一类具有非线性阻尼的非线性电报方程的初边值问题

Initial Boundary Value Problem for A Generalized Nonlinear Telegraph Equation with Nonlinear Damping

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z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(0)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(v3/3-v)xxt,v(0,t)=0,v(1,t)=0,v(x,0)=vo(z),vt(x,0)=v1(x)的整体解的存在性和唯一性.

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(v3/3-v)xxt,v(0,t)=0,v(1,t)=0,v(x,0)=vo(z),vt(x,0)=v1(x)arealso proved.