Zhuohui You | Searcharxiv

arXiv2025

Global well-posedness of the defocusing nonlinear wave equation outside of a ball with radial data for $3<p<5$

We continue the study of the Dirichlet boundary value problem of nonlinear wave equation with radial data in the exterior $Ω= \mathbb{R}^3\backslash \bar{B}(0,1)$. We combine the distorted Fourier truncation method in \cite{Bourgain98:FTM}, the global-in-time (endpoint) Strichartz estimates in \cite{XuYang:NLW} with the energy method in \cite{GallPlan03:NLW} to prove the global well-posedness of the radial solution to the defocusing, energy-subcriticial nonlinear wave equation outside of a ball in $\left(\dot H^{s}_{D}(Ω) \cap L^{p+1}(Ω) \right)\times \dot H^{s-1}_{D}(Ω)$ with $1-\frac{(p+3)(1-s_c)}{4(2p-3)}<s<1$, $s_c=\frac{3}{2}-\frac{2}{p-1} $, which extends the result for the cubic nonlinearity in \cite{XuYang:NLW} to the case $3<p<5$. Except from the argument in \cite{XuYang:NLW}, another new ingredient is that we need make use of the radial Sobolev inequality to deal with the super-conformal nonlinearity in addition to the Sobolev inequality.

math.AP↗

arXiv2024

Several new classes of optimal ternary cyclic codes with two or three zeros

Cyclic codes are a subclass of linear codes and have wide applications in data storage systems, communication systems and consumer electronics due to their efficient encoding and decoding algorithms. Let $α$ be a generator of $\mathbb{F}_{3^m}^*$, where $m$ is a positive integer. Denote by $\mathcal{C}_{(i_1,i_2,\cdots, i_t)}$ the cyclic code with generator polynomial $m_{α^{i_1}}(x)m_{α^{i_2}}(x)\cdots m_{α^{i_t}}(x)$, where ${{m}_{α^{i}}}(x)$ is the minimal polynomial of ${{α}^{i}}$ over ${\mathbb{F}_{3}}$. In this paper, by analyzing the solutions of certain equations over finite fields, we present four classes of optimal ternary cyclic codes $\mathcal{C}_{(0,1,e)}$ and $\mathcal{C}_{(1,e,s)}$ with parameters $[3^m-1,3^m-\frac{3m}{2}-2,4]$, where $s=\frac{3^m-1}{2}$. In addition, by determining the solutions of certain equations and analyzing the irreducible factors of certain polynomials over $\mathbb{F}_{3^m}$, we present four classes of optimal ternary cyclic codes $\mathcal{C}_{(2,e)}$ and $\mathcal{C}_{(1,e)}$ with parameters $[3^m-1,3^m-2m-1,4]$. We show that our new optimal cyclic codes are inequivalent to the known ones.

cs.IT↗