On a class of solutions to the generalized KdV type equation

TL;DR

This paper establishes local well-posedness and regularity propagation for solutions to a generalized KdV equation with low non-linearity degree, expanding understanding of solution behavior in weighted Sobolev spaces.

Contribution

It introduces a novel approach to prove local well-posedness for low-degree non-linear generalized KdV equations and confirms the propagation of regularity for these solutions.

Findings

Proved local well-posedness in weighted Sobolev spaces.

Demonstrated propagation of regularity in solutions.

Extended existing results to equations with low non-linearity.

Abstract

We consider the IVP associated to the generalized KdV equation with low degree of non-linearity \begin{equation*} \partial_t u + \partial_x^3 u \pm |u|^{\alpha}\partial_x u = 0,\; x,t \in \mathbb{R},\;\alpha \in (0,1). \end{equation*} By using an argument similar to that introduced by Cazenave and Naumkin [2] we establish the local well-posedness for a class of data in an appropriate weighted Sobolev space. Also, we show that the solutions obtained satisfy the propagation of regularity principle proven in [3] in solutions of the $k$-generalized KdV equation.

Equations

$${\partial }_{t}u+{\partial }_x^{3}u\pm |u{|}^{\alpha }{\partial }_{x}u=0,x,t\in \mathbb{R},\alpha \in (0,1).$$

$$\{\begin{array}{rl}& {\partial }_{t}u+{\partial }_x^{3}u\pm |u{|}^{\alpha }{\partial }_{x}u=0,x,t\in \mathbb{R},\alpha \in (0,1),\\ & u(x,0)=u_0(x),\end{array}$$

$${\partial }_{t}u+{\partial }_x^{3}u+u{\partial }_{x}u=0,x,t\in \mathbb{R},$$

$${\partial }_{t}u+{\partial }_x^{3}u+u^k{\partial }_{x}u=0,x,t\in \mathbb{R},k\in {\mathbb{Z}}^{+}.$$

$$I_1(u)$$

$$I_3(u)$$

$$\{\begin{array}{l}{\partial }_{t}u=i(\Delta u\pm |u{|}^{\alpha }u),x\in {\mathbb{R}}^n,t\in \mathbb{R},\alpha >0,\\ u(x,0)=u_0(x),\end{array}$$

$$\underset{x\in {\mathbb{R}}^n}{\mathrm{Inf}}{⟨x⟩}^m|e^{it\Delta }{u}_0(x)|>0,$$

$$\underset{x\in {\mathbb{R}}^N}{\mathrm{Inf}}{⟨x⟩}^m|u_0(x)|⩾\lambda >0.$$

$${\partial }_{t}u=i{\partial }_x^{2}u+\mu |u{|}^{\alpha }{\partial }_{x}u,x,t\in \mathbb{R},0<\alpha ⩽1\mathrm{and}|\mu |=1,$$

$$\displaystyle{}\color[rgb]{0,0,0}u_{0}\in H^{s}(\mathbb{R}),\;\left\langle x\right\rangle^{m}u_{0}\in L^{\infty}(\mathbb{R}),\;\left\langle x\right\rangle^{m}\partial_{x}^{j+1}u_{0}\in L^{2}(\mathbb{R}),\;j=0,1,2,3,$$

$$\displaystyle{}\left\lVert u_{0}\right\rVert_{H^{s}}+\left\lVert\left\langle x\right\rangle^{m}u_{0}\right\rVert_{L^{\infty}}+{\color[rgb]{0,0,0}\sum_{j=0}^{3}\left\lVert\left\langle x\right\rangle^{m}\partial_{x}^{j+1}u_{0}\right\rVert_{L^{2}}}<\delta$$

$$\underset{x\in \mathbb{R}}{\inf }{⟨x⟩}^m|u_0(x)|=:\lambda >0.$$

$$\displaystyle{\color[rgb]{0,0,0}u\in C([0,T];H^{s}(\mathbb{R})),\quad\left\langle x\right\rangle^{m}\partial_{x}^{j+1}u\in C([0,T];L^{2}(\mathbb{R})),\quad j=0,1,2,3}$$

$${⟨x⟩}^{m}u\in C([0,T];L^{\infty }(\mathbb{R})),{\partial }_x^{s+1}u\in L^{\infty }(\mathbb{R};L^2([0,T])),$$

$$\underset{0⩽t⩽T}{\sup }{\Vert {⟨x⟩}^m(u(t)-u_0)\Vert }_{L^{\infty }}⩽\frac{\lambda }{2}.$$

$${\color[rgb]{0,0,0}C([0,T];H^{2m+2}(\mathbb{R})),\quad\left\langle x\right\rangle^{m}\partial_{x}^{j+1}u\in C([0,T];L^{2}(\mathbb{R})),\quad j=0,1,}$$

$${⟨x⟩}^{m}u\in L^{\infty }([0,T];L^{\infty }(\mathbb{R})),{\partial }_x^{2m+3}u\in L^{\infty }(\mathbb{R};L^2([0,T])),$$

$$\underset{0⩽t⩽T}{\sup }{\Vert {⟨x⟩}^m(u(t)-u_0)\Vert }_{L^{\infty }}⩽\frac{\lambda }{2}.$$

$$\frac{\lambda }{2}\le -\frac{\lambda }{2}+{⟨x⟩}^m|u_0(x)|⩽{⟨x⟩}^m|u(x,t)|⩽{⟨x⟩}^m|u_0(x)|+\frac{\lambda }{2}$$

$${⟨x⟩}^{m-1/2}{u}_0\notin L^2(\mathbb{R})\text{and}{⟨x⟩}^{m-1/2}u(t)\notin L^2(\mathbb{R}),t\in (0,T].$$

$$u_0(x)=\frac{2\lambda e^{i\theta }}{{⟨x⟩}^m}+\phi (x),\theta \in \mathbb{R},$$

$$\Vert {⟨x⟩}^m\phi {\Vert }_{\infty }\le \lambda .$$

$${\varphi }_{c,\alpha }(x,t)$$

$$\varphi (x)$$

$${\partial }_{t}v+{\partial }_x^{3}v+{\partial }_x(v\log |v|)=0,x,t\in \mathbb{R}$$

$$\displaystyle u_{0}\Big{|}_{(x_{0},\infty)}\in H^{s+l}((x_{0},\infty)).$$

$$\underset{0\le t\le T}{\sup }{\int }_{x_0+{ϵ}^{\prime }-vt}^{\infty }({\partial }_x^{s+j}u(x,t){)}^{2}dx<c^{\ast }=c^{\ast }({ϵ}^{\prime };v;x_0;j;l;s)$$

$${\int }_0^T{\int }_{x_0+{ϵ}^{\prime }-vt}^{x_0+R-vt}({\partial }_x^{s+l+1}u(x,t){)}^{2}dxdt<c^{\ast \ast }=c^{\ast \ast }({ϵ}^{\prime };R;v;x_0;j;l;s).$$

$${\Vert F\Vert }_{L_t^{\infty }{L}_x^2}=\underset{t\in \mathbb{R}}{\sup }\Vert F(t){\Vert }_2,{\Vert F\Vert }_{L_x^{\infty }{L}_t^2}=\underset{x\in \mathbb{R}}{\sup }({\int }_{-\infty }^{\infty }|f(x,t){|}^{2}dt{)}^{1/2},$$

$${\Vert U(t)f\Vert }_{L_t^{\infty }{L}_x^2}+{\Vert {\partial }_{x}U(t)f\Vert }_{L_x^{\infty }{L}_t^2}$$

$${\Vert {⟨x⟩}^{\theta \mu }{\partial }_x^{(1-\theta )r}f\Vert }_{L^2}⩽$$

Full text

[TABLE]

By using an argument similar to that introduced by Cazenave and Naumkin [CaNa] we establish the local well-posedness for a class of data in an appropriate weighted Sobolev space. Also, we show that the solutions obtained satisfy the propagation of regularity principle proven in [ILP] in solutions of the $k$-generalized KdV equation.

1. Introduction

In this work we study the initial value problem (IVP) for the generalized Korteweg-de Vries (KdV) type equation

[TABLE]

where $u=u(x,t)$ is a real-valued or complex-valued unknown function.

The equation in (GK) is a lower nonlinearity version of the celebrate Korteweg-de Vries equation (KdV) [KdV]

[TABLE]

and its $k$-generalized form

[TABLE]

The IVP and the periodic boundary value problem (pbvp) associated to the equation in (1.2) have been extensively studied. In fact, sharp local and global well-posedness, stability of special solutions and blow-up results have been established in several publications (for a more detail account of them we refer to [LP] Chapters 7-8).

Formally, real valued solutions of (GK) satisfy three conservation laws:

[TABLE]

Roughly speaking, the nonlinearity in (GK) is non-Lipchitz in any Sobolev space $H^{s}(\mathbb{R})=(1-\partial_{x}^{2})^{s/2}L^{2}(\mathbb{R}),\,s\in\mathbb{R},$ or in the weighted versions $H^{s}(\mathbb{R})\cap L^{2}(\mathbb{R}:\left\langle x\right\rangle^{r}dx)$, $\,k,\,\,r\in\mathbb{R}$ as a consequence local well-posedness can not be established in these spaces.

Our first goal is to establish the local well-posedness for the IVP (GK) in a class of initial data. To present our result we first describe our motivation and the ideas behind the proofs.

In [CaNa] Cazenave and Naumkin studied the IVP associated to semi-linear Schrödinger equation,

[TABLE]

with initial data $u_{0}\in H^{s}(\mathbb{R}^{n})$. For every $\alpha>0$ they constructed a class of initial data for which there exist unique local solutions for the IVP (1.3). Also, they obtained a class of initial data for $\alpha>\frac{2}{n}$ for which there exist global solutions that scatter.

One of the ingredients in the proofs of their results, is the fact that solutions of the linear problem satisfy

[TABLE]

for $t\in[0,T]$ with $T$ sufficiently small whenever the initial data satisfy

[TABLE]

This is reached for $m=m(\alpha)$ and $u_{0}\in H^{s}(\mathbb{R}^{n})$ with $s$ sufficiently large with appropriate decay. To prove the inequality (1.4) the authors in [CaNa] rely on Taylor’s power expansion to avoid applying the Sobolev embedding since the nonlinear $|u|^{\alpha}u$ is not regular enough and it would restrict the argument to dimensions $n\geqslant 4$.

In [LPS], the arguments introduced in [CaNa] were modified to study the IVP associated to the generalized derivative Schrödinger equation

[TABLE]

establishing local well-posedness for a class of small data in an appropriate weighted Sobolev space.

In the case considered here, the non-linearity is non-Lipschitz. Motivated by the results in [CaNa] and using the smoothing effects of Kato type [Ka] we shall obtain the desired local well-posedness result for the IVP (GK) for a class of data satisfying (1.5).

The first aim of this paper is to show the following:

Theorem 1.1**.**

Fix $m=\left[\frac{1}{\alpha}\right]+1$. Let $s\in\mathbb{Z}^{+}$ satisfy $s\geqslant 2m+4$. Assume $\,u_{0}\;$ is a complex-valued function such that

[TABLE]

for some $\delta>0$ and

[TABLE]

Then there exists $T=T(\alpha;\delta;s;\lambda)>0$ such that (GK) has a unique local solution

[TABLE]

with

[TABLE]

and

[TABLE]

Moreover, the map $u_{0}\mapsto u(t)$ is continuous in the following sense: For any compact $I\subset[0,T]$, there exists a neighborhood $V$ of $u_{0}$ satisfying (1.6) and (1.8) such that the map is Lipschitz continuous from $V$ into the class defined by (1.9) and (1.10).

Above we have used the following notation: $$