一类具有非线性阻尼的非线性电报方程的初边值问题

陈国旺, 达芳

数学学报 ›› 2022, Vol. 65 ›› Issue (6) : 1009-1022.

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数学学报 ›› 2022, Vol. 65 ›› Issue (6) : 1009-1022. DOI: 10.12386/A20200216
论文

一类具有非线性阻尼的非线性电报方程的初边值问题

    陈国旺, 达芳
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Initial Boundary Value Problem for A Generalized Nonlinear Telegraph Equation with Nonlinear Damping

    Guo Wang CHEN, Fang DA
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摘要

本文研究一类具有非线性阻尼的广义非线性电报方程的初边值问题vtt−αvxxtt−vxx+βvxxt=βf(v)xxt,  x∈(0,1),  t>0,v(0,t)=0,  v(1,t)=0,  t>0,v(x,0)=v0(x),  vt(x,0)=v1(x),  x∈(0,1),其中 v(x,t) 是一未知函数,下标 x 和t 分别表示对x 和t 求偏导数, α 和β 是正常数, f(v) 是一给定的非线性函数, v0(x) 和v1(x) 是给定的初值函数, 证明该问题整体广义解和整体古典解的存在性和唯一性. 当 f(v) 为线性函数时, 我们研究初边值问题解的渐近性质, 还证明初边值问题vtt−vxx=αvxxtt+β(v33−v)xxt,v(0,t)=0,  v(1,t)=0,v(x,0)=v0(x),  vt(x,0)=v1(x) 的整体解的存在性和唯一性.

Abstract

In this paper, the existence and uniqueness of the global generalized solution and the global classical solution of the initial boundary value problem for the generalized nonlinear telegraph equation with nonlinear damping vtt−αvxxtt−vxx+βvxxt=βf(v)xxt,  x∈(0,1),  t>0,v(0,t)=0,  v(1,t)=0,  t>0,v(x,0)=v0(x),  vt(x,0)=v1(x),  x∈(0,1), are proved. When f(v) is a linear function, the asymptotic behavior of the solution for the problem is studied. The existence and uniqueness of the global solution for the following initial boundary value problem vtt−vxx=αvxxtt+β(v33−v)xxt,v(0,t)=0,  v(1,t)=0,v(x,0)=v0(x),  vt(x,0)=v1(x) are also proved.

关键词

非线性电报方程 / 初边值问题 / 整体解 / 解的渐近性质

Key words

nonlinear telegraph equation / initial boundary value problem / global solution / asymptotic behavior of solution

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陈国旺, 达芳. 一类具有非线性阻尼的非线性电报方程的初边值问题. 数学学报, 2022, 65(6): 1009-1022 https://doi.org/10.12386/A20200216
Guo Wang CHEN, Fang DA. Initial Boundary Value Problem for A Generalized Nonlinear Telegraph Equation with Nonlinear Damping. Acta Mathematica Sinica, Chinese Series, 2022, 65(6): 1009-1022 https://doi.org/10.12386/A20200216

参考文献

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基金

国家自然科学基金资助项目（12171438，11171311）
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