I have the next equation
$ u_{tt} - k^2 u_{xx} + 2 \alpha u_t = 0 \quad $ with $u = u(x,t)$
where $ x \in (0, L)$ and $t\geq 0$.
this equation is called "telegraph equation" or "damped wave equation" and this equation seems like wave equation, so to prove the stability of the solutions I tried to use the integral energy but I don't how to use it. I prove that the energy decresing:
Suppose that:
$u(L,t)=u(0,t)=0 $ $\forall t\geq 0$
$u(x,0)=f(x)$
$u_t(x,0)=g(x)$
Be $k, \alpha > 0$, and $x \in (-\infty , \infty)$. Let $$ E(t) = \frac{1}{2} \int_{-\infty}^\infty \left(u_t^2 + k^2 u_x^2\right)\; dx $$
Then $$ E'(t) = \int_{-\infty}^\infty \left(u_t u_{tt} + k^2 u_x u_{xt}\right)\; dx$$ Integrate the second term by parts, and apply the PDE:
$$ \eqalign{E'(t) &= \int_{-\infty}^\infty \left(u_t u_{tt} - k^2 u_{xx} u_t\right)\; dx\cr &= - 2 \alpha \int_{-\infty}^\infty u_t^2 \; dx \le 0}$$
and now I don't what to do! some hint or idea? exist another method to prove the stability?
I want stability in the sense that for exemple, if $u$ satisfied
$u_{tt} - k^2 u_{xx} + 2 \alpha u_t = 0$
$u(L,t)=u(0,t)=0 $
$u(x,0)=f1(x)$
$u_t(x,0)=g1(x)$
and v satified
$v_{tt} - k^2 v_{xx} + 2 \alpha v_t = 0$
$v(L,t)=v(0,t)=0 $
$v(x,0)=f2(x)$
$v_t(x,0)=g2(x)$
such that $||f1-f2|| < \epsilon $ and $|||g1-g2|| < \epsilon $ so $||u-v|| < \epsilon $ with the norm in $L_2$