows that \begin{equation} \begin{aligned} &\frac 12 \int_{t_0^*}^{t_1^*} \int_\mathbb{R}(\partial_x^2 w_m)^2(t) \tilde\phi_N^{(1)} \,dx\,dt \\ &\leq \frac 15 \int_\mathbb{R} w_m^2(t_1^*) \tilde\phi_N dx -\frac 15 \int_\mathbb{R} w_m^2(t_0^*)\tilde\phi_N dx\\ &\quad +\frac K2 \int_{t_0^*}^{t_1^*} \int_\mathbb{R} w_m^2(t) |\tilde\phi_N^{(3)}| \,dx\,dt +\frac 3{10} \int_{t_0^*}^{t_1^*} \int_\mathbb{R}w_m^2(t) \tilde\phi_N^{(5)} \,dx\,dt\\ &\quad +2 \int_{t_0^*}^{t_1^*} \int_\mathbb{R} u_m(t) \partial_x u_m(t) w_m^2(t) \tilde\phi_N \,dx\,dt\\ &\quad +\frac 45\int_{t_0^*}^{t_1^*} \int_\mathbb{R}(\partial_x u_m)^3(t) w_m(t) \tilde\phi_N \,dx\,dt\\ &\equiv \frac 15 \int_\mathbb{R} w_m^2(t_1^*) dx -\frac 15 \int_\mathbb{R} w_m^2(t_0^*)\tilde\phi_N dx+I+II+III+IV. \end{aligned} \label{mdes25} \end{equation} Since $\tilde\phi_N^{(3)}$ and $\tilde\phi_N^{(5)}$ are bounded functions, the convergence in \eqref{seq2} implies that \begin{equation} I+II\leq C.\label{mdes26} \end{equation} We now estimate $III+IV$. Using the convergence \eqref{seq2}, and the boundedness of the function $\tilde\phi_N^{(1)}$, we obtain, for $\alpha\in(0,1/4]$, that \begin{equation} \begin{aligned} III+IV &\leq C\int_{t_0^*}^{t_1^*} \| u_m(t)\|_{H^2}^3 \|u_m(t) \tilde\phi_N\|_{L^\infty} dt\\ &\quad +C\int_{t_0^*}^{t_1^*}\| u_m(t)\|_{H^2}^2\int_\mathbb{R} |w_m(t)| |\partial_x u_m(t)| |\tilde\phi_N| \,dx\,dt\\ &\leq C \int_{t_0^*}^{t_1^*} \| u_m(t) \tilde\phi_N \|_{H^1} dt +C\int_{t_0^*}^{t_1^*} \|w_m(t) \|_{L^2}\|\partial_x u_m(t) \tilde\phi_N \|_{L^2}dt\\ &\leq C\int_{t_0^*}^{t_1^*} (\| u_m(t) \tilde\phi_N \|_{H^1} +\| \partial_x u_m(t) \tilde\phi_N \|_{L^2})dt\\ &\leq C \int_{t_0^*}^{t_1^*} (\| u_m(t) \tilde\phi_N\|_{L^2} +\| \partial_x u_m(t) \tilde\phi_N \|_{L^2}+\| u_m(t) \tilde\phi_N^{(1)}\|_{L^2}) dt\\ &\leq C \int_{t_0^*}^{t_1^*} (\| u_m(t) \|_{L^2(\langle x\rangle^{4\alpha}dx)} +\| \partial_x u_m(t) \|_{L^2(\langle x\rangle^{4\alpha}dx)}) dt+C\\ &\leq C+C\int_{t_0^*}^{t_1^*} \| \partial_x u_m(t) \|_{L^2(\langle x\rangle^{4\alpha}dx)}dt. \end{aligned} \label{mdes27} \end{equation} Using Lemmas \ref{dos} and \ref{uno} and the convergence in \eqref{seq2}, for $\alpha\in(0,1/8]$ and $t\in[t_0^*,t_1^*]\subset[0,T]$, it follows that \begin{equation} \begin{aligned} \| \partial_x u_m(t)\|_{L^2(\langle x \rangle^{4\alpha}dx)} &=\|\langle x \rangle^{2\alpha} \partial_x u_m(t) \|_{L^2}\\ &\leq \|\langle x\rangle^{1/4} \partial_x u_m(t) \|_{L^2} \leq C\| J(\langle x \rangle^{1/4} u_m(t))\|_{L^2}\\ &\leq C\| \langle x \rangle^{1/2} u_m(t)\|_{L^2}^{1/2}\| J^2 u_m(t)\|_{L^2}^{1/2} \leq C. \end{aligned}\label{mdes28} \end{equation} From \eqref{mdes25}-\eqref{mdes28} we conclude that, if $\alpha\in(0,1/8]$ then \begin{equation} \frac 12 \int_{t_0^*}^{t_1^*} \int_\mathbb{R} (\partial_x^2 w_m)^2(t) \tilde\phi_N^{(1)} \,dx\,dt \leq \frac 15 \int_\mathbb{R} w_m^2(t_1^*)\tilde\phi_N -\frac 15 \int_\mathbb{R} w_m^2(t_0^*)\tilde\phi_N +C, \label{mdes29} \end{equation} where $C=C(T)$ is a constant indepent of $N$ and $m$. Let us observe that from the convergence in \eqref{seq3} we can conclude that $$ \sup_{x\in\mathbb{R}}\int_0^T |\partial_x^2 w_m(x,t) -\partial_x^2 w(x,t)|^2dt=\| \partial_x^4 u_m -\partial_x^4 u\|_{L^\infty_x L^2_T}^2\to 0, $$ as $m\to\infty$. Hence, denoting by $|A|$ the measure of a set $A\subset \mathbb{R}$, we have \begin{align*} &\Big|\int_{t_0^*}^{t_1^*}\int_\mathbb{R}[(\partial_x^2 w_m)^2(x,t) -(\partial_x^2 w)^2(x,t)] \tilde \phi_N^{(1)} \,dx\,dt \Big| \\ &\leq C \int_{supp\, \tilde\phi_N^{(1)}} \int_{t_0^*}^{t_1^*} |\partial_x^2 w_m-\partial_x^2 w| |\partial_x^2 w_m+\partial_x^2 w| dt dx\\ &\leq C \int _{supp\, \tilde\phi_N^{(1)}} \| \partial_x^2 w_m(x,\cdot) -\partial_x^2 w(x,\cdot)\|_{L^2_T}\| \partial_x^2 w_m(x,\cdot) +\partial_x^2 w(x,\cdot)\|_{L^2_T }dx\\ &\leq C \| \partial_x^2 w_m-\partial_x^2 w\|_{L^\infty_x L^2_T} \| \partial_x^2 w_m +\partial_x^2 w\|_{L^\infty_x L^2_T} |\operatorname{supp} \tilde\phi_N^{(1)}| \\ &\leq C \| \partial_x^2 w_m-\partial_x^2 w\|_{L^\infty_x L^2_T}\to 0, \end{align*} as $m\to \infty$. Therefore, passing to the limit in \eqref{mdes29}, as $m\to\infty$, we obtain \begin{equation} \begin{aligned} &\frac 12 \int_{t_0^*}^{t_1^*} \int_\mathbb{R} (\partial_x^2 w)^2(t) \tilde\phi_N ^{(1)} \,dx\,dt\\ &\leq \frac 15 \int_\mathbb{R} w^2(t_1^*) \tilde\phi_N -\frac 15 \int_\mathbb{R} w^2(t_0^*) \tilde\phi_N +C\\ &\leq C\int_\mathbb{R} w^2(t_1^*) \langle x\rangle^{2\alpha} dx +C\int_\mathbb{R} w^2(t_0^*)\langle x \rangle ^{2\alpha} +C\equiv M. \end{aligned}\label{mdes30} \end{equation} Since $\tilde\phi_N^{(1)}$ is an even function, $\tilde\phi_N^{(1)}\geq 0$ and, for $x\geq 1$, $$ \tilde\phi_N^{(1)}(x)\to 2\alpha \langle x \rangle^{2\alpha-1} \langle x \rangle' \sim \langle x \rangle^{2\alpha-1}, $$ as $N\to\infty$, applying Fatou's Lemma in \eqref{mdes30} we can conclude that $$ \int_{t_0^*}^{t_1^*} \int_{|x|\geq 1} (\partial_x^2 w)^2(t) \langle x\rangle^{2\alpha-1} \,dx\,dt\leq CM. $$ On the other hand, since $2\alpha-1\leq 0$, \begin{align*} \int_{t_0^*}^{t_1^*} \int_{|x|\leq 1} (\partial_x^2 w)^2(t) \langle x\rangle^{2\alpha-1} \,dx\,dt &\leq \int_{t_0^*}^{t_1^*} \int_{|x|\leq 1} (\partial_x^2 w)^2(t) \,dx\,dt\\ &\leq 2 \| \partial_x^2 w\|^2_{L^\infty_x L^2_T} =2\|\partial_x^4 u \|^2_{L^\infty_x L^2_T}<\infty. \end{align*} In consequence \begin{equation} \int_{t_0^*}^{t_1^*} \int_\mathbb{R} (\partial_x^2 w)^2(t) \langle x \rangle^{2\alpha-1} \,dx\,dt <\infty.\label{mdes31} \end{equation} From \eqref{mdes31} it follows that $$ \partial_x^2 w(t) \langle x \rangle ^{\alpha-1/2}\in L^2(\mathbb{R}), \quad \text{ a.e. } t\in[t_0^*,t_1^*]. $$ This fact and \eqref{mdes21} imply that $$ \langle x \rangle^\alpha w(t) \in L^2 \text{ and } J^2(\langle x\rangle^{\alpha-1/2} w(t))\in L^2, \quad\text{ a.e. } t\in[t_0^*,t_1^*]. $$ Let us define $f:=\langle x \rangle^{\alpha-1/2} w(t)$. Then $$ \langle x \rangle^{1/2} f\in L^2\text{ and } J^2(f)\in L^2. $$ Using Lemma \ref{uno} with $a=2$ and $b=1/2$ we conclude that, for $\theta\in[0,1]$, $$ \|J^{2\theta}(\langle x\rangle^{(1-\theta)/2} f) \|_{L^2} \leq C\| \langle x \rangle^{1/2} f\|_{L^2}^{(1-\theta)}\|J^2(f) \|^\theta_{L^2}. $$ Taking $\theta=2\alpha$, we have that $\| J^{4\alpha} w(t)\|_{L^2}<\infty$, a.e. $t\in [t_0^*,t_1^*],$ i.e., $w(t)\in H^{4\alpha}$, a.e. $t\in[t_0^*,t_1^*]$, i.e., $u(t)\in H^{2+4\alpha}$. The rest of the proof is as in \cite{ILP}, in consequence we omit the details. \subsection*{Acknowledgments} This research was supported by Facultad de Ciencias, Universidad Nacional de Colombia Sede Medell\'in. \begin{thebibliography}{00} \bibitem{CP} Cunha, A.; Pastor, A.; \emph{The IVP for the Benjamin-Ono-Zakharov-Kuznetsov equation in weighted Sobolev spaces}, J. Math. Anal. 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