$ u_{tt}-u_{t}-u_{xx} =0\;,\;t>0\;,\;x\in(0,\pi) \\ u(0,t)=u(\pi,t)=0,\; t>0 \\ E(t):= \frac{1}{2}\int_0^\pi(u_t^2(x,t)+u_x^2(x,t))dx. \\ $
I solved this equation and I end up with something like that $ E'(t)=\int_0^\pi (u_tu_{tt}+u_xu_{xt})\mathrm dx=\int_0^\pi (u_tu_{xx}+u_xu_{xt})\mathrm dx+\int_0^\pi (u_t)^2\mathrm dx$. Since $\int_0^\pi (u_tu_{xx}+u_xu_{xt})\mathrm dx = \int_0^\pi(u_tu_x)_xdx=0$ we ended with $E'(t)=\int_0^\pi (u_t)^2\mathrm dx$. How can I show that it is decreasing? thanks in advance