Improved lower bound for the radius of analyticity for the modified KdV equation
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| Format: | Preprint |
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2024
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| _version_ | 1866909222463078400 |
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| author | Figueira, Renata O. Panthee, Mahendra |
| author_facet | Figueira, Renata O. Panthee, Mahendra |
| contents | We investigate the initial value problem (IVP) associated to the modified Korteweg-de Vries equation (mKdV) in the defocusing scenario: \begin{equation*} \left\{\begin{array}{l} \partial_t u+ \partial_x^3u-u^2\partial_x(u) = 0, \quad x,t\in\mathbb{R}, \\ u(x,0) = u_0(x), \end{array}\right. \end{equation*} where $u$ is a real valued function and the initial data $u_0$ is analytic on $\mathbb{R}$ and has uniform radius of analyticity $σ_0$ in the spatial variable.
It is well-known that the solution $u$ preserves its analyticity with the same radius $σ_0$ for at least some time span $0<T_0\le 1$. This local result was obtained in [Nonlinear Differ. Equ. Appl. (2024), 31--68] by proving a trilinear estimate in the Gevrey spaces $G^{σ, s}$, $s\geq \frac14$. Global in time behaviour of the solution and algebraic lower bound of the evolution of the radius of analyticity was also studied in authors' earlier works in [Nonlinear Differ. Equ. Appl. (2024), 31--68] and [J. Evol. Equ. 24 No. 42 (2024)] by constructing almost conserved quantities in the classical Gevrey space with $H^1$ and $H^2$ levels of Sobolev regularities. The present study aims to construct a new almost conservation law in the Gevrey space defined with a weight function $\cosh(σ|ξ|)$ and use it demonstrate that the local solution $u$ extends globally in time, and the radius of spatial analyticity is bounded from below by $c T^{-\frac{1}{2}}$, for any time $T\geq T_0$. The outcome of this paper represents an improvement on the one achieved by the authors' previous work in [J. Evol. Equ. 24 No. 42 (2024)]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_08400 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Improved lower bound for the radius of analyticity for the modified KdV equation Figueira, Renata O. Panthee, Mahendra Analysis of PDEs We investigate the initial value problem (IVP) associated to the modified Korteweg-de Vries equation (mKdV) in the defocusing scenario: \begin{equation*} \left\{\begin{array}{l} \partial_t u+ \partial_x^3u-u^2\partial_x(u) = 0, \quad x,t\in\mathbb{R}, \\ u(x,0) = u_0(x), \end{array}\right. \end{equation*} where $u$ is a real valued function and the initial data $u_0$ is analytic on $\mathbb{R}$ and has uniform radius of analyticity $σ_0$ in the spatial variable. It is well-known that the solution $u$ preserves its analyticity with the same radius $σ_0$ for at least some time span $0<T_0\le 1$. This local result was obtained in [Nonlinear Differ. Equ. Appl. (2024), 31--68] by proving a trilinear estimate in the Gevrey spaces $G^{σ, s}$, $s\geq \frac14$. Global in time behaviour of the solution and algebraic lower bound of the evolution of the radius of analyticity was also studied in authors' earlier works in [Nonlinear Differ. Equ. Appl. (2024), 31--68] and [J. Evol. Equ. 24 No. 42 (2024)] by constructing almost conserved quantities in the classical Gevrey space with $H^1$ and $H^2$ levels of Sobolev regularities. The present study aims to construct a new almost conservation law in the Gevrey space defined with a weight function $\cosh(σ|ξ|)$ and use it demonstrate that the local solution $u$ extends globally in time, and the radius of spatial analyticity is bounded from below by $c T^{-\frac{1}{2}}$, for any time $T\geq T_0$. The outcome of this paper represents an improvement on the one achieved by the authors' previous work in [J. Evol. Equ. 24 No. 42 (2024)]. |
| title | Improved lower bound for the radius of analyticity for the modified KdV equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2406.08400 |