For the damped string($u_{tt}-c^2u_{xx}+\gamma u_t=0,c=\sqrt{\frac{T}{p}}$), show that the energy decreases.
$E = \dfrac{\int_{-\infty}^{\infty}p{u_t}^2+T{u_x}^2 dx}{2}$
$\dfrac{dE}{dt}=\dfrac{\int_{-\infty}^{\infty}2pu_tu_{tt}+2T{u_x}u_{xt} dx}{2}$ $=\dfrac{\int_{-\infty}^{\infty}2pu_t(c^2u_{xx}-\gamma u_t)+2T{u_x}u_{xt} dx}{2} = \dfrac{\int_{-\infty}^{\infty}2T(u_tu_x)_x-2p\gamma {u_t}^2 dx}{2}= {Tu_tu_x|_{-\infty}^\infty}-p\gamma \int_{-\infty}^\infty{u_t}^2dx$
Then , how can I get further?
First of all a string is usually modelled as a fixed interval $[a,b]$, not $(-\infty,\infty)$. Secondly you don't mention any boundary conditions. Standard boundary conditions are to fix the end-points of the string (so $u_t = 0$ at $x=a$ and $x=b$). This would make the boundary term you end up with zero and we are left with
$$\frac{dE}{dt} = - p\gamma \int_a^b u_t^2{\rm d}x$$
Finally if $p\gamma > 0$ then the right hand side is negative (the integral of a positive function over a positive directed interval is positive) so $\frac{dE}{dt} < 0$.