Guo Wang CHEN, Fang DA
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 \begin{align*} &v_{tt}-\alpha v_{xxtt}-v_{xx}+ \beta v_{xxt}=\beta f(v)_{xxt}, \ \ x\in(0,1),\ \ t>0,\\ &v(0,t)=0, \ \ v(1,t)=0, \ \ t>0,\\ &v(x,0)=v_{0}(x),\ \ v_{t}(x,0)=v_{1}(x), \ \ x\in(0,1), \end{align*} 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 \begin{align} &v_{tt}-v_{xx}=\alpha v_{xxtt}+\beta\Big(\frac{v^{3}}{3}-v\Big)_{xxt},\nonumber\\ &v(0,t)=0, \ \ v(1,t)=0,\nonumber\\ &v(x,0)=v_{0}(x),\ \ v_{t}(x,0)=v_{1}(x)\nonumber \end{align} are also proved.